research topics for masters in mathematics

251+ Math Research Topics [2024 Updated]

Mathematics, often dubbed as the language of the universe, holds immense significance in shaping our understanding of the world around us. It’s not just about crunching numbers or solving equations; it’s about unraveling mysteries, making predictions, and creating innovative solutions to complex problems. In this blog, we embark on a journey into the realm of math research topics, exploring various branches of mathematics and their real-world applications.

How Do You Write A Math Research Topic?

Writing a math research topic involves several steps to ensure clarity, relevance, and feasibility. Here’s a guide to help you craft a compelling math research topic:

• Identify Your Interests: Start by exploring areas of mathematics that interest you. Whether it’s pure mathematics, applied mathematics, or interdisciplinary topics, choose a field that aligns with your passion and expertise.
• Narrow Down Your Focus: Mathematics is a broad field, so it’s essential to narrow down your focus to a specific area or problem. Consider the scope of your research and choose a topic that is manageable within your resources and time frame.
• Review Existing Literature: Conduct a thorough literature review to understand the current state of research in your chosen area. Identify gaps, controversies, or unanswered questions that could form the basis of your research topic.
• Formulate a Research Question: Based on your exploration and literature review, formulate a clear and concise research question. Your research question should be specific, measurable, achievable, relevant, and time-bound (SMART).
• Consider Feasibility: Assess the feasibility of your research topic in terms of available resources, data availability, and research methodologies. Ensure that your topic is realistic and achievable within the constraints of your project.
• Consult with Experts: Seek feedback from mentors, advisors, or experts in the field to validate your research topic and refine your ideas. Their insights can help you identify potential challenges and opportunities for improvement.
• Refine and Iterate: Refine your research topic based on feedback and further reflection. Iterate on your ideas to ensure clarity, coherence, and relevance to the broader context of mathematics research.
• Craft a Title: Once you have finalized your research topic, craft a compelling title that succinctly summarizes the essence of your research. Your title should be descriptive, engaging, and reflective of the key themes of your study.
• Write a Research Proposal: Develop a comprehensive research proposal outlining the background, objectives, methodology, and expected outcomes of your research. Your research proposal should provide a clear roadmap for your study and justify the significance of your research topic.

By following these steps, you can effectively write a math research topic that is well-defined, relevant, and poised to make a meaningful contribution to the field of mathematics.

251+ Math Research Topics: Beginners To Advanced

• Prime Number Distribution in Arithmetic Progressions
• Diophantine Equations and their Solutions
• Applications of Modular Arithmetic in Cryptography
• The Riemann Hypothesis and its Implications
• Graph Theory: Exploring Connectivity and Coloring Problems
• Knot Theory: Unraveling the Mathematics of Knots and Links
• Fractal Geometry: Understanding Self-Similarity and Dimensionality
• Differential Equations: Modeling Physical Phenomena and Dynamical Systems
• Chaos Theory: Investigating Deterministic Chaos and Strange Attractors
• Combinatorial Optimization: Algorithms for Solving Optimization Problems
• Computational Complexity: Analyzing the Complexity of Algorithms
• Game Theory: Mathematical Models of Strategic Interactions
• Number Theory: Exploring Properties of Integers and Primes
• Algebraic Topology: Studying Topological Invariants and Homotopy Theory
• Analytic Number Theory: Investigating Properties of Prime Numbers
• Algebraic Geometry: Geometry Arising from Algebraic Equations
• Galois Theory: Understanding Field Extensions and Solvability of Equations
• Representation Theory: Studying Symmetry in Linear Spaces
• Harmonic Analysis: Analyzing Functions on Groups and Manifolds
• Mathematical Logic: Foundations of Mathematics and Formal Systems
• Set Theory: Exploring Infinite Sets and Cardinal Numbers
• Real Analysis: Rigorous Study of Real Numbers and Functions
• Complex Analysis: Analytic Functions and Complex Integration
• Measure Theory: Foundations of Lebesgue Integration and Probability
• Topological Groups: Investigating Topological Structures on Groups
• Lie Groups and Lie Algebras: Geometry of Continuous Symmetry
• Differential Geometry: Curvature and Topology of Smooth Manifolds
• Algebraic Combinatorics: Enumerative and Algebraic Aspects of Combinatorics
• Ramsey Theory: Investigating Structure in Large Discrete Structures
• Analytic Geometry: Studying Geometry Using Analytic Methods
• Hyperbolic Geometry: Non-Euclidean Geometry of Curved Spaces
• Nonlinear Dynamics: Chaos, Bifurcations, and Strange Attractors
• Homological Algebra: Studying Homology and Cohomology of Algebraic Structures
• Topological Vector Spaces: Vector Spaces with Topological Structure
• Representation Theory of Finite Groups: Decomposition of Group Representations
• Category Theory: Abstract Structures and Universal Properties
• Operator Theory: Spectral Theory and Functional Analysis of Operators
• Algebraic Number Theory: Study of Algebraic Structures in Number Fields
• Cryptanalysis: Breaking Cryptographic Systems Using Mathematical Methods
• Discrete Mathematics: Combinatorics, Graph Theory, and Number Theory
• Mathematical Biology: Modeling Biological Systems Using Mathematical Tools
• Population Dynamics: Mathematical Models of Population Growth and Interaction
• Epidemiology: Mathematical Modeling of Disease Spread and Control
• Mathematical Ecology: Dynamics of Ecological Systems and Food Webs
• Evolutionary Game Theory: Evolutionary Dynamics and Strategic Behavior
• Mathematical Neuroscience: Modeling Brain Dynamics and Neural Networks
• Mathematical Physics: Mathematical Models in Physical Sciences
• Quantum Mechanics: Foundations and Applications of Quantum Theory
• Statistical Mechanics: Statistical Methods in Physics and Thermodynamics
• Fluid Dynamics: Modeling Flow of Fluids Using Partial Differential Equations
• Mathematical Finance: Stochastic Models in Finance and Risk Management
• Option Pricing Models: Black-Scholes Model and Beyond
• Portfolio Optimization: Maximizing Returns and Minimizing Risk
• Stochastic Calculus: Calculus of Stochastic Processes and Itô Calculus
• Financial Time Series Analysis: Modeling and Forecasting Financial Data
• Operations Research: Optimization of Decision-Making Processes
• Linear Programming: Optimization Problems with Linear Constraints
• Integer Programming: Optimization Problems with Integer Solutions
• Network Flow Optimization: Modeling and Solving Flow Network Problems
• Combinatorial Game Theory: Analysis of Games with Perfect Information
• Algorithmic Game Theory: Computational Aspects of Game-Theoretic Problems
• Fair Division: Methods for Fairly Allocating Resources Among Parties
• Auction Theory: Modeling Auction Mechanisms and Bidding Strategies
• Voting Theory: Mathematical Models of Voting Systems and Social Choice
• Social Network Analysis: Mathematical Analysis of Social Networks
• Algorithm Analysis: Complexity Analysis of Algorithms and Data Structures
• Machine Learning: Statistical Learning Algorithms and Data Mining
• Deep Learning: Neural Network Models with Multiple Layers
• Reinforcement Learning: Learning by Interaction and Feedback
• Natural Language Processing: Statistical and Computational Analysis of Language
• Computer Vision: Mathematical Models for Image Analysis and Recognition
• Computational Geometry: Algorithms for Geometric Problems
• Symbolic Computation: Manipulation of Mathematical Expressions
• Numerical Analysis: Algorithms for Solving Numerical Problems
• Finite Element Method: Numerical Solution of Partial Differential Equations
• Monte Carlo Methods: Statistical Simulation Techniques
• High-Performance Computing: Parallel and Distributed Computing Techniques
• Quantum Computing: Quantum Algorithms and Quantum Information Theory
• Quantum Information Theory: Study of Quantum Communication and Computation
• Quantum Error Correction: Methods for Protecting Quantum Information from Errors
• Topological Quantum Computing: Using Topological Properties for Quantum Computation
• Quantum Algorithms: Efficient Algorithms for Quantum Computers
• Quantum Cryptography: Secure Communication Using Quantum Key Distribution
• Topological Data Analysis: Analyzing Shape and Structure of Data Sets
• Persistent Homology: Topological Invariants for Data Analysis
• Mapper Algorithm: Method for Visualization and Analysis of High-Dimensional Data
• Algebraic Statistics: Statistical Methods Based on Algebraic Geometry
• Tropical Geometry: Geometric Methods for Studying Polynomial Equations
• Model Theory: Study of Mathematical Structures and Their Interpretations
• Descriptive Set Theory: Study of Borel and Analytic Sets
• Ergodic Theory: Study of Measure-Preserving Transformations
• Combinatorial Number Theory: Intersection of Combinatorics and Number Theory
• Additive Combinatorics: Study of Additive Properties of Sets
• Arithmetic Geometry: Interplay Between Number Theory and Algebraic Geometry
• Proof Theory: Study of Formal Proofs and Logical Inference
• Reverse Mathematics: Study of Logical Strength of Mathematical Theorems
• Nonstandard Analysis: Alternative Approach to Analysis Using Infinitesimals
• Computable Analysis: Study of Computable Functions and Real Numbers
• Graph Theory: Study of Graphs and Networks
• Random Graphs: Probabilistic Models of Graphs and Connectivity
• Spectral Graph Theory: Analysis of Graphs Using Eigenvalues and Eigenvectors
• Algebraic Graph Theory: Study of Algebraic Structures in Graphs
• Metric Geometry: Study of Geometric Structures Using Metrics
• Geometric Measure Theory: Study of Measures on Geometric Spaces
• Discrete Differential Geometry: Study of Differential Geometry on Discrete Spaces
• Algebraic Coding Theory: Study of Error-Correcting Codes
• Information Theory: Study of Information and Communication
• Coding Theory: Study of Error-Correcting Codes
• Cryptography: Study of Secure Communication and Encryption
• Finite Fields: Study of Fields with Finite Number of Elements
• Elliptic Curves: Study of Curves Defined by Cubic Equations
• Hyperelliptic Curves: Study of Curves Defined by Higher-Degree Equations
• Modular Forms: Analytic Functions with Certain Transformation Properties
• L-functions: Analytic Functions Associated with Number Theory
• Zeta Functions: Analytic Functions with Special Properties
• Analytic Number Theory: Study of Number Theoretic Functions Using Analysis
• Dirichlet Series: Analytic Functions Represented by Infinite Series
• Euler Products: Product Representations of Analytic Functions
• Arithmetic Dynamics: Study of Iterative Processes on Algebraic Structures
• Dynamics of Rational Maps: Study of Dynamical Systems Defined by Rational Functions
• Julia Sets: Fractal Sets Associated with Dynamical Systems
• Mandelbrot Set: Fractal Set Associated with Iterations of Complex Quadratic Polynomials
• Arithmetic Geometry: Study of Algebraic Geometry Over Number Fields
• Diophantine Geometry: Study of Solutions of Diophantine Equations Using Geometry
• Arithmetic of Elliptic Curves: Study of Elliptic Curves Over Number Fields
• Rational Points on Curves: Study of Rational Solutions of Algebraic Equations
• Galois Representations: Study of Representations of Galois Groups
• Automorphic Forms: Analytic Functions with Certain Transformation Properties
• L-functions: Analytic Functions Associated with Automorphic Forms
• Selberg Trace Formula: Tool for Studying Spectral Theory and Automorphic Forms
• Langlands Program: Program to Unify Number Theory and Representation Theory
• Hodge Theory: Study of Harmonic Forms on Complex Manifolds
• Riemann Surfaces: One-dimensional Complex Manifolds
• Shimura Varieties: Algebraic Varieties Associated with Automorphic Forms
• Modular Curves: Algebraic Curves Associated with Modular Forms
• Hyperbolic Manifolds: Manifolds with Constant Negative Curvature
• Teichmüller Theory: Study of Moduli Spaces of Riemann Surfaces
• Mirror Symmetry: Duality Between Calabi-Yau Manifolds
• Kähler Geometry: Study of Hermitian Manifolds with Special Symmetries
• Algebraic Groups: Linear Algebraic Groups and Their Representations
• Lie Algebras: Study of Algebraic Structures Arising from Lie Groups
• Representation Theory of Lie Algebras: Study of Representations of Lie Algebras
• Quantum Groups: Deformation of Lie Groups and Lie Algebras
• Algebraic Topology: Study of Topological Spaces Using Algebraic Methods
• Homotopy Theory: Study of Continuous Deformations of Spaces
• Homology Theory: Study of Algebraic Invariants of Topological Spaces
• Cohomology Theory: Study of Dual Concepts to Homology Theory
• Singular Homology: Homology Theory Defined Using Simplicial Complexes
• Sheaf Theory: Study of Sheaves and Their Cohomology
• Differential Forms: Study of Multilinear Differential Forms
• De Rham Cohomology: Cohomology Theory Defined Using Differential Forms
• Morse Theory: Study of Critical Points of Smooth Functions
• Symplectic Geometry: Study of Symplectic Manifolds and Their Geometry
• Floer Homology: Study of Symplectic Manifolds Using Pseudoholomorphic Curves
• Gromov-Witten Invariants: Invariants of Symplectic Manifolds Associated with Pseudoholomorphic Curves
• Mirror Symmetry: Duality Between Symplectic and Complex Geometry
• Calabi-Yau Manifolds: Ricci-Flat Complex Manifolds
• Moduli Spaces: Spaces Parameterizing Geometric Objects
• Donaldson-Thomas Invariants: Invariants Counting Sheaves on Calabi-Yau Manifolds
• Algebraic K-Theory: Study of Algebraic Invariants of Rings and Modules
• Homological Algebra: Study of Homology and Cohomology of Algebraic Structures
• Derived Categories: Categories Arising from Homological Algebra
• Stable Homotopy Theory: Homotopy Theory with Stable Homotopy Groups
• Model Categories: Categories with Certain Homotopical Properties
• Higher Category Theory: Study of Higher Categories and Homotopy Theory
• Higher Topos Theory: Study of Higher Categorical Structures
• Higher Algebra: Study of Higher Categorical Structures in Algebra
• Higher Algebraic Geometry: Study of Higher Categorical Structures in Algebraic Geometry
• Higher Representation Theory: Study of Higher Categorical Structures in Representation Theory
• Higher Category Theory: Study of Higher Categorical Structures
• Homotopical Algebra: Study of Algebraic Structures in Homotopy Theory
• Homotopical Groups: Study of Groups with Homotopical Structure
• Homotopical Categories: Study of Categories with Homotopical Structure
• Homotopy Groups: Algebraic Invariants of Topological Spaces
• Homotopy Type Theory: Study of Foundations of Mathematics Using Homotopy Theory

In conclusion, the world of mathematics is vast and multifaceted, offering endless opportunities for exploration and discovery. Whether delving into the abstract realms of pure mathematics or applying mathematical principles to solve real-world problems, mathematicians play a vital role in advancing human knowledge and shaping the future of our world.

By embracing diverse math research topics and interdisciplinary collaborations, we can unlock new possibilities and harness the power of mathematics to address the challenges of today and tomorrow. So, let’s embark on this journey together as we unravel the mysteries of numbers and explore the boundless horizons of mathematical inquiry.

Related Posts

Step by Step Guide on The Best Way to Finance Car

The Best Way on How to Get Fund For Business to Grow it Efficiently

• Write my thesis
• Thesis writers
• Buy thesis papers
• Bachelor thesis
• Master's thesis
• Thesis editing services
• Thesis proofreading services
• Buy a thesis online
• Write my dissertation
• Dissertation proposal help
• Pay for dissertation
• Custom dissertation
• Dissertation help online
• Buy dissertation online
• Cheap dissertation
• Dissertation editing services
• Write my research paper
• Buy research paper online
• Pay for research paper
• Research paper help
• Order research paper
• Custom research paper
• Cheap research paper
• Research papers for sale
• Thesis subjects
• How It Works

181 Mathematics Research Topics From PhD Experts

If you are reading this blog post, it means you are looking for some exceptional math research topics. You want them to be original, unique even. If you manage to find topics like this, you can be sure your professor will give you a top grade (if you write a decent paper, that is). The good news is that you have arrived at just the right place – at the right time. We have just finished updating our list of topics, so you will find plenty of original ideas right on this page. All our topics are 100 percent free to use as you see fit. You can reword them and you don’t need to give us any credit.

And remember: if you need assistance from a professional, don’t hesitate to reach out to us. We are not just the best place for math research topics for high school students; we are also the number one choice for students looking for top-notch research paper writing services.

Our Newest Research Topics in Math

We know you probably want the best and most recent research topics in math. You want your paper to stand out from all the rest. After all, this is the best way to get some bonus points from your professor. On top of this, finding some great topics for your next paper makes it easier for you to write the essay. As long as you know at least something about the topic, you’ll find that writing a great paper or buy phd thesis isn’t as difficult as you previously thought.

So, without further ado, here are the 181 brand new topics for your next math research paper:

Cool Math Topics to Research

Are you looking for some cool math topics to research? We have a list of original topics for your right here. Pick the one you like and start writing now:

• Roll two dice and calculate a probability
• Discuss ancient Greek mathematics
• Is math really important in school?
• Discuss the binomial theorem
• The math behind encryption
• Game theory and its real-life applications
• Analyze the Bernoulli scheme
• What are holomorphic functions and how do they work?
• Describe big numbers
• Solving the Tower of Hanoi problem

Undergraduate Math Research Topics

If you are an undergraduate looking for some research topics for your next math paper, you will surely appreciate our list of interesting undergraduate math research topics:

• Methods to count discrete objects
• The origins of Greek symbols in mathematics
• Methods to solve simultaneous equations
• Real-world applications of the theorem of Pythagoras
• Discuss the limits of diffusion
• Use math to analyze the abortion data in the UK over the last 100 years
• Discuss the Knot theory
• Analyze predictive models (take meteorology as an example)
• In-depth analysis of the Monte Carlo methods for inverse problems
• Squares vs. rectangles (compare and contrast)

Number Theory Topics to Research

Interested in writing about number theory? It is not an easy subject to discuss, we know. However, we are sure you will appreciate these number theory topics:

• Discuss the greatest common divisor
• Explain the extended Euclidean algorithm
• What are RSA numbers?
• Discuss Bézout’s lemma
• In-depth analysis of the square-free polynomial
• Discuss the Stern-Brocot tree
• Analyze Fermat’s little theorem
• What is a discrete logarithm?
• Gauss’s lemma in number theory
• Analyze the Pentagonal number theorem

Math Research Topics for High School

High school students shouldn’t be too worried about their math papers because we have some unique, and quite interesting, math research topics for high school right here:

• Discuss Brun’s constant
• An in-depth look at the Brahmagupta–Fibonacci identity
• What is derivative algebra?
• Describe the Symmetric Boolean function
• Discuss orders of approximation in limits
• Solving Regiomontanus’ angle maximization problem
• What is a Quadratic integral?
• Define and describe complementary angles
• Analyze the incircle and excircles of a triangle
• Analyze the Bolyai–Gerwien theorem in geometry
• Math in our everyday life

Complex Math Topics

If you want to give some complex math topics a try, we have the best examples below. Remember, these topics should only be attempted by students who are proficient in mathematics:

• Mathematics and its appliance in Artificial Intelligence
• Try to solve an unsolved problem in math
• Discuss Kolmogorov’s zero-one law
• What is a discrete random variable?
• Analyze the Hewitt–Savage zero-one law
• What is a transferable belief model?
• Discuss 3 major mathematical theorems
• Describe and analyze the Dempster-Shafer theory
• An in-depth analysis of a continuous stochastic process
• Identify and analyze Gauss-Markov processes

Easy Math Research Paper Topics

Perhaps you don’t want to spend too much time working on your next research paper. Who can blame you? Check out these easy math research paper topics:

• Define the hyperbola
• Do we need to use a calculator during math class?
• The binomial theorem and its real-world applications
• What is a parabola in geometry?
• How do you calculate the slope of a curve?
• Define the Jacobian matrix
• Solving matrix problems effectively
• Why do we need differential equations?
• Should math be mandatory in all schools?
• What is a Hessian matrix?

Logic Topics to Research

We have some interesting logical topics for research papers. These are perfect for students interested in writing about math logic. Pick one right now:

• Discuss the reductio ad absurdum approach
• Discuss Boolean algebra
• What is consistency proof?
• Analyze Trakhtenbrot’s theorem (the finite model theory)
• Discuss the Gödel completeness theorem
• An in-depth analysis of Morley’s categoricity theorem
• How does the Back-and-forth method work?
• Discuss the Ehrenfeucht–Fraïssé game technique
• Discuss Aleph numbers (Aleph-null and Aleph-one)
• Solving the Suslin problem

Algebra Topics for a Research Paper

Would you like to write about an algebra topic? No problem, our seasoned writers have compiled a list of the best algebra topics for a research paper:

• Discuss the differential equation
• Analyze the Jacobson density theorem
• The 4 properties of a binary operation in algebra
• Analyze the unary operator in depth
• Analyze the Abel–Ruffini theorem
• Epimorphisms vs. monomorphisms: compare and contrast
• Discuss the Morita duality in algebraic structures
• Idempotent vs. nilpotent in Ring theory
• Discuss the Artin-Wedderburn theorem
• What is a commutative ring in algebra?
• Analyze and describe the Noetherian ring

Math Education Research Topics

There is nothing wrong with writing about math education, especially if your professor did not give you writing prompts. Here are some very nice math education research topics:

• What are the goals a mathematics professor should have?
• What is math anxiety in the classroom?
• Teaching math in UK schools: the difficulties
• Computer programming or math in high school?
• Is math education in Europe at a high enough level?
• Common Core Standards and their effects on math education
• Culture and math education in Africa
• What is dyscalculia and how does it manifest itself?
• When was algebra first thought in schools?
• Math education in the United States versus the United Kingdom

Computability Theory Topics to Research

Writing about computability theory can be a very interesting adventure. Give it a try! Here are some of our most interesting computability theory topics to research:

• What is a multiplication table?
• Analyze the Scholz conjecture
• Explain exponentiating by squaring
• Analyze the Myhill-Nerode theorem
• What is a tree automaton?
• Compare and contrast the Pushdown automaton and the Büchi automaton
• Discuss the Markov algorithm
• What is a Turing machine?
• Analyze the post correspondence problem
• Discuss the linear speedup theorem
• Discuss the Boolean satisfiability problem

Interesting Math Research Topics

We know you want topics that are interesting and relatively easy to write about. This is why we have a separate list of our most interesting math research topics:

• What is two-element Boolean algebra?
• The life of Gauss
• The life of Isaac Newton
• What is an orthodiagonal quadrilateral?
• Tessellation in Euclidean plane geometry
• Describe a hyperboloid in 3D geometry
• What is a sphericon?
• Discuss the peculiarities of Borel’s paradox
• Analyze the De Finetti theorem in statistics
• What are Martingales?
• The basics of stochastic calculus

Applied Math Research Topics

Interested in writing about applied mathematics? Our team managed to create a list of awesome applied math research topics from scratch for you:

• Discuss Newton’s laws of motion
• Analyze the perpendicular axes rule
• How is a Galilean transformation done?
• The conservation of energy and its applications
• Discuss Liouville’s theorem in Hamiltonian mechanics
• Analyze the quantum field theory
• Discuss the main components of the Lorentz symmetry
• An in-depth look at the uncertainty principle

Geometry Topics for a Research Paper

Geometry can be a very captivating subject, especially when you know plenty about it. Check out our list of geometry topics for a research paper and pick the best one today:

• Most useful trigonometry functions in math
• The life of Archimedes and his achievements
• Trigonometry in computer graphics
• Using Vincenty’s formulae in geodesy
• Define and describe the Heronian tetrahedron
• The math behind the parabolic microphone
• Discuss the Japanese theorem for concyclic polygons
• Analyze Euler’s theorem in geometry

Math Research Topics for Middle School

Yes, even middle school children can write about mathematics. We have some original math research topics for middle school right here:

• Finding critical points in a graph
• The basics of calculus
• What makes a graph ultrahomogeneous?
• How do you calculate the area of different shapes?
• What contributions did Euclid have to the field of mathematics?
• What is Diophantine geometry?
• What makes a graph regular?
• Analyze a full binary tree

Math Research Topics for College Students

As you’ve probably already figured out, college students should pick topics that are a bit more complex. We have some of the best math research topics for college students right here:

• What are extremal problems and how do you solve them?
• Discuss an unsolvable math problem
• How can supercomputers solve complex mathematical problems?
• An in-depth analysis of fractals
• Discuss the Boruvka’s algorithm (related to the minimum spanning tree)
• Discuss the Lorentz–FitzGerald contraction hypothesis in relativity
• An in-depth look at Einstein’s field equation
• The math behind computer vision and object recognition

Calculus Topics for a Research Paper

Let’s face it: calculus is not a very difficult field. So, why don’t you pick one of our excellent calculus topics for a research paper and start writing your essay right away:

• When do we need to apply the L’Hôpital rule?
• Discuss the Leibniz integral rule
• Calculus in ancient Egypt
• Discuss and analyze linear approximations
• The applications of calculus in real life
• The many uses of Stokes’ theorem
• Discuss the Borel regular measure
• An in-depth analysis of Lebesgue’s monotone convergence theorem

Simple Math Research Paper Topics for High School

This is the place where you can find some pretty simple topics if you are a high school student. Check out our simple math research paper topics for high school:

• The life and work of the famous Pierre de Fermat
• What are limits and why are they useful in calculus?
• Explain the concept of congruency
• The life and work of the famous Jakob Bernoulli
• Analyze the rhombicosidodecahedron and its applications
• Calculus and the Egyptian pyramids
• The life and work of the famous Jean d’Alembert
• Discuss the hyperplane arrangement in combinatorial computational geometry
• The smallest enclosing sphere method in combinatorics

Business Math Topics

If you want to surprise your professor, why don’t you write about business math? We have some exceptional topics that nobody has thought about right here:

• Is paying a loan with another loan a good approach?
• Discuss the major causes of a stock market crash
• Best debt amortization methods in the US
• How do bank loans work in the UK?
• Calculating interest rates the easy way
• Discuss the pros and cons of annuities
• Basic business math skills everyone should possess
• Business math in United States schools
• Analyze the discount factor

Probability and Statistics Topics for Research

Probability and statistics are not easy fields. However, you can impress your professor with one of our unique probability and statistics topics for research:

• What is the autoregressive conditional duration?
• Applying the ANOVA method to ranks
• Discuss the practical applications of the Bates distribution
• Explain the principle of maximum entropy
• Discuss Skorokhod’s representation theorem in random variables
• What is the Factorial moment in the Theory of Probability?
• Compare and contrast Cochran’s C test and his Q test
• Analyze the De Moivre-Laplace theorem
• What is a negative probability?

Need Help With Research Paper?

We offer the absolute best high school and college research paper writing service on the Internet. When you need any kind of research paper help, our experienced ENL writers and professional editors are here to help. With years of experience under their belts, our experts can get your research paper done in as little as 3 hours.

Getting cheap online help with research papers has never been easier. College students should just get in touch with us and tell us what they need. We will assign them our most affordable and experienced math writer in minutes, even during the night. We are the best-rated online writing company on the Internet because we always deliver high-quality academic content at the most competitive prices. Give us a try today!

Leave a Reply Cancel reply

Research Areas

Analysis & pde, applied math, combinatorics, financial math, number theory, probability, representation theory, symplectic geometry & topology.

• Welcome from the Chair
• Michalik Distinguished Lecture Series
• Pittsburgh Mathematical Horizons Lecture Series
• Open Faculty Positions
• Advising & Support
• Calculus Curriculum
• Degree Programs/Requirements
• Extracurricular Activities
• Math Placement Assessment
• Math Assistance Center/Posvar Computing Lab
• Research/Career Opportunities
• Admissions & Financial Aid
• Degree Programs
• Graduate Employment
• Graduate Handbook
• Information for Incoming Graduate Students
• Organizations
• Research Opportunities
• Teaching Opportunities

Research Areas

• Graduate Research
• Undergraduate Research
• Mathematics Research Center
• Technical Reports
• Publications
• Gallery of Research Images
• Faculty Admin
• Adjunct Faculty
• Part-Time Faculty
• Emeritus Faculty
• Post-Doctoral Associates
• Graduate Students
• Stay in Touch
• Newsletter Archive
• Upcoming Events
• Past Events
• Prospective Students

Department members engage in cutting-edge research on a wide variety of topics in mathematics and its applications. Topics continually evolve to reflect emerging interests and developments, but can roughly grouped into the following areas.

Algebra, Combinatorics, and Geometry

Algebra, combinatorics, and geometry are areas of very active research at the University of Pittsburgh.

Analysis and Partial Differential Equations

The research of the analysis group covers functional analysis, harmonic analysis, several complex variables, partial differential equations, and analysis on metric and Carnot-Caratheodory spaces.

Applied Analysis

The department is a leader in the analysis of systems of nonlinear differential equations and dynamical systems  that arise in modeling a variety of physical phenomena. They include problems in biology, chemistry, phase transitions, fluid flow, flame propagation, diffusion processes, and pattern formation in nonlinear stochastic partial differential equations.

Mathematical Biology

The biological world stands as the next great frontier for mathematical modeling and analysis. This group studies complex systems and dynamics arising in various biological phenomena.

Mathematical Finance

A rapidly growing area of mathematical finance is Quantitative Behavioral Finance. The high-tech boom and bust of the late 1990s followed by the housing and financial upheavals of 2008 have made a convincing case for the necessity of adopting broader assumptions in finance.

Numerical Analysis and Scientific Computing

The diversity of this group is reflected in its research interests: numerical analysis of partial differential equations , adaptive methods for scientific computing, computational methods of fluid dynamics and turbulence, numerical solution of nonlinear problems arising from porous media flow and transport, optimal control, and simulation of stochastic reaction diffusion systems.

Topology and Differential Geometry

Research in analytic topology continues in the broad area of generalized metric spaces. This group studies relativity theory and differential geometry, with emphasis on twistor methods, as well as geometric and topological aspects of quantum field theory, string theory, and M-theory.

Guide to Graduate Studies

The PhD Program The Ph.D. program of the Harvard Department of Mathematics is designed to help motivated students develop their understanding and enjoyment of mathematics. Enjoyment and understanding of the subject, as well as enthusiasm in teaching it, are greater when one is actively thinking about mathematics in one’s own way. For this reason, a Ph.D. dissertation involving some original research is a fundamental part of the program. The stages in this program may be described as follows:

• Acquiring a broad basic knowledge of mathematics on which to build a future mathematical culture and more detailed knowledge of a field of specialization.
• Choosing a field of specialization within mathematics and obtaining enough knowledge of this specialized field to arrive at the point of current thinking.
• Making a first original contribution to mathematics within this chosen special area.

Students are expected to take the initiative in pacing themselves through the Ph.D. program. In theory, a future research mathematician should be able to go through all three stages with the help of only a good library. In practice, many of the more subtle aspects of mathematics, such as a sense of taste or relative importance and feeling for a particular subject, are primarily communicated by personal contact. In addition, it is not at all trivial to find one’s way through the ever-burgeoning literature of mathematics, and one can go through the stages outlined above with much less lost motion if one has some access to a group of older and more experienced mathematicians who can guide one’s reading, supplement it with seminars and courses, and evaluate one’s first attempts at research. The presence of other graduate students of comparable ability and level of enthusiasm is also very helpful.

University Requirements

The University requires a minimum of two years of academic residence (16 half-courses) for the Ph.D. degree. On the other hand, five years in residence is the maximum usually allowed by the department. Most students complete the Ph.D. in four or five years. Please review the program requirements timeline .

There is no prescribed set of course requirements, but students are required to register and enroll in four courses each term to maintain full-time status with the Harvard Kenneth C. Griffin Graduate School of Arts and Sciences.

Qualifying Exam

The department gives the qualifying examination at the beginning of the fall and spring terms. The qualifying examination covers algebra, algebraic geometry, algebraic topology, complex analysis, differential geometry, and real analysis. Students are required to take the exam at the beginning of the first term. More details about the qualifying exams can be found here .

Students are expected to pass the qualifying exam before the end of their second year. After passing the qualifying exam students are expected to find a Ph.D. dissertation advisor.

Minor Thesis

The minor thesis is complementary to the qualifying exam. In the course of mathematical research, students will inevitably encounter areas in which they have gaps in knowledge. The minor thesis is an exercise in confronting those gaps to learn what is necessary to understand a specific area of math. Students choose a topic outside their area of expertise and, working independently, learns it well and produces a written exposition of the subject.

The topic is selected in consultation with a faculty member, other than the student’s Ph.D. dissertation advisor, chosen by the student. The topic should not be in the area of the student’s Ph.D. dissertation. For example, students working in number theory might do a minor thesis in analysis or geometry. At the end of three weeks time (four if teaching), students submit to the faculty member a written account of the subject and are prepared to answer questions on the topic.

The minor thesis must be completed before the start of the third year in residence.

Language Exam

Mathematics is an international subject in which the principal languages are English, French, German, and Russian. Almost all important work is published in one of these four languages. Accordingly, students are required to demonstrate the ability to read mathematics in French, German, or Russian by passing a two-hour, written language examination. Students are asked to translate one page of mathematics into English with the help of a dictionary. Students may request to substitute the Italian language exam if it is relevant to their area of mathematics. The language requirement should be fulfilled by the end of the second year. For more information on the graduate program requirements, a timeline can be viewed at here .

Non-native English speakers who have received a Bachelor’s degree in mathematics from an institution where classes are taught in a language other than English may request to waive the language requirement.

Upon completion of the language exam and eight upper-level math courses, students can apply for a continuing Master’s Degree.

Teaching Requirement

Most research mathematicians are also university teachers. In preparation for this role, all students are required to participate in the department’s teaching apprenticeship program and to complete two semesters of classroom teaching experience, usually as a teaching fellow. During the teaching apprenticeship, students are paired with a member of the department’s teaching staff. Students attend some of the advisor’s classes and then prepare (with help) and present their own class, which will be videotaped. Apprentices will receive feedback both from the advisor and from members of the class.

Teaching fellows are responsible for teaching calculus to a class of about 25 undergraduates. They meet with their class three hours a week. They have a course assistant (an advanced undergraduate) to grade homework and to take a weekly problem session. Usually, there are several classes following the same syllabus and with common exams. A course head (a member of the department teaching staff) coordinates the various classes following the same syllabus and is available to advise teaching fellows. Other teaching options are available: graduate course assistantships for advanced math courses and tutorials for advanced undergraduate math concentrators.

Final Stages

How students proceed through the second and third stages of the program varies considerably among individuals. While preparing for the qualifying examination or immediately after, students should begin taking more advanced courses to help with choosing a field of specialization. Unless prepared to work independently, students should choose a field that falls within the interests of a member of the faculty who is willing to serve as dissertation advisor. Members of the faculty vary in the way that they go about dissertation supervision; some faculty members expect more initiative and independence than others and some variation in how busy they are with current advisees. Students should consider their own advising needs as well as the faculty member’s field when choosing an advisor. Students must take the initiative to ask a professor if she or he will act as a dissertation advisor. Students having difficulty deciding under whom to work, may want to spend a term reading under the direction of two or more faculty members simultaneously. The sooner students choose an advisor, the sooner they can begin research. Students should have a provisional advisor by the second year.

It is important to keep in mind that there is no technique for teaching students to have ideas. All that faculty can do is to provide an ambiance in which one’s nascent abilities and insights can blossom. Ph.D. dissertations vary enormously in quality, from hard exercises to highly original advances. Many good research mathematicians begin very slowly, and their dissertations and first few papers could be of minor interest. The ideal attitude is: (1) a love of the subject for its own sake, accompanied by inquisitiveness about things which aren’t known; and (2) a somewhat fatalistic attitude concerning “creative ability” and recognition that hard work is, in the end, much more important.

Applied Mathematics

Faculty and students interested in the applications of mathematics are an integral part of the department of mathematics; there is no formal separation between pure and applied mathematics, and the department takes pride in the many ways in which they enrich each other. we also benefit tremendously from close collaborations with faculty and students in other departments at uc berkeley as well as scientists at  lawrence berkeley national laboratory  and visitors to the  mathematical sciences research institute.

The Department regularly offers courses in ordinary and partial differential equations and their numerical solution, discrete applied mathematics, the methods of mathematical physics, mathematical biology, the mathematical aspects of fluid and solid mechanics, approximation theory, scientific computing, numerical linear algebra, and mathematical aspects of computer science. Courses in probability theory, stochastic processes, data analysis and bioinformatics are offered by the Department of Statistics, while courses in combinatorial and convex optimization are offered by the Department of Industrial Engineering and Operations Research. Our students are encouraged to take courses of mathematical interest in these and many other departments. Topics explored intensively by our faculty and students in recent years include scientific computation and the mathematical aspects of quantum theory, computational genomics, image processing and medical imaging, inverse problems, combinatorial optimization, control, robotics, shape optimization, turbulence, hurricanes, microchip failure, MEMS, biodemography, population genetics, phylogenetics, and computational approaches to historical linguistics. Within the department we also have a  Laboratory for Mathematical and Computational Biology .

Chair:   Per-Olof Persson

Applied Mathematics Faculty, Courses, Dissertations

Senate faculty, graduate students, visiting faculty, meet our faculty, mina aganagic, david aldous, robert m. anderson, sunčica čanić, jennifer chayes, alexandre j. chorin, paul concus, james w. demmel, l. craig evans, steven n. evans, f. alberto grünbaum, venkatesan guruswami, ole h. hald, william m. kahan, richard karp, michael j. klass, hendrik w. lenstra, jr., lin lin (林霖), michael j. lindsey, c. keith miller, john c. neu, beresford n. parlett.

• 1 of 2 Grid: People (Current page)
• 2 of 2 Grid: People
• next › Grid: People
• last » Grid: People

MSc by Research in Mathematics

• Entry requirements
• Funding and costs

College preference

• How to apply

About the course

The MSc by Research is an advanced research degree which provides the opportunity to investigate a project in depth and write a thesis which makes a significant contribution in the field. The research project is however designed to take less time than a Doctorate degree (normally two years, though it is possible to complete the requirements in a single year). It is not intended as a first step towards a DPhil, but rather as an alternative to a DPhil. Few students opt to apply for the MSc by Research unless there are limitations on the time or funding available.

There are no associated lectures, classes or written examinations. Your project can be in any of the subject areas for which supervision is available. The department’s research covers the entire spectrum of mathematics, with subject areas including:

• Algebra (primarily group theory and representation theory)
• Mathematical biology and ecology
• Industrial and applied mathematics (including fluid and solid mechanics, geosciences, mathematical physiology, methodologies, and networks)
• Numerical analysis.

You will be asked to outline your research interests when you apply by listing at least one but no more than three of the fields of research above on your application form.  More information about the Research Groups in the Mathematical Institute  can be found on the department's website. Full instructions for completing this section of the application form can be found in the How to apply section of this page.

You will be expected to acquire transferable skills as part of your training, which will require you to attend courses, lectures, workshops and colloquia. You will have the opportunity to develop other valuable skills and to contribute to the teaching work of the department, both by marking students’ work and later by leading classes of around eight to twelve students.

Supervision

You will be invited to suggest a specific supervisor or supervisors in your application form, and your preferences will be taken into account in allocating you a supervisor (which will be done before your arrival). The allocation of graduate supervision for this course is the responsibility of the Mathematical Institute and it is not always possible to accommodate the preferences of incoming graduate students to work with a particular member of staff. Under exceptional circumstances a co-supervisor may be found outside the Mathematical Institute.

Students are expected to meet with their supervisors at least four times per term. A more typical pattern is weekly, at least until you reach the stage of writing up your thesis.

Initially, you will be admitted as a Probationer Research Student in the same way as those intending to do a DPhil.

Within four terms of admission as a PRS student you will be expected to apply for transfer of status from Probationer Research Student to MSc by Research status.

A successful transfer of status from PRS to MSc by Research status will require satisfactory attendance and a successful performance in an oral examination in front of two appointed assessors.

You will be expected to submit a substantial original thesis after two or, at most, three years from the date of admission. To be successfully awarded a MSc by Research in Mathematics you will need to defend your thesis orally ( viva voce ) in front of two appointed examiners.

Graduate destinations

The department, working alongside the University’s Careers Service, supports graduate students as they move from the MSc by Research to the next stage of their career. Many graduate students stay in academia, by taking up a postdoctoral position, and many move into employment in a range of industries and sectors where the expertise and skills developed during the degree are highly valued.

Changes to this course and your supervision

The University will seek to deliver this course in accordance with the description set out in this course page. However, there may be situations in which it is desirable or necessary for the University to make changes in course provision, either before or after registration. The safety of students, staff and visitors is paramount and major changes to delivery or services may have to be made in circumstances of a pandemic, epidemic or local health emergency. In addition, in certain circumstances, for example due to visa difficulties or because the health needs of students cannot be met, it may be necessary to make adjustments to course requirements for international study.

Where possible your academic supervisor will not change for the duration of your course. However, it may be necessary to assign a new academic supervisor during the course of study or before registration for reasons which might include illness, sabbatical leave, parental leave or change in employment.

For further information please see our page on changes to courses and the provisions of the student contract regarding changes to courses.

Entry requirements for entry in 2024-25

Proven and potential academic excellence.

The requirements described below are specific to this course and apply only in the year of entry that is shown. You can use our interactive tool to help you  evaluate whether your application is likely to be competitive .

Please be aware that any studentships that are linked to this course may have different or additional requirements and you should read any studentship information carefully before applying. 

Degree-level qualifications

As a minimum, applicants should hold or be predicted to achieve the following UK qualifications or their equivalent:

• a first-class undergraduate degree with honours in mathematics or a related discipline. 

A previous master's degree is not required, though the requirement for a first-class undergraduate degree with honours can be alternatively demonstrated by strong performance in a master's degree.

For applicants with a degree from the USA, the minimum GPA sought is 3.7 out of 4.0.

If your degree is not from the UK or another country specified above, visit our International Qualifications page for guidance on the qualifications and grades that would usually be considered to meet the University’s minimum entry requirements.

GRE General Test scores

No Graduate Record Examination (GRE) or GMAT scores are sought.

Other qualifications, evidence of excellence and relevant experience

• Research or working experience in the proposed research may be an advantage.
• Publications are not expected.

English language proficiency

This course requires proficiency in English at the University's  standard level . If your first language is not English, you may need to provide evidence that you meet this requirement. The minimum scores required to meet the University's standard level are detailed in the table below.

*Previously known as the Cambridge Certificate of Advanced English or Cambridge English: Advanced (CAE) † Previously known as the Cambridge Certificate of Proficiency in English or Cambridge English: Proficiency (CPE)

Your test must have been taken no more than two years before the start date of your course. Our Application Guide provides further information about the English language test requirement .

Declaring extenuating circumstances

If your ability to meet the entry requirements has been affected by the COVID-19 pandemic (eg you were awarded an unclassified/ungraded degree) or any other exceptional personal circumstance (eg other illness or bereavement), please refer to the guidance on extenuating circumstances in the Application Guide for information about how to declare this so that your application can be considered appropriately.

You will need to register three referees who can give an informed view of your academic ability and suitability for the course. The  How to apply  section of this page provides details of the types of reference that are required in support of your application for this course and how these will be assessed.

Supporting documents

You will be required to supply supporting documents with your application. The  How to apply  section of this page provides details of the supporting documents that are required as part of your application for this course and how these will be assessed.

Performance at interview

Technical interviews are normally held as part of the admissions process.  

If invited you can expect to be interviewed by at least two people and for the interview to last around 30 minutes. The interview could take place face-to-face or remotely.

It is expected that interviews will take place around three to five weeks after an application deadline.

How your application is assessed

Your application will be assessed purely on your proven and potential academic excellence and other entry requirements described under that heading.

References  and  supporting documents  submitted as part of your application, and your performance at interview (if interviews are held) will be considered as part of the assessment process. Whether or not you have secured funding will not be taken into consideration when your application is assessed.

An overview of the shortlisting and selection process is provided below. Our ' After you apply ' pages provide  more information about how applications are assessed . 

Shortlisting and selection

Students are considered for shortlisting and selected for admission without regard to age, disability, gender reassignment, marital or civil partnership status, pregnancy and maternity, race (including colour, nationality and ethnic or national origins), religion or belief (including lack of belief), sex, sexual orientation, as well as other relevant circumstances including parental or caring responsibilities or social background. However, please note the following:

• socio-economic information may be taken into account in the selection of applicants and award of scholarships for courses that are part of  the University’s pilot selection procedure  and for  scholarships aimed at under-represented groups ;
• country of ordinary residence may be taken into account in the awarding of certain scholarships; and
• protected characteristics may be taken into account during shortlisting for interview or the award of scholarships where the University has approved a positive action case under the Equality Act 2010.

Initiatives to improve access to graduate study

This course is taking part in a continuing pilot programme to improve the selection procedure for graduate applications, in order to ensure that all candidates are evaluated fairly.

For this course, socio-economic data (where it has been provided in the application form) will be used to contextualise applications at the different stages of the selection process.  Further information about how we use your socio-economic data  can be found in our page about initiatives to improve access to graduate study.

If you wish, you may submit an additional contextual statement (using the instructions in the How to apply section of this page) to provide further information on your socio-economic background or personal circumstances in support of your application.  Further information about how your contextual statement will be used  can be found in our page about initiatives to improve access to graduate study.

Processing your data for shortlisting and selection

Information about  processing special category data for the purposes of positive action  and  using your data to assess your eligibility for funding , can be found in our Postgraduate Applicant Privacy Policy.

Admissions panels and assessors

All recommendations to admit a student involve the judgement of at least two members of the academic staff with relevant experience and expertise, and must also be approved by the Director of Graduate Studies or Admissions Committee (or equivalent within the department).

Admissions panels or committees will always include at least one member of academic staff who has undertaken appropriate training.

Other factors governing whether places can be offered

The following factors will also govern whether candidates can be offered places:

• the ability of the University to provide the appropriate supervision for your studies, as outlined under the 'Supervision' heading in the  About  section of this page;
• the ability of the University to provide appropriate support for your studies (eg through the provision of facilities, resources, teaching and/or research opportunities); and
• minimum and maximum limits to the numbers of students who may be admitted to the University's taught and research programmes.

Offer conditions for successful applications

If you receive an offer of a place at Oxford, your offer will outline any conditions that you need to satisfy and any actions you need to take, together with any associated deadlines. These may include academic conditions, such as achieving a specific final grade in your current degree course. These conditions will usually depend on your individual academic circumstances and may vary between applicants. Our ' After you apply ' pages provide more information about offers and conditions . 

In addition to any academic conditions which are set, you will also be required to meet the following requirements:

Financial Declaration

If you are offered a place, you will be required to complete a  Financial Declaration  in order to meet your financial condition of admission.

Disclosure of criminal convictions

In accordance with the University’s obligations towards students and staff, we will ask you to declare any  relevant, unspent criminal convictions  before you can take up a place at Oxford.

Academic Technology Approval Scheme (ATAS)

Some postgraduate research students in science, engineering and technology subjects will need an Academic Technology Approval Scheme (ATAS) certificate prior to applying for a  Student visa (under the Student Route) . For some courses, the requirement to apply for an ATAS certificate may depend on your research area.

The Mathematical Institute's home is the purpose-built Andrew Wiles Building, opened in 2013. This provides ample teaching facilities for lectures, classes and seminars. Each research student is allocated an office in the Andrew Wiles Building that they will share with 3 or 4 other students: each student has their own desk, with a computer. The Mathematical Institute provides IT support, and students can use the department's Whitehead Library, with an extensive range of books and journals.

In addition to the common room, where graduate students regularly gather for coffee and other social occasions, there is also a café in the Andrew Wiles Building.

The department offers extensive support to students, from regular skills training and career development sessions to a variety of social events in a welcoming and inclusive atmosphere. You will have the opportunity to interact with fellow students and other members of your research groups, and more widely across the department. The department is committed to offering you the best supervision and to providing a stimulating research environment.

Mathematics

Mathematics has been studied in Oxford since the University was first established in the 12th century. The Mathematical Institute aims to preserve and expand mathematical culture through excellence in teaching and research.

The Mathematical Institute offers a wide range of graduate courses, including both taught master’s courses and research degrees. Research and teaching covers the spectrum of pure and applied mathematics with researchers working in fields including:

• number theory
• combinatorics
• mathematical physics
• mathematical finance
• mathematical modelling
• mathematical biology
• numerical analysis.

Graduate students are an integral part of the department, interacting with each other and with academic staff as part of a vibrant community that strives to further mathematical study. As a graduate student at Oxford you will benefit from excellent resources, extensive training opportunities and supportive guidance from your supervisor or course director.

The Mathematical Institute has strong ties with other University departments including Computer Science, Statistics and Physics, teaching several courses jointly. Strong links with industrial and other partners are also central to the department.

View all courses   View taught courses View research courses

The University expects to be able to offer over 1,000 full or partial graduate scholarships across the collegiate University in 2024-25. You will be automatically considered for the majority of Oxford scholarships , if you fulfil the eligibility criteria and submit your graduate application by the relevant December or January deadline. Most scholarships are awarded on the basis of academic merit and/or potential. 

For further details about searching for funding as a graduate student visit our dedicated Funding pages, which contain information about how to apply for Oxford scholarships requiring an additional application, details of external funding, loan schemes and other funding sources.

Please ensure that you visit individual college websites for details of any college-specific funding opportunities using the links provided on our college pages or below:

Please note that not all the colleges listed above may accept students on this course. For details of those which do, please refer to the College preference section of this page.

Annual fees for entry in 2024-25

Further details about fee status eligibility can be found on the fee status webpage.

Information about course fees

Course fees are payable each year, for the duration of your fee liability (your fee liability is the length of time for which you are required to pay course fees). For courses lasting longer than one year, please be aware that fees will usually increase annually. For details, please see our guidance on changes to fees and charges .

Course fees cover your teaching as well as other academic services and facilities provided to support your studies. Unless specified in the additional information section below, course fees do not cover your accommodation, residential costs or other living costs. They also don’t cover any additional costs and charges that are outlined in the additional information below.

Continuation charges

Following the period of fee liability , you may also be required to pay a University continuation charge and a college continuation charge. The University and college continuation charges are shown on the Continuation charges page.

Where can I find further information about fees?

The Fees and Funding  section of this website provides further information about course fees , including information about fee status and eligibility  and your length of fee liability .

Additional information

There are no compulsory elements of this course that entail additional costs beyond fees (or, after fee liability ends, continuation charges) and living costs. However, please note that, depending on your choice of research topic and the research required to complete it, you may incur additional expenses, such as travel expenses, research expenses, and field trips. You will need to meet these additional costs, although you may be able to apply for small grants from your department and/or college to help you cover some of these expenses.

Living costs

In addition to your course fees, you will need to ensure that you have adequate funds to support your living costs for the duration of your course.

For the 2024-25 academic year, the range of likely living costs for full-time study is between c. £1,345 and £1,955 for each month spent in Oxford. Full information, including a breakdown of likely living costs in Oxford for items such as food, accommodation and study costs, is available on our living costs page. The current economic climate and high national rate of inflation make it very hard to estimate potential changes to the cost of living over the next few years. When planning your finances for any future years of study in Oxford beyond 2024-25, it is suggested that you allow for potential increases in living expenses of around 5% each year – although this rate may vary depending on the national economic situation. UK inflationary increases will be kept under review and this page updated.

Students enrolled on this course will belong to both a department/faculty and a college. Please note that ‘college’ and ‘colleges’ refers to all 43 of the University’s colleges, including those designated as societies and permanent private halls (PPHs). 

If you apply for a place on this course you will have the option to express a preference for one of the colleges listed below, or you can ask us to find a college for you. Before deciding, we suggest that you read our brief  introduction to the college system at Oxford  and our  advice about expressing a college preference . For some courses, the department may have provided some additional advice below to help you decide.

The following colleges accept students on the MSc by Research in Mathematics:

• Balliol College
• Brasenose College
• Christ Church
• Corpus Christi College
• Exeter College
• Green Templeton College
• Hertford College
• Jesus College
• Keble College
• Kellogg College
• Lady Margaret Hall
• Linacre College
• Lincoln College
• Magdalen College
• Mansfield College
• Merton College
• Oriel College
• Pembroke College
• The Queen's College
• Reuben College
• St Anne's College
• St Catherine's College
• St Cross College
• St Edmund Hall
• St Hilda's College
• St Hugh's College
• St Peter's College
• Somerville College
• University College
• Wadham College
• Wolfson College
• Worcester College
• Wycliffe Hall

Before you apply

Our  guide to getting started  provides general advice on how to prepare for and start your application. You can use our interactive tool to help you  evaluate whether your application is likely to be competitive .

If it's important for you to have your application considered under a particular deadline – eg under a December or January deadline in order to be considered for Oxford scholarships – we recommend that you aim to complete and submit your application at least two weeks in advance . Check the deadlines on this page and the  information about deadlines and when to apply  in our Application Guide.

Application fee waivers

An application fee of £75 is payable per course application. Application fee waivers are available for the following applicants who meet the eligibility criteria:

• applicants from low-income countries;
• refugees and displaced persons; 
• UK applicants from low-income backgrounds; and 
• applicants who applied for our Graduate Access Programmes in the past two years and met the eligibility criteria.

You are encouraged to  check whether you're eligible for an application fee waiver  before you apply.

Readmission for current Oxford graduate taught students

If you're currently studying for an Oxford graduate taught course and apply to this course with no break in your studies, you may be eligible to apply to this course as a readmission applicant. The application fee will be waived for an eligible application of this type. Check whether you're eligible to apply for readmission .

Do I need to contact anyone before I apply?

You do not need to make contact with the department before you apply but you are encouraged to visit the relevant departmental webpages to read any further information about your chosen course.

However, if you would like to speak to an academic member of staff involved in your preferred area of research, you may get in touch with them directly or via the course administrator using the contact details provided on this page.

Completing your application

You should refer to the information below when completing the application form, paying attention to the specific requirements for the supporting documents .

For this course, the application form will include questions that collect information that would usually be included in a CV/résumé. You should not upload a separate document. If a separate CV/résumé is uploaded, it will be removed from your application .

If any document does not meet the specification, including the stipulated word count, your application may be considered incomplete and not assessed by the academic department. Expand each section to show further details.

Proposed field and title of research project

Under the section titled 'Field and title of research project', you are strongly encouraged to name at least one but no more than three research groups that you would like your application to be seen by.  More information about the Research Groups in the Mathematical Institute  can be found on the department's website.

You should not use this field to type out a full research proposal. You will be able to upload your research supporting materials separately if they are required (as described below).

Proposed supervisor

If known, under 'Proposed supervisor name' enter the name of the academic(s) whom you would like to supervise your research. Otherwise, leave this field blank.

You can enter up to four names and you should list them in order of preference or indicate equal preference.

Referees: Three overall, academic preferred

Whilst you must register three referees, the department may start the assessment of your application if two of the three references are submitted by the course deadline and your application is otherwise complete. Please note that you may still be required to ensure your third referee supplies a reference for consideration.

Your references should generally be academic, though up to one professional reference will be accepted.

Your references should describe your intellectual ability, academic achievement, motivation, and aptitude for advanced research.

Official transcript(s)

Your transcripts should give detailed information of the individual grades received in your university-level qualifications to date. You should only upload official documents issued by your institution and any transcript not in English should be accompanied by a certified translation.

More information about the transcript requirement is available in the Application Guide.

Contextual statement

If you wish to provide a contextual statement with your application, you may also submit an additional statement to provide contextual information on your socio-economic background or personal circumstances in support of your application.

Submit a contextual statement

It is not necessary to anonymise this document, as we recognise that it may be necessary for you to disclose certain information in your statement. This statement will not be used as part of the initial academic assessment of applications at shortlisting, but may be used in combination with socio-economic data to provide contextual information during decision-making processes.

Please note, this statement is in addition to  completing the 'Extenuating circumstances’ section of the standard application form .

You can find  more information about the contextual statement  on our page that provides details of the continuing pilot programme to improve the assessment procedure for graduate applications.

Statement of purpose/personal statement: A maximum of 1,000 words

Your statement should be written in English and explain your reasons for applying for the course at Oxford, your relevant experience and education, and the specific areas that interest you and/or you intend to specialise in. This will be assessed for evidence of motivation for and understanding of the proposed area of study and whether a suitable supervisor can be provided.

If possible, please ensure that the word count is clearly displayed on the document.

Start or continue your application

You can start or return to an application using the relevant link below. As you complete the form, please  refer to the requirements above  and  consult our Application Guide for advice . You'll find the answers to most common queries in our FAQs.

Application Guide   Apply

ADMISSION STATUS

Closed to applications for entry in 2024-25

Register to be notified via email when the next application cycle opens (for entry in 2025-26)

12:00 midday UK time on:

Friday 5 January 2024 Latest deadline for most Oxford scholarships Final application deadline for entry in 2024-25

^Included in 2024/25 places for the DPhil in Mathematics *Three-year average (applications for entry in 2021-22 to 2023-24)

Further information and enquiries

This course is offered by the Mathematical Institute

• Course page on the Mathematical Institute's website
• Academic and research staff
• Departmental research
• Mathematical, Physical and Life Sciences
• Residence requirements for full-time courses
• Postgraduate applicant privacy policy

Course-related enquiries

Advice about contacting the department can be found in the How to apply section of this page

✉ [email protected] ☎ +44 (0)1865 615208

Application-process enquiries

See the application guide

Quick links

• Directories

Research Areas

It is possible to apply mathematics to almost any field of human endeavor. Here are some of the fields we’re working on now.

Scientific Computing and Numerical Analysis

Researchers : Loyce Adams , Bernard Deconinck , Randy LeVeque , Ioana Dumitriu , Anne Greenbaum , James Riley

Many practical problems in science and engineering cannot be solved completely by analytical means. Research in the area of numerical analysis and scientific computation is concerned with the development and analysis of numerical algorithms, the implementation of these algorithms on modern computer architectures, and the use of numerical methods in conjunction with mathematical modeling to solve large-scale practical problems. Major research areas in this department include computational fluid dynamics (CFD), interface and front tracking methods, iterative methods in numerical linear algebra, and algorithms for parallel computers.Current research topics in CFD include: 

• high resolution methods for solving nonlinear conservation laws with shock wave solutions
• numerical methods for atmospheric flows, particularly cloud formation
• Cartesian grid methods for solving multidimensional problems in complicated geometries on uniform grids
• spectral methods for fluid stability problems
• front tracking methods for fluid flow problems with free surfaces or immersed interfaces in the context of porous media flow (ground water or oil reservoir simulation) and in physiological flows with elastic membranes.
• nonequilibrium flows in combustion and astrophysical simulation
• immersed interface methods for solidification or melting problems and seismic wave equations with discontinuous coefficients that arise in modeling the geological structure of the earth.

Another research focus is the development of methods for large-scale scientific computations that are suited to implementation on parallel computer architectures. Current interests include:

• preconditioners for the iterative solution of large linear or nonlinear systems
• methods for the symmetric and nonsymmetric eigenvalue problems
• methods for general interface problems in complicated domains.

The actual implementation and testing of methods on parallel architectures is possible through collaboration with the Department of Computer Science, the Boeing Company, and the Pacific Northwest Labs.

Nonlinear Waves and Coherent Structures

Researchers : Bernard Deconinck , Nathan Kutz ,  Randy LeVeque

Most problems in applied mathematics are inherently nonlinear. The effects due to nonlinearities may become important under the right circumstances. The area of nonlinear waves and coherent structures considers how nonlinear effects influence problems involving wave propagation. Sometimes these effects are desirable and lead to new applications (mode-locked lasers, optical solitons and nonlinear optics). Other times one has no choice but to consider their impact (water waves). The area of nonlinear waves encompasses a large collection of phenomena, such as the formation and propagation of shocks and solitary waves. The area received renewed interest starting in the 1960s with the development of soliton theory, which examines completely integrable systems and classes of their special solutions.

Mathematical Biology

Researchers : Mark Kot , Hong Qian , Eric Shea-Brown , Elizabeth Halloran , Suresh Moolgavkar , Eli Shlizerman , Ivana Bozic

Mathematical biology is an increasingly large and well-established branch of applied mathematics. This growth reflects both the increasing importance of the biological and biomedical sciences and an appreciation for the mathematical subtleties and challenges that arise in the modelling of complex biological systems. Our interest, as a group, lies in understanding the spatial and temporal patterns that arise in dynamic biological systems. Our mathematical activities range from reaction-diffusion equations, to nonlinear and chaotic dynamics, to optimization. We employ a variety of tools and models to study problems that arise in development, epidemiology, ecology, neuroscience, resource management, and biomechanics; and we maintain active collaborations with a large number and variety of biologists and biomedical departments both in the University and elsewhere. For more information, please see the  Mathematical Biology page .

Atmospheric Sciences and Climate Modeling

Researchers : Chris Bretherton , Ka-Kit Tung , Dale Durran

Mathematical models play a crucial role in our understanding of the fluid dynamics of the atmosphere and oceans. Our interests include mathematical methods for studying the hydrodynamical instability of shear flows, transition from laminar flow to turbulence, applications of fractals to turbulence, two-dimensional and quasi-geostrophic turbulence theory and computation, and large-scale nonlinear wave mechanics.We also develop and apply realistic coupled radiative- chemical-dynamical models for studying stratospheric chemistry, and coupled radiative-microphysical-dynamical models for studying the interaction of atmospheric turbulence and cloud systems These two topics are salient for understanding how man is changing the earth’s climate.Our work involves a strong interaction of computer modelling and classical applied analysis. This research group actively collaborates with scientists in the Atmospheric Science, Oceanography, and Geophysics department, and trains students in the emerging interdisciplinary area of earth system modeling, in addition to providing a traditional education in classical fluid dynamics.

Mathematical Methods

Researchers : Bernard Deconinck , Robert O'Malley ,  Jim Burke , Archis Ghate , John Sylvester , Gunther Uhlmann

The department maintains active research in fundamental methods of applied mathematics. These methods can be broadly applied to a vast number of problems in the engineering, physical and biological sciences. The particular strengths of the department of applied mathematics are in asymptotic and perturbation methods, applied analysis, optimization and control, and inverse problems.

Mathematical Finance

Researchers : Tim Leung , Matt Lorig , Doug Martin

The department’s growing financial math group is active in the areas of derivative pricing & hedging, algorithmic trading, portfolio optimization, insurance, risk measures, credit risk, and systemic risk. Research includes collaboration with students as well as partners from both academia and industry.

•   LinkedIn
•   Mailing List
•   YouTube
•   News Feed

• Student intranet /
• Staff intranet

Department of Mathematics

Research projects

Find a postgraduate research project in your area of interest by exploring the research projects we offer in the Department of Mathematics.

Programme directors

If you are not sure which supervisors are the best match for your interests, contact the postgraduate programme directors:

• Sean Holman  (applied mathematics and numerical analysis)
• Olatunji Johnson (probability, statistics and financial mathematics)
• Marcus Tressl  (pure mathematics)

You can also get in touch with the postgraduate research leads through our  research themes  page.

Opportunities within the department are advertised by supervisors as either:

• Specific, well-defined individual projects : which you can apply for directly after contacting the named supervisor
• Research fields with suggestions for possible projects : where you can discuss a range of potential projects available in a specific area with the supervisor.

Choosing the right PhD project depends on matching your interests to those of your supervisor.

Our  research themes  page gives an overview of the research taking place in the Department and contacts for each area. Potential supervisors can also be contacted directly through the academic staff list . They will be able to tell you more about the type of projects they offer and/or you can suggest a research project yourself.

Please note that all PhD projects are eligible for funding via a variety of scholarships from the Department, the Faculty of Science and Engineering and/or the University; see our  funding page  for further details. All scholarships are awarded competitively by the relevant postgraduate funding committees.

Academics regularly apply for research grants and may therefore be able to offer funding for specific projects without requiring approval from these committees. Some specific funded projects are listed below, but many of our students instead arrive at a project through discussion with potential supervisors.

Specific, individual projects

Browse all of our specific, individual projects listed on FindAPhD:

Research field projects

In addition to individual projects listed on FindAPhD, we are also looking for postgraduate researchers for potential projects within a number of other research fields.

Browse these fields below and get in contact with the named supervisor to find out more.

Continuum mechanics

Maths: continuum mechanics, maths: mathematics in the life sciences, numerical analytics and scientific computing, statistics, inverse problems, uncertainty quantification and data science, algebra, logic and number theory, maths: algebra, logic and number theory, mathematics in the life sciences, maths: statistics, inverse problems, uncertainty quantification and data science, probability, financial mathematics and actuarial science, applied mathematics and numerical analysis, complex deformations of biological soft tissues.

Supervisor: [email protected] | [email protected]

The answers to many open questions in medicine depend on understanding the mechanical behaviour of biological soft tissues. For example, which tendon is most appropriate to replace the anterior cruciate ligament in reconstruction surgery? what causes the onset of aneurysms in the aorta? and how does the mechanics of the bladder wall affect afferent nerve firing? Current work at The University of Manchester seeks to understand how the microstructure of a biological soft tissue affects its macroscale mechanical properties. We have previously focused on developing non-linear elastic models of tendons and are now seeking to incorporate more complex physics such as viscoelasticity, and to consider other biological soft tissues, using our “in house” finite element software oomph-lib. The work will require development and implementation of novel constitutive equations as well as formulation of non-standard problems in solid mechanics. The project is likely to appeal to students with an interest in continuum mechanics, computational mathematics and interdisciplinary science.

Fluid flow, interfaces, bifurcations, continuation and control

Supervisor: [email protected]

My research interests are in fluid dynamical systems with deformable interfaces, for example bubbles in very viscous fluid, or inkjet printed droplets. The deformability of the interface can lead to complex nonlinear behaviour, and often occurs in configurations where full numerical simulation of the three-dimensional system is computationally impossible. This computational difficulty leaves an important role for mathematical modelling, in using asymptotic or physical arguments to devise simpler models which can help us understand underlying physical mechanisms, make testable predictions, and to directly access control problems for active (feedback) or passive control mechanisms. Most recently I am interested in how different modelling methodologies affect whether models are robust in the forward or control problems. I am also interested in how control-based continuation methods can be used in continuum mechanics to directly observe unstable dynamical behaviour in experiments, even without access to a physical model. This research combines fluid dynamics, mathematical modelling, computational methods (e.g. with the finite-element library oomph-lib), experiments conducted in the Manchester Centre for Nonlinear Dynamics, control theory and nonlinear dynamics. I would not expect any student to have experience in all these areas and there is scope to shape any project to your interests.

Granular materials in industry and nature

Supervisor: [email protected]

The field of granular materials encompasses a vast range of materials and processes, from the formation of sand dunes on a beach and snow avalanches in the mountains, to the roasting of coffee beans and the manufacture of pharmaceutical tablets. The science of granular materials is still in its relative infancy, and many aspects of flowing grains cannot yet be predicted with a continuum rheology. Insights into granular material behaviour come from a range of methods, and my research therefore combines mathematical modelling, computation, and laboratory experiments, undertaken at the Manchester Centre for Nonlinear Dynamics laboratories. Some example areas of work suitable for a PhD project include: - Debris flows and their deposits Debris flows are rapid avalanches of rock and water, which are triggered on mountainsides when erodible sediment is destabilised by heavy rainfall or snowmelt. These flows cause loss of life and infrastructure across the world, but many of the physical mechanisms underlying their motion remain poorly understood. Because it is difficult to predict where and when a debris flow will occur, scientific observations are rarely made on an active flow. More often, all we have to work from is the deposit left behind, and some detective work is required to infer properties of the flow (such as its speed and composition) from this deposit. This project focuses on developing theoretical models for debris flows that predict both a debris flow and its deposit -- in particular the way in which grains of different sizes are distributed throughout the deposit. The aim is then to invert such models, allowing observations of a deposit, when combined with model simulations, to constrain what must have happened during the flow. - Modelling polydispersity Much of the current theory of granular materials has been formulated with the assumption of a single type of grain. When grains vary in size, shape or density, it opens up the possibility that such grains with different properties separate from one another, a process called segregation. A fundamental question in this area is predicting the rate of segregation from a description of a granular material, such as the distribution of particle sizes. Thanks to some recent developments, we are approaching a point where this can be done for very simple granular materials (in particular those containing only two, similar, sizes sizes of grain), but many practical granular materials are much more complex. For example, it is common for mixtures of grains used in industry to vary in diameter by a factor of more than 100, and the complex segregation that can occur in these mixtures is poorly understood. This project will make measurements of the segregation behaviour of such mixtures and use these to put together a theoretical framework for describing segregation in complex granular materials.

Mathematical modelling of nano-reinforced foams

Supervisor: [email protected]

Complex materials are important in almost every aspect of our lives, whether that is using a cell phone, insulating a house, ensuring that transport is environmentally friendly or that packaging is sustainable. An important facet of this is to ensure that materials are optimal in some sense. This could be an optimal stiffness for a given weight or an optimal conductivity for a given stiffness. Foams are an important class of material that are lightweight but also have the potential for unprecedented mechanical properties by adding nano-reinforcements (graphene flakes or carbon nanotubes) into the background or matrix material from which the foam is fabricated. When coupled with experimentation such as imaging and mechanical testing, mathematical models allow us to understand how to improve the design and properties of such foams. A number of projects are available in this broad area and interested parties can discuss these by making contact with the supervisor.

Wave manipulation using metamaterials

The ability to control electromagnetic waves, sound, vibration has been of practical interest for decades. Over the last century a number of materials have been designed to assist with the attenuation of unwanted noise and vibration. However, recently there has been an explosion of interest in the topic of metamaterials and metasurfaces. Such media have special microstructures, designed to provide overall (dynamic) material properties that natural materials can never hope to attain and lead to the potential of negative refraction, wave redirection and the holy grail of cloaking. Many of the mechanisms to create these artificial materials rely on low frequency resonance. Frequently we are interested in the notion of homogenisation of these microstructures and this requires a mathematical framework. A number of projects are available in this broad area and interested parties can discuss these by making contact with the supervisor.

Multiscale modelling of structure-function relationships in biological tissues

Supervisor: [email protected]

Biological tissues have an intrinsically multiscale structure. They contain components that range in size from individual molecules to the scale of whole organs. The organisation of individual components of a tissue, which often has a stochastic component, is intimately connected to biological function. Examples include exchange organs such as the lung and placenta, and developing multicellular tissues where mechanical forces play an crucial role in growth. To describe such materials mathematically, new multiscale approaches are needed that retain essential elements of tissue organisation at small scales, while providing tractable descriptions of function at larger scales. Projects are available in these areas that offer opportunities to collaborate with life scientists while developing original mathematical models relating tissue structure to its biological function.

Adaptive finite element approximation strategies

Supervisor: [email protected]

I would be happy to supervise projects in the general area of efficient solution of elliptic and parabolic partial differential equations using finite elements. PhD projects would involve a mix of theoretical analysis and the development of proof-of-concept software written in MATLAB or Python. The design of robust and efficient error estimators is an open problem in computational fluid dynamics. Recent papers on this topic include Alex Bespalov, Leonardo Rocchi and David Silvester, T--IFISS: a toolbox for adaptive FEM computation, Computers and Mathematics with Applications, 81: 373--390, 2021. https://doi.org/10.1016/j.camwa.2020.03.005 Arbaz Khan, Catherine Powell and David Silvester, Robust a posteriori error estimators for mixed approximation of nearly incompressible elasticity, International Journal for Numerical Methods in Engineering, 119: 1--20, 2019. https://doi.org/10.1002/nme.6040 John Pearson, Jen Pestana and David Silvester, Refined saddle-point preconditioners for discretized Stokes problems, Numerische Mathematik, 138: 331--363, 2018. https://doi.org/10.1007/s00211-017-0908-4

Efficient solution for PDEs with random data

I would be happy to supervise projects in the general area of efficient solution of elliptic and parabolic partial differential equations with random data. PhD projects would involve a mix of theoretical analysis and the development of proof-of-concept software written in MATLAB or Python. The design of robust and efficient error estimators for stochastic collocation approximation methods is an active area of research within the uncertainty quantification community. Recent papers on this topic include Alex Bespalov, David Silvester and Feng Xu. Error estimation and adaptivity for stochastic collocation finite elements Part I: single-level approximation, SIAM J. Scientific Computing, 44: A3393--A3412, 2022. {\tt https://doi.org/10.1137/21M1446745} Arbaz Khan, Alex Bespalov, Catherine Powell and David Silvester, Robust a posteriori error estimators for stochastic Galerkin formulations of parameter-dependent linear elasticity equations, Mathematics of Computation, 90: 613--636, 2021. https://doi.org/10.1090/mcom/3572 Jens Lang, Rob Scheichl and David Silvester, A fully adaptive multilevel collocation strategy for solving elliptic PDEs with random data, J. Computational Physics, 419, 109692, 2020. https://doi.org/10.1016/j.jcp.2020.109692

Bayesian and machine learning methods for statistical inverse problems

Supervisor: [email protected]

A range of projects are available on the topic of statistical inverse problems, in particular with application to problems in applied mathematics. Our aim is to construct new methods for the solution of statistical inverse problems, and to apply them to real problems from science, biology, engineering, etc. These may be more traditional Markov chain Monte Carlo (MCMC) methods, Piecewise-deterministic Markov processes (PDMPs), gradient flows (e.g. Stein gradient descent), or entirely new families of methods. Where possible the methods will be flexible and widely applicable, which will enable us to also apply them to real problems and datasets. Some recent applications involve cell matching in biology, and characterisation of physical properties of materials, for example the thermal properties of a manmade material, or the Young's modulus of a tendon or artery. The project will require the candidate to be proficient in a modern programming language (e.g. Python).

Machine learning with partial differential equations

Supervisor: [email protected]

Machine learning and artificial intelligence play a major part in our everyday life. Self-driving cars, automatic diagnoses from medical images, face recognition, or fraud detection, all profit especially from the universal applicability of deep neural networks. Their use in safety critical applications, however, is problematic: no interpretability, missing mathematical guarantees for network or learning process, and no quantification of the uncertainties in the neural network output. Recently, models that are based on partial differential equations (PDEs) have gained popularity in machine learning. In a classification problem, for instance, a PDE is constructed whose solution correctly classifies the training data and gives a suitable model to classify unlabelled feature vectors. In practice, feature vectors tend to be high dimensional and the natural space on which they live tends to have a complicated geometry. Therefore, partial differential equations on graphs are particularly suitable and popular. The resulting models are interpretable, mathematically well-understood, and uncertainty quantification is possible. In addition, they can be employed in a semi-supervised fashion, making them highly applicable in small data settings. I am interested in various mathematical, statistical, and computational aspects of PDE-based machine learning. Many of those aspects translate easily into PhD projects; examples are - Efficient algorithms for p-Laplacian-based regression and clustering - Bayesian identification of graphs from flow data - PDEs on random graphs - Deeply learned PDEs in data science Depending on the project, applicants should be familiar with at least one of: (a) numerical analysis and numerical linear algebra; (b) probability theory and statistics; (c) machine learning and deep learning.

Pure Mathematics and Logic

Algebraic differential equations and model theory.

Supervisor: [email protected]

Differential rings and algebraic differential equations have been a crucial source of examples for model theory (more specifically, geometric stability theory), and have had numerous application in number theory, algebraic geometry, and combinatorics (to name a few). In this project we propose to establish and analyse deep structural results on the model theory of (partial) differential fields. In particular, in the setup of differentially large fields. There are interesting questions around inverse problems in differential Galois theory that can be address as part of this project. On the other hand, there are (still open) questions related to the different notions of rank in differentially closed fields; for instance: are there infinite dimensional types that are also strongly minimal? This is somewhat related to the understanding of regular types, which interestingly are quite far from being fully classified. A weak version of Zilber's dichotomy have been established for such types, but is the full dichotomy true?

Algebraic invariants of abelian varieties

Supervisor: [email protected]

Project: Algebraic invariants of abelian varieties Abelian varieties are higher-dimensional generalisations of elliptic curves, objects of algebraic geometry which are of great interest to number theorists. There are various open questions about how properties of abelian varieties vary across a families of abelian varieties. The aim of this project is to study the variation of algebraic objects attached to abelian varieties, such as endomorphism algebras, Mumford-Tate groups or isogenies. These algebraic objects control much of the behaviour of the abelian variety. We aim to bound their complexity in terms of the equations defining the abelian variety. Potential specific projects include: (1) Constructing "relations between periods" from the Mumford-Tate group. This involves concrete calculations of polynomials, similar in style to classical invariant theory of reductive groups. (2) Understanding the interactions between isogenies and polarisations of abelian varieties. This involves calculations with fundamental sets for arithmetic group actions, generalising reduction theory for quadratic forms. A key tool is the theory of reductive groups and their finite-dimensional representations (roots and weights).

Algebraic Model Theory of Fields with Operators

Model theory is a branch of Mathematical Logic that has had several remarkable applications with other areas of mathematics, such as Combinatorics, Algebraic Geometry, Number Theory, Arithmetic Geometry, Complex and Real Analysis, Functional Analysis, and Algebra (to name a few). Some of these applications have come from the study of model-theoretic properties of fields equipped with a family of operators. For instance, this includes differential/difference fields. In this project, we look at the model theory of fields equipped with general classes of operators and also within certain natural classes of arithmetic fields (such as large fields). Several foundational questions remain open around what is called "model-companion", "elimination of imaginaries", and the "trichotomy", this is a small sample of the problems that can be tackled.

Homeomorphism groups from a geometric viewpoint

Supervisor: [email protected]

A powerful technique for studying groups is to use their actions by isometries on metric spaces. Properties of the action can be translated into algebraic properties of the group, and vice versa. This is called geometric group theory, and has played a key role in different fields of mathematics e.g. random groups, mapping class groups of surfaces, fundamental groups of 3-manifolds, the Cremona group. In this project we will study the homeomorphisms of a surface by using geometric group theoretic techniques recently introduced by Bowden, Hensel, and myself. This is a new research initiative at the frontier between dynamics, topology, and geometric group theory, and there are many questions waiting to be explored using these tools. These range from new questions on the relationship between the topology/dynamics of homeomorphisms and their action on metric spaces, to older questions regarding the algebraic structure of the homeomorphism group.

The Existential Closedness problem for exponential and automorphic functions

Supervisor: [email protected]

The Existential Closedness problem asks when systems of equations involving field operations and certain classical functions of a complex variable, such as exponential and modular functions, have solutions in the complex numbers. There are conjectures predicting when such systems should have solutions. The general philosophy is that when a system is not "overdetermined" (e.g. more equations than variables) then it should have a solution. The notion of an overdetermined system of equations is related to Schanuel's conjecture and its analogues and is captured by some purely algebraic conditions. The aim of this project is to make progress towards the Existential Closedness conjectures (EC for short) for exponential and automorphic functions (and the derivatives of automorphic functions). These include the usual complex exponential function, as well as the exponential functions of semi-abelian varieties, and modular functions such as the j-invariant. Significant progress has been made towards EC in recent years, but the full conjectures are open. There are many special cases which are within reach and could be tackled as part of a PhD project. Methods used to approach EC come from complex analysis and geometry, differential algebra, model theory (including o-minimality), tropical geometry. Potential specific projects are: (1) proving EC in low dimensions (e.g. for 2 or 3 variables), (2) proving EC for certain relations defined in terms of the function under consideration, e.g. establishing new EC results for "blurred" exponential and/or modular functions, (3) proving EC under additional geometric assumptions on the system of equations, (4) using EC to study the model-theoretic properties of exponential and automorphic functions.

Statistics and Probability

Mathematical epidemiology.

Supervisor: [email protected]

Understanding patterns of disease at the population level - Epidemiology - is inherently a quantitative problem, and increasingly involves sophisticated research-level mathematics and statistics in both infectious and chronic diseases. The details of which diseases and mathematics offer the best PhD directions are likely to vary over time, but this broad area is available for PhD research.

Spatial and temporal modelling for crime

Supervisor: [email protected] | [email protected]

A range of projects are available on the topic of statistical spatial and temporal modelling for crime. These projects will focus on developing novel methods for modelling crime related events in space and time, and applying these to real world datasets, mostly within the UK, but with the possibility to use international datasets. Some examples of recent applications include spatio-temporal modelling of drug overdoses and related crime. These projects will aim to use statistical spatio-temporalpoint processes methods, Bayesian methods, and machine learning methods. The project will require the candidate to be proficient in a modern programming language (e.g., R or Python). Applicants should have achieved a first-class degree in Statistics or Mathematics, with a significant component of Statistics, and be proficient in a statistical programming language (e.g., R, Python, Stata, S). We strongly recommend that you contact the supervisor(s) for this project before you apply. Please send your CV and a brief cover letter to [email protected] before you apply. At Manchester we offer a range of scholarships, studentships and awards at university, faculty and department level, to support both UK and overseas postgraduate researchers. For more information, visit our funding page or search our funding database for specific scholarships, studentships and awards you may be eligible for.

Distributional approximation by Stein's method Theme

Supervisor: [email protected]

Stein's method is a powerful (and elegant) technique for deriving bounds on the distance between two probability distributions with respect to a probability metric. Such bounds are of interest, for example, in statistical inference when samples sizes are small; indeed, obtaining bounds on the rate of convergence of the central limit theorem was one of the most important problems in probability theory in the first half of the 20th century. The method is based on differential or difference equations that in a sense characterise the limit distribution and coupling techniques that allow one to derive approximations whilst retaining the probabilistic intuition. There is an active area of research concerning the development of Stein's method as a probabilistic tool and its application in areas as diverse as random graph theory, statistical mechanics and queuing theory. There is an excellent survey of Stein's method (see below) and, given a strong background in probability, the basic method can be learnt quite quickly, so it would be possible for the interested student to make progress on new problems relatively shortly into their PhD. Possible directions for research (although not limited) include: extend Stein's method to new limit distributions; generalisations of the central limit theorem; investigate `faster than would be expected' convergence rates and establish necessary and sufficient conditions under which they occur; applications of Stein's method to problems from, for example, statistical inference. Literature: Ross, N. Fundamentals of Stein's method. Probability Surveys 8 (2011), pp. 210-293.

Long-term behaviour of Markov Chains

Supervisor: [email protected]

Several projects are available, studying idealised Markovian models of epidemic, population and network processes. The emphasis will mostly be on theoretical aspects of the models, involving advanced probability theory. For instance, there are a number of stochastic models of epidemics where the course of the epidemic is known to follow the solution of a differential equation over short time intervals, but where little or nothing has been proved about the long-term behaviour of the stochastic process. Techniques have been developed for studying such problems, and a project might involve adapting these methods to new settings. Depending on the preference of the candidate, a project might involve a substantial computational component, gaining insights into the behaviour of a model, via simulations, ahead of proving rigorous theoretical results.

Secondary Menu

• Math Intranet
• Computational Mathematics

Computational Mathematics involves mathematical research in areas of science and engineering where computing plays a central and essential role. Topics include for example developing accurate and efficient numerical methods for solving physical or biological models, analysis of numerical approximations to differential and integral equations, developing computational tools to better understand data and structure, etc. Computational mathematics is a field closely connected with a variety of other mathematical branches, as for often times a better mathematical understanding of the problem leads to innovative numerical techniques.

Duke's Mathematics Department has a large group of mathematicians whose research involves scientific computing, numerical analysis, machine learning, computational topology, and algorithmic algebraic geometry. The computational mathematics research of our faculty has applications in data analysis and signal processing, fluid and solid mechanics, electronic structure theory, biological networks, and many other topics.

Future themes of mathematics education research: an international survey before and during the pandemic

• Open access
• Published: 06 April 2021
• Volume 107 , pages 1–24, ( 2021 )

Cite this article

You have full access to this open access article

• Arthur Bakker   ORCID: orcid.org/0000-0002-9604-3448 1 ,
• Jinfa Cai   ORCID: orcid.org/0000-0002-0501-3826 2 &
• Linda Zenger 1  

32k Accesses

98 Citations

18 Altmetric

Explore all metrics

Before the pandemic (2019), we asked: On what themes should research in mathematics education focus in the coming decade? The 229 responses from 44 countries led to eight themes plus considerations about mathematics education research itself. The themes can be summarized as teaching approaches, goals, relations to practices outside mathematics education, teacher professional development, technology, affect, equity, and assessment. During the pandemic (November 2020), we asked respondents: Has the pandemic changed your view on the themes of mathematics education research for the coming decade? If so, how? Many of the 108 respondents saw the importance of their original themes reinforced (45), specified their initial responses (43), and/or added themes (35) (these categories were not mutually exclusive). Overall, they seemed to agree that the pandemic functions as a magnifying glass on issues that were already known, and several respondents pointed to the need to think ahead on how to organize education when it does not need to be online anymore. We end with a list of research challenges that are informed by the themes and respondents’ reflections on mathematics education research.

Similar content being viewed by others

Learning from Research, Advancing the Field

The Narcissism of Mathematics Education

Educational Research on Learning and Teaching Mathematics

Explore related subjects.

• Artificial Intelligence
• Medical Ethics

Avoid common mistakes on your manuscript.

1 An international survey in two rounds

Around the time when Educational Studies in Mathematics (ESM) and the Journal for Research in Mathematics Education (JRME) were celebrating their 50th anniversaries, Arthur Bakker (editor of ESM) and Jinfa Cai (editor of JRME) saw a need to raise the following future-oriented question for the field of mathematics education research:

Q2019: On what themes should research in mathematics education focus in the coming decade?

To that end, we administered a survey with just this one question between June 17 and October 16, 2019.

When we were almost ready with the analysis, the COVID-19 pandemic broke out, and we were not able to present the results at the conferences we had planned to attend (NCTM and ICME in 2020). Moreover, with the world shaken up by the crisis, we wondered if colleagues in our field might think differently about the themes formulated for the future due to the pandemic. Hence, on November 26, 2020, we asked a follow-up question to those respondents who in 2019 had given us permission to approach them for elaboration by email:

Q2020: Has the pandemic changed your view on the themes of mathematics education research for the coming decade? If so, how?

In this paper, we summarize the responses to these two questions. Similar to Sfard’s ( 2005 ) approach, we start by synthesizing the voices of the respondents before formulating our own views. Some colleagues put forward the idea of formulating a list of key themes or questions, similar to the 23 unsolved mathematical problems that David Hilbert published around 1900 (cf. Schoenfeld, 1999 ). However, mathematics and mathematics education are very different disciplines, and very few people share Hilbert’s formalist view on mathematics; hence, we do not want to suggest that we could capture the key themes of mathematics education in a similar way. Rather, our overview of themes drawn from the survey responses is intended to summarize what is valued in our global community at the time of the surveys. Reasoning from these themes, we end with a list of research challenges that we see worth addressing in the future (cf. Stephan et al., 2015 ).

2 Methodological approach

2.1 themes for the coming decade (2019).

We administered the 1-question survey through email lists that we were aware of (e.g., Becker, ICME, PME) and asked mathematics education researchers to spread it in their national networks. By October 16, 2019, we had received 229 responses from 44 countries across 6 continents (Table 1 ). Although we were happy with the larger response than Sfard ( 2005 ) received (74, with 28 from Europe), we do not know how well we have reached particular regions, and if potential respondents might have faced language or other barriers. We did offer a few Chinese respondents the option to write in Chinese because the second author offered to translate their emails into English. We also received responses in Spanish, which were translated for us.

Ethical approval was given by the Ethical Review Board of the Faculties of Science and Geo-science of Utrecht University (Bèta L-19247). We asked respondents to indicate if they were willing to be quoted by name and if we were allowed to approach them for subsequent information. If they preferred to be named, we mention their name and country; otherwise, we write “anonymous.” In our selection of quotes, we have focused on content, not on where the response came from. On March 2, 2021, we approached all respondents who were quoted to double-check if they agreed to be quoted and named. One colleague preferred the quote and name to be deleted; three suggested small changes in wording; the others approved.

On September 20, 2019, the three authors met physically at Utrecht University to analyze the responses. After each individual proposal, we settled on a joint list of seven main themes (the first seven in Table 2 ), which were neither mutually exclusive nor exhaustive. The third author (Zenger, then still a student in educational science) next color coded all parts of responses belonging to a category. These formed the basis for the frequencies and percentages presented in the tables and text. The first author (Bakker) then read all responses categorized by a particular code to identify and synthesize the main topics addressed within each code. The second author (Cai) read all of the survey responses and the response categories, and commented. After the initial round of analysis, we realized it was useful to add an eighth theme: assessment (including evaluation).

Moreover, given that a large number of respondents made comments about mathematics education research itself, we decided to summarize these separately. For analyzing this category of research, we used the following four labels to distinguish types of comments on our discipline of mathematics education research: theory, methodology, self-reflection (including ethical considerations), interdisciplinarity, and transdisciplinarity. We then summarized the responses per type of comment.

It has been a daunting and humbling experience to study the huge coverage and diversity of topics that our colleagues care about. Any categorization felt like a reduction of the wealth of ideas, and we are aware of the risks of “sorting things out” (Bowker & Star, 2000 ), which come with foregrounding particular challenges rather than others (Stephan et al., 2015 ). Yet the best way to summarize the bigger picture seemed by means of clustering themes and pointing to their relationships. As we identified these eight themes of mathematics education research for the future, a recurring question during the analysis was how to represent them. A list such as Table 2 does not do justice to the interrelations between the themes. Some relationships are very clear, for example, educational approaches (theme 2) working toward educational or societal goals (theme 1). Some themes are pervasive; for example, equity and (positive) affect are both things that educators want to achieve but also phenomena that are at stake during every single moment of learning and teaching. Diagrams we considered to represent such interrelationships were either too specific (limiting the many relevant options, e.g., a star with eight vertices that only link pairs of themes) or not specific enough (e.g., a Venn diagram with eight leaves such as the iPhone symbol for photos). In the end, we decided to use an image and collaborated with Elisabeth Angerer (student assistant in an educational sciences program), who eventually made the drawing in Fig. 1 to capture themes in their relationships.

Artistic impression of the future themes

2.2 Has the pandemic changed your view? (2020)

On November 26, 2020, we sent an email to the colleagues who responded to the initial question and who gave permission to be approached by email. We cited their initial response and asked: “Has the pandemic changed your view on the themes of mathematics education research for the coming decade? If so, how?” We received 108 responses by January 12, 2021. The countries from which the responses came included China, Italy, and other places that were hit early by the COVID-19 virus. The length of responses varied from a single word response (“no”) to elaborate texts of up to 2215 words. Some people attached relevant publications. The median length of the responses was 87 words, with a mean length of 148 words and SD = 242. Zenger and Bakker classified them as “no changes” (9 responses) or “clearly different views” (8); the rest of the responses saw the importance of their initial themes reinforced (45), specified their initial responses (43), or added new questions or themes (35). These last categories were not mutually exclusive, because respondents could first state that they thought the initial themes were even more relevant than before and provide additional, more specified themes. We then used the same themes that had been identified in the first round and identified what was stressed or added in the 2020 responses.

3 The themes

The most frequently mentioned theme was what we labeled approaches to teaching (64% of the respondents, see Table 2 ). Next was the theme of goals of mathematics education on which research should shed more light in the coming decade (54%). These goals ranged from specific educational goals to very broad societal ones. Many colleagues referred to mathematics education’s relationships with other practices (communities, institutions…) such as home, continuing education, and work. Teacher professional development is a key area for research in which the other themes return (what should students learn, how, how to assess that, how to use technology and ensure that students are interested?). Technology constitutes its own theme but also plays a key role in many other themes, just like affect. Another theme permeating other ones is what can be summarized as equity, diversity, and inclusion (also social justice, anti-racism, democratic values, and several other values were mentioned). These values are not just societal and educational goals but also drivers for redesigning teaching approaches, using technology, working on more just assessment, and helping learners gain access, become confident, develop interest, or even love for mathematics. To evaluate if approaches are successful and if goals have been achieved, assessment (including evaluation) is also mentioned as a key topic of research.

In the 2020 responses, many wise and general remarks were made. The general gist is that the pandemic (like earlier crises such as the economic crisis around 2008–2010) functioned as a magnifying glass on themes that were already considered important. Due to the pandemic, however, systemic societal and educational problems were said to have become better visible to a wider community, and urge us to think about the potential of a “new normal.”

3.1 Approaches to teaching

We distinguish specific teaching strategies from broader curricular topics.

3.1.1 Teaching strategies

There is a widely recognized need to further design and evaluate various teaching approaches. Among the teaching strategies and types of learning to be promoted that were mentioned in the survey responses are collaborative learning, critical mathematics education, dialogic teaching, modeling, personalized learning, problem-based learning, cross-curricular themes addressing the bigger themes in the world, embodied design, visualization, and interleaved learning. Note, however, that students can also enhance their mathematical knowledge independently from teachers or parents through web tutorials and YouTube videos.

Many respondents emphasized that teaching approaches should do more than promote cognitive development. How can teaching be entertaining or engaging? How can it contribute to the broader educational goals of developing students’ identity, contribute to their empowerment, and help them see the value of mathematics in their everyday life and work? We return to affect in Section 3.7 .

In the 2020 responses, we saw more emphasis on approaches that address modeling, critical thinking, and mathematical or statistical literacy. Moreover, respondents stressed the importance of promoting interaction, collaboration, and higher order thinking, which are generally considered to be more challenging in distance education. One approach worth highlighting is challenge-based education (cf. Johnson et al. 2009 ), because it takes big societal challenges as mentioned in the previous section as its motivation and orientation.

3.1.2 Curriculum

Approaches by which mathematics education can contribute to the aforementioned goals can be distinguished at various levels. Several respondents mentioned challenges around developing a coherent mathematics curriculum, smoothing transitions to higher school levels, and balancing topics, and also the typical overload of topics, the influence of assessment on what is taught, and what teachers can teach. For example, it was mentioned that mathematics teachers are often not prepared to teach statistics. There seems to be little research that helps curriculum authors tackle some of these hard questions as well as how to monitor reform (cf. Shimizu & Vithal, 2019 ). Textbook analysis is mentioned as a necessary research endeavor. But even if curricula within one educational system are reasonably coherent, how can continuity between educational systems be ensured (cf. Jansen et al., 2012 )?

In the 2020 responses, some respondents called for free high-quality curriculum resources. In several countries where Internet access is a problem in rural areas, a shift can be observed from online resources to other types of media such as radio and TV.

3.2 Goals of mathematics education

The theme of approaches is closely linked to that of the theme of goals. For example, as Fulvia Furinghetti (Italy) wrote: “It is widely recognized that critical thinking is a fundamental goal in math teaching. Nevertheless it is still not clear how it is pursued in practice.” We distinguish broad societal and more specific educational goals. These are often related, as Jane Watson (Australia) wrote: “If Education is to solve the social, cultural, economic, and environmental problems of today’s data-driven world, attention must be given to preparing students to interpret the data that are presented to them in these fields.”

3.2.1 Societal goals

Respondents alluded to the need for students to learn to function in the economy and in society more broadly. Apart from instrumental goals of mathematics education, some emphasized goals related to developing as a human being, for instance learning to see the mathematics in the world and develop a relation with the world. Mathematics education in these views should empower students to combat anti-expertise and post-fact tendencies. Several respondents mentioned even larger societal goals such as avoiding extinction as a human species and toxic nationalism, resolving climate change, and building a sustainable future.

In the second round of responses (2020), we saw much more emphasis on these bigger societal issues. The urgency to orient mathematics education (and its research) toward resolving these seemed to be felt more than before. In short, it was stressed that our planet needs to be saved. The big question is what role mathematics education can play in meeting these challenges.

3.2.2 Educational goals

Several respondents expressed a concern that the current goals of mathematics education do not reflect humanity’s and societies’ needs and interests well. Educational goals to be stressed more were mathematical literacy, numeracy, critical, and creative thinking—often with reference to the changing world and the planet being at risk. In particular, the impact of technology was frequently stressed, as this may have an impact on what people need to learn (cf. Gravemeijer et al., 2017 ). If computers can do particular things much better than people, what is it that students need to learn?

Among the most frequently mentioned educational goals for mathematics education were statistical literacy, computational and algorithmic thinking, artificial intelligence, modeling, and data science. More generally, respondents expressed that mathematics education should help learners deploy evidence, reasoning, argumentation, and proof. For example, Michelle Stephan (USA) asked:

What mathematics content should be taught today to prepare students for jobs of the future, especially given growth of the digital world and its impact on a global economy? All of the mathematics content in K-12 can be accomplished by computers, so what mathematical procedures become less important and what domains need to be explored more fully (e.g., statistics and big data, spatial geometry, functional reasoning, etc.)?

One challenge for research is that there is no clear methodology to arrive at relevant and feasible learning goals. Yet there is a need to choose and formulate such goals on the basis of research (cf. Van den Heuvel-Panhuizen, 2005 ).

Several of the 2020 responses mentioned the sometimes problematic way in which numbers, data, and graphs are used in the public sphere (e.g., Ernest, 2020 ; Kwon et al., 2021 ; Yoon et al., 2021 ). Many respondents saw their emphasis on relevant educational goals reinforced, for example, statistical and data literacy, modeling, critical thinking, and public communication. A few pandemic-specific topics were mentioned, such as exponential growth.

3.3 Relation of mathematics education to other practices

Many responses can be characterized as highlighting boundary crossing (Akkerman & Bakker, 2011 ) with disciplines or communities outside mathematics education, such as in science, technology, engineering, art, and mathematics education (STEM or STEAM); parents or families; the workplace; and leisure (e.g., drama, music, sports). An interesting example was the educational potential of mathematical memes—“humorous digital objects created by web users copying an existing image and overlaying a personal caption” (Bini et al., 2020 , p. 2). These boundary crossing-related responses thus emphasize the movements and connections between mathematics education and other practices.

In the 2020 responses, we saw that during the pandemic, the relationship between school and home has become much more important, because most students were (and perhaps still are) learning at home. Earlier research on parental involvement and homework (Civil & Bernier, 2006 ; de Abreu et al., 2006 ; Jackson, 2011 ) proves relevant in the current situation where many countries are still or again in lockdown. Respondents pointed to the need to monitor students and their work and to promote self-regulation. They also put more stress on the political, economic, and financial contexts in which mathematics education functions (or malfunctions, in many respondents’ views).

3.4 Teacher professional development

Respondents explicitly mentioned teacher professional development as an important domain of mathematics education research (including teacher educators’ development). For example, Loide Kapenda (Namibia) wrote, “I am supporting UNESCO whose idea is to focus on how we prepare teachers for the future we want.” (e.g., UNESCO, 2015 ) And, Francisco Rojas (Chile) wrote:

Although the field of mathematics education is broad and each time faced with new challenges (socio-political demands, new intercultural contexts, digital environments, etc.), all of them will be handled at school by the mathematics teacher, both in primary as well as in secondary education. Therefore, from my point of view, pre-service teacher education is one of the most relevant fields of research for the next decade, especially in developing countries.

It is evident from the responses that teaching mathematics is done by a large variety of people, not only by people who are trained as primary school teachers, secondary school mathematics teachers, or mathematicians but also parents, out-of-field teachers, and scientists whose primary discipline is not mathematics but who do use mathematics or statistics. How teachers of mathematics are trained varies accordingly. Respondents frequently pointed to the importance of subject-matter knowledge and particularly noted that many teachers seem ill-prepared to teach statistics (e.g., Lonneke Boels, the Netherlands).

Key questions were raised by several colleagues: “How to train mathematics teachers with a solid foundation in mathematics, positive attitudes towards mathematics teaching and learning, and wide knowledge base linking to STEM?” (anonymous); “What professional development, particularly at the post-secondary level, motivates changes in teaching practices in order to provide students the opportunities to engage with mathematics and be successful?” (Laura Watkins, USA); “How can mathematics educators equip students for sustainable, equitable citizenship? And how can mathematics education equip teachers to support students in this?” (David Wagner, Canada)

In the 2020 responses, it was clear that teachers are incredibly important, especially in the pandemic era. The sudden change to online teaching means that

higher requirements are put forward for teachers’ educational and teaching ability, especially the ability to carry out education and teaching by using information technology should be strengthened. Secondly, teachers’ ability to communicate and cooperate has been injected with new connotation. (Guangming Wang, China)

It is broadly assumed that education will stay partly online, though more so in higher levels of education than in primary education. This has implications for teachers, for instance, they will have to think through how they intend to coordinate teaching on location and online. Hence, one important focus for professional development is the use of technology.

3.5 Technology

Technology deserves to be called a theme in itself, but we want to emphasize that it ran through most of the other themes. First of all, some respondents argued that, due to technological advances in society, the societal and educational goals of mathematics education need to be changed (e.g., computational thinking to ensure employability in a technological society). Second, responses indicated that the changed goals have implications for the approaches in mathematics education. Consider the required curriculum reform and the digital tools to be used in it. Students do not only need to learn to use technology; the technology can also be used to learn mathematics (e.g., visualization, embodied design, statistical thinking). New technologies such as 3D printing, photo math, and augmented and virtual reality offer new opportunities for learning. Society has changed very fast in this respect. Third, technology is suggested to assist in establishing connections with other practices , such as between school and home, or vocational education and work, even though there is a great disparity in how successful these connections are.

In the 2020 responses, there was great concern about the current digital divide (cf. Hodgen et al., 2020 ). The COVID-19 pandemic has thus given cause for mathematics education research to understand better how connections across educational and other practices can be improved with the help of technology. Given the unequal distribution of help by parents or guardians, it becomes all the more important to think through how teachers can use videos and quizzes, how they can monitor their students, how they can assess them (while respecting privacy), and how one can compensate for the lack of social, gestural, and embodied interaction that is possible when being together physically.

Where mobile technology was considered very innovative before 2010, smartphones have become central devices in mathematics education in the pandemic with its reliance on distance learning. Our direct experience showed that phone applications such as WhatsApp and WeChat have become key tools in teaching and learning mathematics in many rural areas in various continents where few people have computers (for a report on podcasts distributed through WhatsApp, community loudspeakers, and local radio stations in Colombia, see Saenz et al., 2020 ).

3.6 Equity, diversity, and inclusion

Another cross-cutting theme can be labeled “equity, diversity, and inclusion.” We use this triplet to cover any topic that highlights these and related human values such as equality, social and racial justice, social emancipation, and democracy that were also mentioned by respondents (cf. Dobie & Sherin, 2021 ). In terms of educational goals , many respondents stressed that mathematics education should be for all students, including those who have special needs, who live in poverty, who are learning the instruction language, who have a migration background, who consider themselves LGBTQ+, have a traumatic or violent history, or are in whatever way marginalized. There is broad consensus that everyone should have access to high-quality mathematics education. However, as Niral Shah (USA) notes, less attention has been paid to “how phenomena related to social markers (e.g., race, class, gender) interact with phenomena related to the teaching and learning of mathematical content.”

In terms of teaching approaches , mathematics education is characterized by some respondents from particular countries as predominantly a white space where some groups feel or are excluded (cf. Battey, 2013 ). There is a general concern that current practices of teaching mathematics may perpetuate inequality, in particular in the current pandemic. In terms of assessment , mathematics is too often used or experienced as a gatekeeper rather than as a powerful resource (cf. Martin et al., 2010 ). Steve Lerman (UK) “indicates that understanding how educational opportunities are distributed inequitably, and in particular how that manifests in each end every classroom, is a prerequisite to making changes that can make some impact on redistribution.” A key research aim therefore is to understand what excludes students from learning mathematics and what would make mathematics education more inclusive (cf. Roos, 2019 ). And, what does professional development of teachers that promotes equity look like?

In 2020, many respondents saw their emphasis on equity and related values reinforced in the current pandemic with its risks of a digital divide, unequal access to high-quality mathematics education, and unfair distribution of resources. A related future research theme is how the so-called widening achievement gaps can be remedied (cf. Bawa, 2020 ). However, warnings were also formulated that thinking in such deficit terms can perpetuate inequality (cf. Svensson et al., 2014 ). A question raised by Dor Abrahamson (USA) is, “What roles could digital technology play, and in what forms, in restoring justice and celebrating diversity?”

Though entangled with many other themes, affect is also worth highlighting as a theme in itself. We use the term affect in a very broad sense to point to psychological-social phenomena such as emotion, love, belief, attitudes, interest, curiosity, fun, engagement, joy, involvement, motivation, self-esteem, identity, anxiety, alienation, and feeling of safety (cf. Cobb et al., 2009 ; Darragh, 2016 ; Hannula, 2019 ; Schukajlow et al., 2017 ). Many respondents emphasized the importance of studying these constructs in relation to (and not separate from) what is characterized as cognition. Some respondents pointed out that affect is not just an individual but also a social phenomenon, just like learning (cf. Chronaki, 2019 ; de Freitas et al., 2019 ; Schindler & Bakker, 2020 ).

Among the educational goals of mathematics education, several participants mentioned the need to generate and foster interest in mathematics. In terms of approaches , much emphasis was put on the need to avoid anxiety and alienation and to engage students in mathematical activity.

In the 2020 responses, more emphasis was put on the concern about alienation, which seems to be of special concern when students are socially distanced from peers and teachers as to when teaching takes place only through technology . What was reiterated in the 2020 responses was the importance of students’ sense of belonging in a mathematics classroom (cf. Horn, 2017 )—a topic closely related to the theme of equity, diversity, and inclusion discussed before.

3.8 Assessment

Assessment and evaluation were not often mentioned explicitly, but they do not seem less important than the other related themes. A key challenge is to assess what we value rather than valuing what we assess. In previous research, the assessment of individual students has received much attention, but what seems to be neglected is the evaluation of curricula. As Chongyang Wang (China) wrote, “How to evaluate the curriculum reforms. When we pay much energy in reforming our education and curriculum, do we imagine how to ensure it will work and there will be pieces of evidence found after the new curricula are carried out? How to prove the reforms work and matter?” (cf. Shimizu & Vithal, 2019 )

In the 2020 responses, there was an emphasis on assessment at a distance. Distance education generally is faced with the challenge of evaluating student work, both formatively and summatively. We predict that so-called e-assessment, along with its privacy challenges, will generate much research interest in the near future (cf. Bickerton & Sangwin, 2020 ).

4 Mathematics education research itself

Although we only asked for future themes, many respondents made interesting comments about research in mathematics education and its connections with other disciplines and practices (such as educational practice, policy, home settings). We have grouped these considerations under the subheadings of theory, methodology, reflection on our discipline, and interdisciplinarity and transdisciplinarity. As with the previous categorization into themes, we stress that these four types are not mutually exclusive as theoretical and methodological considerations can be intricately intertwined (Radford, 2008 ).

Several respondents expressed their concern about the fragmentation and diversity of theories used in mathematics education research (cf. Bikner-Ahsbahs & Prediger, 2014 ). The question was raised how mathematics educators can “work together to obtain valid, reliable, replicable, and useful findings in our field” and “How, as a discipline, can we encourage sustained research on core questions using commensurable perspectives and methods?” (Keith Weber, USA). One wish was “comparing theoretical perspectives for explanatory power” (K. Subramaniam, India). At the same time, it was stressed that “we cannot continue to pretend that there is just one culture in the field of mathematics education, that all the theoretical framework may be applied in whichever culture and that results are universal” (Mariolina Bartolini Bussi, Italy). In addition, the wish was expressed to deepen theoretical notions such as numeracy, equity, and justice as they play out in mathematics education.

4.2 Methodology

Many methodological approaches were mentioned as potentially useful in mathematics education research: randomized studies, experimental studies, replication, case studies, and so forth. Particular attention was paid to “complementary methodologies that bridge the ‘gap’ between mathematics education research and research on mathematical cognition” (Christian Bokhove, UK), as, for example, done in Gilmore et al. ( 2018 ). Also, approaches were mentioned that intend to bridge the so-called gap between educational practice and research, such as lesson study and design research. For example, Kay Owens (Australia) pointed to the challenge of studying cultural context and identity: “Such research requires a multi-faceted research methodology that may need to be further teased out from our current qualitative (e.g., ethnographic) and quantitative approaches (‘paper and pencil’ (including computing) testing). Design research may provide further possibilities.”

Francisco Rojas (Chile) highlighted the need for more longitudinal and cross-sectional research, in particular in the context of teacher professional development:

It is not enough to investigate what happens in pre-service teacher education but understand what effects this training has in the first years of the professional career of the new teachers of mathematics, both in primary and secondary education. Therefore, increasingly more longitudinal and cross-sectional studies will be required to understand the complexity of the practice of mathematics teachers, how the professional knowledge that articulates the practice evolves, and what effects have the practice of teachers on the students’ learning of mathematics.

4.3 Reflection on our discipline

Calls were made for critical reflection on our discipline. One anonymous appeal was for more self-criticism and scientific modesty: Is research delivering, or is it drawing away good teachers from teaching? Do we do research primarily to help improve mathematics education or to better understand phenomena? (cf. Proulx & Maheux, 2019 ) The general gist of the responses was a sincere wish to be of value to the world and mathematics education more specifically and not only do “research for the sake of research” (Zahra Gooya, Iran). David Bowers (USA) expressed several reflection-inviting views about the nature of our discipline, for example:

We must normalize (and expect) the full taking up the philosophical and theoretical underpinnings of all of our work (even work that is not considered “philosophical”). Not doing so leads to uncritical analysis and implications.

We must develop norms wherein it is considered embarrassing to do “uncritical” research.

There is no such thing as “neutral.” Amongst other things, this means that we should be cultivating norms that recognize the inherent political nature of all work, and norms that acknowledge how superficially “neutral” work tends to empower the oppressor.

We must recognize the existence of but not cater to the fragility of privilege.

In terms of what is studied, some respondents felt that the mathematics education research “literature has been moving away from the original goals of mathematics education. We seem to have been investigating everything but the actual learning of important mathematics topics.” (Lyn English, Australia) In terms of the nature of our discipline, Taro Fujita (UK) argued that our discipline can be characterized as a design science, with designing mathematical learning environments as the core of research activities (cf. Wittmann, 1995 ).

A tension that we observe in different views is the following: On the one hand, mathematics education research has its origin in helping teachers teach particular content better. The need for such so-called didactical, topic-specific research is not less important today but perhaps less fashionable for funding schemes that promote innovative, ground-breaking research. On the other hand, over time it has become clear that mathematics education is a multi-faceted socio-cultural and political endeavor under the influence of many local and global powers. It is therefore not surprising that the field of mathematics education research has expanded so as to include an increasingly wide scope of themes that are at stake, such as the marginalization of particular groups. We therefore highlight Niral Shah’s (USA) response that “historically, these domains of research [content-specific vs socio-political] have been decoupled. The field would get closer to understanding the experiences of minoritized students if we could connect these lines of inquiry.”

Another interesting reflective theme was raised by Nouzha El Yacoubi (Morocco): To what extent can we transpose “research questions from developed to developing countries”? As members of the plenary panel at PME 2019 (e.g., Kazima, 2019 ; Kim, 2019 ; Li, 2019 ) conveyed well, adopting interventions that were successful in one place in another place is far from trivial (cf. Gorard, 2020 ).

Juan L. Piñeiro (Spain in 2019, Chile in 2020) highlighted that “mathematical concepts and processes have different natures. Therefore, can it be characterized using the same theoretical and methodological tools?” More generally, one may ask if our theories and methodologies—often borrowed from other disciplines—are well suited to the ontology of our own discipline. A discussion started by Niss ( 2019 ) on the nature of our discipline, responded to by Bakker ( 2019 ) and Cai and Hwang ( 2019 ), seems worth continuing.

An important question raised in several comments is how close research should be to existing curricula. One respondent (Benjamin Rott, Germany) noted that research on problem posing often does “not fit into school curricula.” This makes the application of research ideas and findings problematic. However, one could argue that research need not always be tied to existing (local) educational contexts. It can also be inspirational, seeking principles of what is possible (and how) with a longer-term view on how curricula may change in the future. One option is, as Simon Zell (Germany) suggests, to test designs that cover a longer timeframe than typically done. Another way to bridge these two extremes is “collaboration between teachers and researchers in designing and publishing research” (K. Subramaniam, India) as is promoted by facilitating teachers to do PhD research (Bakx et al., 2016 ).

One of the responding teacher-researchers (Lonneke Boels, the Netherlands) expressed the wish that research would become available “in a more accessible form.” This wish raises the more general questions of whose responsibility it is to do such translation work and how to communicate with non-researchers. Do we need a particular type of communication research within mathematics education to learn how to convey particular key ideas or solid findings? (cf. Bosch et al., 2017 )

4.4 Interdisciplinarity and transdisciplinarity

Many respondents mentioned disciplines which mathematics education research can learn from or should collaborate with (cf. Suazo-Flores et al., 2021 ). Examples are history, mathematics, philosophy, psychology, psychometry, pedagogy, educational science, value education (social, emotional), race theory, urban education, neuroscience/brain research, cognitive science, and computer science didactics. “A big challenge here is how to make diverse experts approach and talk to one another in a productive way.” (David Gómez, Chile)

One of the most frequently mentioned disciplines in relation to our field is history. It is a common complaint in, for instance, the history of medicine that historians accuse medical experts of not knowing historical research and that medical experts accuse historians of not understanding the medical discipline well enough (Beckers & Beckers, 2019 ). This tension raises the question who does and should do research into the history of mathematics or of mathematics education and to what broader purpose.

Some responses go beyond interdisciplinarity, because resolving the bigger issues such as climate change and a more equitable society require collaboration with non-researchers (transdisciplinarity). A typical example is the involvement of educational practice and policy when improving mathematics education (e.g., Potari et al., 2019 ).

Let us end this section with a word of hope, from an anonymous respondent: “I still believe (or hope?) that the pandemic, with this making-inequities-explicit, would help mathematics educators to look at persistent and systemic inequalities more consistently in the coming years.” Having learned so much in the past year could indeed provide an opportunity to establish a more equitable “new normal,” rather than a reversion to the old normal, which one reviewer worried about.

5 The themes in their coherence: an artistic impression

As described above, we identified eight themes of mathematics education research for the future, which we discussed one by one. The disadvantage of this list-wise discussion is that the entanglement of the themes is backgrounded. To compensate for that drawback, we here render a brief interpretation of the drawing of Fig. 1 . While doing so, we invite readers to use their own creative imagination and perhaps use the drawing for other purposes (e.g., ask researchers, students, or teachers: Where would you like to be in this landscape? What mathematical ideas do you spot?). The drawing mainly focuses on the themes that emerged from the first round of responses but also hints at experiences from the time of the pandemic, for instance distance education. In Appendix 1 , we specify more of the details in the drawing and we provide a link to an annotated image (available at https://www.fisme.science.uu.nl/toepassingen/28937/ ).

The boat on the river aims to represent teaching approaches. The hand drawing of the boat hints at the importance of educational design: A particular approach is being worked out. On the boat, a teacher and students work together toward educational and societal goals, further down the river. The graduation bridge is an intermediate educational goal to pass, after which there are many paths leading to other goals such as higher education, citizenship, and work in society. Relations to practices outside mathematics education are also shown. In the left bottom corner, the house and parents working and playing with children represent the link of education with the home situation and leisure activity.

The teacher, represented by the captain in the foreground of the ship, is engaged in professional development, consulting a book, but also learning by doing (cf. Bakkenes et al., 2010 , on experimenting, using resources, etc.). Apart from graduation, there are other types of goals for teachers and students alike, such as equity, positive affect, and fluent use of technology. During their journey (and partially at home, shown in the left bottom corner), students learn to orient themselves in the world mathematically (e.g., fractal tree, elliptical lake, a parabolic mountain, and various platonic solids). On their way toward various goals, both teacher and students use particular technology (e.g., compass, binoculars, tablet, laptop). The magnifying glass (representing research) zooms in on a laptop screen that portrays distance education, hinting at the consensus that the pandemic magnifies some issues that education was already facing (e.g., the digital divide).

Equity, diversity, and inclusion are represented with the rainbow, overarching everything. On the boat, students are treated equally and the sailing practice is inclusive in the sense that all perform at their own level—getting the support they need while contributing meaningfully to the shared activity. This is at least what we read into the image. Affect is visible in various ways. First of all, the weather represents moods in general (rainy and dark side on the left; sunny bright side on the right). Second, the individual students (e.g., in the crow’s nest) are interested in, anxious about, and attentive to the things coming up during their journey. They are motivated to engage in all kinds of tasks (handling the sails, playing a game of chance with a die, standing guard in the crow’s nest, etc.). On the bridge, the graduates’ pride and happiness hints at positive affect as an educational goal but also represents the exam part of the assessment. The assessment also happens in terms of checks and feedback on the boat. The two people next to the house (one with a camera, one measuring) can be seen as assessors or researchers observing and evaluating the progress on the ship or the ship’s progress.

More generally, the three types of boats in the drawing represent three different spaces, which Hannah Arendt ( 1958 ) would characterize as private (paper-folded boat near the boy and a small toy boat next to the girl with her father at home), public/political (ships at the horizon), and the in-between space of education (the boat with the teacher and students). The students and teacher on the boat illustrate school as a special pedagogic form. Masschelein and Simons ( 2019 ) argue that the ancient Greek idea behind school (σχολή, scholè , free time) is that students should all be treated as equal and should all get equal opportunities. At school, their descent does not matter. At school, there is time to study, to make mistakes, without having to work for a living. At school, they learn to collaborate with others from diverse backgrounds, in preparation for future life in the public space. One challenge of the lockdown situation as a consequence of the pandemic is how to organize this in-between space in a way that upholds its special pedagogic form.

6 Research challenges

Based on the eight themes and considerations about mathematics education research itself, we formulate a set of research challenges that strike us as deserving further discussion (cf. Stephan et al., 2015 ). We do not intend to suggest these are more important than others or that some other themes are less worthy of investigation, nor do we suggest that they entail a research agenda (cf. English, 2008 ).

6.1 Aligning new goals, curricula, and teaching approaches

There seems to be relatively little attention within mathematics education research for curricular issues, including topics such as learning goals, curriculum standards, syllabi, learning progressions, textbook analysis, curricular coherence, and alignment with other curricula. Yet we feel that we as mathematics education researchers should care about these topics as they may not necessarily be covered by other disciplines. For example, judging from Deng’s ( 2018 ) complaint about the trends in the discipline of curriculum studies, we cannot assume scholars in that field to address issues specific to the mathematics-focused curriculum (e.g., the Journal of Curriculum Studies and Curriculum Inquiry have published only a limited number of studies on mathematics curricula).

Learning goals form an important element of curricula or standards. It is relatively easy to formulate important goals in general terms (e.g., critical thinking or problem solving). As a specific example, consider mathematical problem posing (Cai & Leikin, 2020 ), which curriculum standards have specifically pointed out as an important educational goal—developing students’ problem-posing skills. Students should be provided opportunities to formulate their own problems based on situations. However, there are few problem-posing activities in current mathematics textbooks and classroom instruction (Cai & Jiang, 2017 ). A similar observation can be made about problem solving in Dutch primary textbooks (Kolovou et al., 2009 ). Hence, there is a need for researchers and educators to align problem posing in curriculum standards, textbooks, classroom instruction, and students’ learning.

The challenge we see for mathematics education researchers is to collaborate with scholars from other disciplines (interdisciplinarity) and with non-researchers (transdisciplinarity) in figuring out how the desired societal and educational goals can be shaped in mathematics education. Our discipline has developed several methodological approaches that may help in formulating learning goals and accompanying teaching approaches (cf. Van den Heuvel-Panhuizen, 2005 ), including epistemological analyses (Sierpinska, 1990 ), historical and didactical phenomenology (Bakker & Gravemeijer, 2006 ; Freudenthal, 1986 ), and workplace studies (Bessot & Ridgway, 2000 ; Hoyles et al., 2001 ). However, how should the outcomes of such research approaches be weighed against each other and combined to formulate learning goals for a balanced, coherent curriculum? What is the role of mathematics education researchers in relation to teachers, policymakers, and other stakeholders (Potari et al., 2019 )? In our discipline, we seem to lack a research-informed way of arriving at the formulation of suitable educational goals without overloading the curricula.

6.2 Researching mathematics education across contexts

Though methodologically and theoretically challenging, it is of great importance to study learning and teaching mathematics across contexts. After all, students do not just learn at school; they can also participate in informal settings (Nemirovsky et al., 2017 ), online forums, or affinity networks (Ito et al., 2018 ) where they may share for instance mathematical memes (Bini et al., 2020 ). Moreover, teachers are not the only ones teaching mathematics: Private tutors, friends, parents, siblings, or other relatives can also be involved in helping children with their mathematics. Mathematics learning could also be situated on streets or in museums, homes, and other informal settings. This was already acknowledged before 2020, but the pandemic has scattered learners and teachers away from the typical central school locations and thus shifted the distribution of labor.

In particular, physical and virtual spaces of learning have been reconfigured due to the pandemic. Issues of timing also work differently online, for example, if students can watch online lectures or videos whenever they like (asynchronously). Such reconfigurations of space and time also have an effect on the rhythm of education and hence on people’s energy levels (cf. Lefebvre, 2004 ). More specifically, the reconfiguration of the situation has affected many students’ levels of motivation and concentration (e.g., Meeter et al., 2020 ). As Engelbrecht et al. ( 2020 ) acknowledged, the pandemic has drastically changed the teaching and learning model as we knew it. It is quite possible that some existing theories about teaching and learning no longer apply in the same way. An interesting question is whether and how existing theoretical frameworks can be adjusted or whether new theoretical orientations need to be developed to better understand and promote productive ways of blended or online teaching, across contexts.

6.3 Focusing teacher professional development

Professional development of teachers and teacher educators stands out from the survey as being in need of serious investment. How can teachers be prepared for the unpredictable, both in terms of beliefs and actions? During the pandemic, teachers have been under enormous pressure to make quick decisions in redesigning their courses, to learn to use new technological tools, to invent creative ways of assessment, and to do what was within their capacity to provide opportunities to their students for learning mathematics—even if technological tools were limited (e.g., if students had little or no computer or internet access at home). The pressure required both emotional adaption and instructional adjustment. Teachers quickly needed to find useful information, which raises questions about the accessibility of research insights. Given the new situation, limited resources, and the uncertain unfolding of education after lockdowns, focusing teacher professional development on necessary and useful topics will need much attention. In particular, there is a need for longitudinal studies to investigate how teachers’ learning actually affects teachers’ classroom instruction and students’ learning.

In the surveys, respondents mainly referred to teachers as K-12 school mathematics teachers, but some also stressed the importance of mathematics teacher educators (MTEs). In addition to conducting research in mathematics education, MTEs are acting in both the role of teacher educators and of mathematics teachers. There has been increased research on MTEs as requiring professional development (Goos & Beswick, 2021 ). Within the field of mathematics education, there is an emerging need and interest in how mathematics teacher educators themselves learn and develop. In fact, the changing situation also provides an opportunity to scrutinize our habitual ways of thinking and become aware of what Jullien ( 2018 ) calls the “un-thought”: What is it that we as educators and researchers have not seen or thought about so much about that the sudden reconfiguration of education forces us to reflect upon?

6.4 Using low-tech resources

Particular strands of research focus on innovative tools and their applications in education, even if they are at the time too expensive (even too labor intensive) to use at large scale. Such future-oriented studies can be very interesting given the rapid advances in technology and attractive to funding bodies focusing on innovation. Digital technology has become ubiquitous, both in schools and in everyday life, and there is already a significant body of work capitalizing on aspects of technology for research and practice in mathematics education.

However, as Cai et al. ( 2020 ) indicated, technology advances so quickly that addressing research problems may not depend so much on developing a new technological capability as on helping researchers and practitioners learn about new technologies and imagine effective ways to use them. Moreover, given the millions of students in rural areas who during the pandemic have only had access to low-tech resources such as podcasts, radio, TV, and perhaps WhatsApp through their parents’ phones, we would like to see more research on what learning, teaching, and assessing mathematics through limited tools such as Whatsapp or WeChat look like and how they can be improved. In fact, in China, a series of WeChat-based mini-lessons has been developed and delivered through the WeChat video function during the pandemic. Even when the pandemic is under control, mini-lessons are still developed and circulated through WeChat. We therefore think it is important to study the use and influence of low-tech resources in mathematics education.

6.5 Staying in touch online

With the majority of students learning at home, a major ongoing challenge for everyone has been how to stay in touch with each other and with mathematics. With less social interaction, without joint attention in the same physical space and at the same time, and with the collective only mediated by technology, becoming and staying motivated to learn has been a widely felt challenge. It is generally expected that in the higher levels of education, more blended or distant learning elements will be built into education. Careful research on the affective, embodied, and collective aspects of learning and teaching mathematics is required to overcome eventually the distance and alienation so widely experienced in online education. That is, we not only need to rethink social interactions between students and/or teachers in different settings but must also rethink how to engage and motivate students in online settings.

6.6 Studying and improving equity without perpetuating inequality

Several colleagues have warned, for a long time, that one risk of studying achievement gaps, differences between majority and minority groups, and so forth can also perpetuate inequity. Admittedly, pinpointing injustice and the need to invest in particular less privileged parts of education is necessary to redirect policymakers’ and teachers’ attention and gain funding. However, how can one reorient resources without stigmatizing? For example, Svensson et al. ( 2014 ) pointed out that research findings can fuel political debates about groups of people (e.g., parents with a migration background), who then may feel insecure about their own capacities. A challenge that we see is to identify and understand problematic situations without legitimizing problematic stereotyping (Hilt, 2015 ).

Furthermore, the field of mathematics education research does not have a consistent conceptualization of equity. There also seem to be regional differences: It struck us that equity is the more common term in the responses from the Americas, whereas inclusion and diversity were more often mentioned in the European responses. Future research will need to focus on both the conceptualization of equity and on improving equity and related values such as inclusion.

6.7 Assessing online

A key challenge is how to assess online and to do so more effectively. This challenge is related to both privacy, ethics, and performance issues. It is clear that online assessment may have significant advantages to assess student mathematics learning, such as more flexibility in test-taking and fast scoring. However, many teachers have faced privacy concerns, and we also have the impression that in an online environment it is even more challenging to successfully assess what we value rather than merely assessing what is relatively easy to assess. In particular, we need to systematically investigate any possible effect of administering assessments online as researchers have found a differential effect of online assessment versus paper-and-pencil assessment (Backes & Cowan, 2019 ). What further deserves careful ethical attention is what happens to learning analytics data that can and are collected when students work online.

6.8 Doing and publishing interdisciplinary research

When analyzing the responses, we were struck by a discrepancy between what respondents care about and what is typically researched and published in our monodisciplinary journals. Most of the challenges mentioned in this section require interdisciplinary or even transdisciplinary approaches (see also Burkhardt, 2019 ).

An overarching key question is: What role does mathematics education research play in addressing the bigger and more general challenges mentioned by our respondents? The importance of interdisciplinarity also raises a question about the scope of journals that focus on mathematics education research. Do we need to broaden the scope of monodisciplinary journals so that they can publish important research that combines mathematics education research with another disciplinary perspective? As editors, we see a place for interdisciplinary studies as long as there is one strong anchor in mathematics education research. In fact, there are many researchers who do not identify themselves as mathematics education researchers but who are currently doing high-quality work related to mathematics education in fields such as educational psychology and the cognitive and learning sciences. Encouraging the reporting of high-quality mathematics education research from a broader spectrum of researchers would serve to increase the impact of the mathematics education research journals in the wider educational arena. This, in turn, would serve to encourage further collaboration around mathematics education issues from various disciplines. Ultimately, mathematics education research journals could act as a hub for interdisciplinary collaboration to address the pressing questions of how mathematics is learned and taught.

7 Concluding remarks

In this paper, based on a survey conducted before and during the pandemic, we have examined how scholars in the field of mathematics education view the future of mathematics education research. On the one hand, there are no major surprises about the areas we need to focus on in the future; the themes are not new. On the other hand, the responses also show that the areas we have highlighted still persist and need further investigation (cf. OECD, 2020 ). But, there are a few areas, based on both the responses of the scholars and our own discussions and views, that stand out as requiring more attention. For example, we hope that these survey results will serve as propelling conversation about mathematics education research regarding online assessment and pedagogical considerations for virtual teaching.

The survey results are limited in two ways. The set of respondents to the survey is probably not representative of all mathematics education researchers in the world. In that regard, perhaps scholars in each country could use the same survey questions to survey representative samples within each country to understand how the scholars in that country view future research with respect to regional needs. The second limitation is related to the fact that mathematics education is a very culturally dependent field. Cultural differences in the teaching and learning of mathematics are well documented. Given the small numbers of responses from some continents, we did not break down the analysis for regional comparison. Representative samples from each country would help us see how scholars from different countries view research in mathematics education; they will add another layer of insights about mathematics education research to complement the results of the survey presented here. Nevertheless, we sincerely hope that the findings from the surveys will serve as a discussion point for the field of mathematics education to pursue continuous improvement.

Akkerman, S. F., & Bakker, A. (2011). Boundary crossing and boundary objects. Review of Educational Research , 81 (2), 132–169. https://doi.org/10.3102/0034654311404435

Article   Google Scholar  

Arendt, H. (1958/1998). The human condition (2nd ed.). University of Chicago Press.

Backes, B., & Cowan, J. (2019). Is the pen mightier than the keyboard? The effect of online testing on measured student achievement. Economics of Education Review , 68 , 89–103. https://doi.org/10.1016/j.econedurev.2018.12.007

Bakkenes, I., Vermunt, J. D., & Wubbels, T. (2010). Teacher learning in the context of educational innovation: Learning activities and learning outcomes of experienced teachers. Learning and Instruction , 20 (6), 533–548. https://doi.org/10.1016/j.learninstruc.2009.09.001

Bakker, A. (2019). What is worth publishing? A response to Niss. For the Learning of Mathematics , 39 (3), 43–45.

Google Scholar  

Bakker, A., & Gravemeijer, K. P. (2006). An historical phenomenology of mean and median. Educational Studies in Mathematics , 62 (2), 149–168. https://doi.org/10.1007/s10649-006-7099-8

Bakx, A., Bakker, A., Koopman, M., & Beijaard, D. (2016). Boundary crossing by science teacher researchers in a PhD program. Teaching and Teacher Education , 60 , 76–87. https://doi.org/10.1016/j.tate.2016.08.003

Battey, D. (2013). Access to mathematics: “A possessive investment in whiteness”. Curriculum Inquiry , 43 (3), 332–359.

Bawa, P. (2020). Learning in the age of SARS-COV-2: A quantitative study of learners’ performance in the age of emergency remote teaching. Computers and Education Open , 1 , 100016. https://doi.org/10.1016/j.caeo.2020.100016

Beckers, D., & Beckers, A. (2019). ‘Newton was heel exact wetenschappelijk – ook in zijn chemische werk’. Nederlandse wetenschapsgeschiedenis in niet-wetenschapshistorische tijdschriften, 1977–2017. Studium , 12 (4), 185–197. https://doi.org/10.18352/studium.10203

Bessot, A., & Ridgway, J. (Eds.). (2000). Education for mathematics in the workplace . Springer.

Bickerton, R. T., & Sangwin, C. (2020). Practical online assessment of mathematical proof. arXiv preprint:2006.01581 . https://arxiv.org/pdf/2006.01581.pdf .

Bikner-Ahsbahs, A., & Prediger, S. (Eds.). (2014). Networking of theories as a research practice in mathematics education . Springer.

Bini, G., Robutti, O., & Bikner-Ahsbahs, A. (2020). Maths in the time of social media: Conceptualizing the Internet phenomenon of mathematical memes. International Journal of Mathematical Education in Science and Technology , 1–40. https://doi.org/10.1080/0020739x.2020.1807069

Bosch, M., Dreyfus, T., Primi, C., & Shiel, G. (2017, February). Solid findings in mathematics education: What are they and what are they good for? CERME 10 . Ireland: Dublin https://hal.archives-ouvertes.fr/hal-01849607

Bowker, G. C., & Star, S. L. (2000). Sorting things out: Classification and its consequences . MIT Press. https://doi.org/10.7551/mitpress/6352.001.0001

Burkhardt, H. (2019). Improving policy and practice. Educational Designer , 3 (12) http://www.educationaldesigner.org/ed/volume3/issue12/article46/

Cai, J., & Hwang, S. (2019). Constructing and employing theoretical frameworks in (mathematics) education research. For the Learning of Mathematics , 39 (3), 44–47.

Cai, J., & Jiang, C. (2017). An analysis of problem-posing tasks in Chinese and U.S. elementary mathematics textbooks. International Journal of Science and Mathematics Education , 15 (8), 1521–1540. https://doi.org/10.1007/s10763-016-9758-2

Cai, J., & Leikin, R. (2020). Affect in mathematical problem posing: Conceptualization, advances, and future directions for research. Educational Studies in Mathematics , 105 , 287–301. https://doi.org/10.1007/s10649-020-10008-x

Cai, J., Morris, A., Hohensee, C., Hwang, S., Robison, V., Cirillo, M., … Hiebert, J. (2020). Improving the impact of research on practice: Capitalizing on technological advances for research. Journal for Research in Mathematics Education , 51 (5), 518–529 https://pubs.nctm.org/view/journals/jrme/51/5/article-p518.xml

Chronaki, A. (2019). Affective bodying of mathematics, children and difference: Choreographing ‘sad affects’ as affirmative politics in early mathematics teacher education. ZDM-Mathematics Education , 51 (2), 319–330. https://doi.org/10.1007/s11858-019-01045-9

Civil, M., & Bernier, E. (2006). Exploring images of parental participation in mathematics education: Challenges and possibilities. Mathematical Thinking and Learning , 8 (3), 309–330. https://doi.org/10.1207/s15327833mtl0803_6

Cobb, P., Gresalfi, M., & Hodge, L. L. (2009). An interpretive scheme for analyzing the identities that students develop in mathematics classrooms. Journal for Research in Mathematics Education , 40 ( 1 ), 40–68 https://pubs.nctm.org/view/journals/jrme/40/1/article-p40.xml

Darragh, L. (2016). Identity research in mathematics education. Educational Studies in Mathematics , 93 (1), 19–33. https://doi.org/10.1007/s10649-016-9696-5

de Abreu, G., Bishop, A., & Presmeg, N. C. (Eds.). (2006). Transitions between contexts of mathematical practices . Kluwer.

de Freitas, E., Ferrara, F., & Ferrari, G. (2019). The coordinated movements of collaborative mathematical tasks: The role of affect in transindividual sympathy. ZDM-Mathematics Education , 51 (2), 305–318. https://doi.org/10.1007/s11858-018-1007-4

Deng, Z. (2018). Contemporary curriculum theorizing: Crisis and resolution. Journal of Curriculum Studies , 50 (6), 691–710. https://doi.org/10.1080/00220272.2018.1537376

Dobie, T. E., & Sherin, B. (2021). The language of mathematics teaching: A text mining approach to explore the zeitgeist of US mathematics education. Educational Studies in Mathematics .  https://doi.org/10.1007/s10649-020-10019-8

Eames, C., & Eames, R. (1977). Powers of Ten [Film]. YouTube. https://www.youtube.com/watch?v=0fKBhvDjuy0

Engelbrecht, J., Borba, M. C., Llinares, S., & Kaiser, G. (2020). Will 2020 be remembered as the year in which education was changed? ZDM-Mathematics Education , 52 (5), 821–824. https://doi.org/10.1007/s11858-020-01185-3

English, L. (2008). Setting an agenda for international research in mathematics education. In L. D. English (Ed.), Handbook of international research in mathematics education (2nd ed., pp. 3–19). Routledge.

Ernest, P. (2020). Unpicking the meaning of the deceptive mathematics behind the COVID alert levels. Philosophy of Mathematics Education Journal , 36 http://socialsciences.exeter.ac.uk/education/research/centres/stem/publications/pmej/pome36/index.html

Freudenthal, H. (1986). Didactical phenomenology of mathematical structures . Springer.

Gilmore, C., Göbel, S. M., & Inglis, M. (2018). An introduction to mathematical cognition . Routledge.

Goos, M., & Beswick, K. (Eds.). (2021). The learning and development of mathematics teacher educators: International perspectives and challenges . Springer. https://doi.org/10.1007/978-3-030-62408-8

Gorard, S. (Ed.). (2020). Getting evidence into education. Evaluating the routes to policy and practice . Routledge.

Gravemeijer, K., Stephan, M., Julie, C., Lin, F.-L., & Ohtani, M. (2017). What mathematics education may prepare students for the society of the future? International Journal of Science and Mathematics Education , 15 (1), 105–123. https://doi.org/10.1007/s10763-017-9814-6

Hannula, M. S. (2019). Young learners’ mathematics-related affect: A commentary on concepts, methods, and developmental trends. Educational Studies in Mathematics , 100 (3), 309–316. https://doi.org/10.1007/s10649-018-9865-9

Hilt, L. T. (2015). Included as excluded and excluded as included: Minority language pupils in Norwegian inclusion policy. International Journal of Inclusive Education , 19 (2), 165–182.

Hodgen, J., Taylor, B., Jacques, L., Tereshchenko, A., Kwok, R., & Cockerill, M. (2020). Remote mathematics teaching during COVID-19: Intentions, practices and equity . UCL Institute of Education https://discovery.ucl.ac.uk/id/eprint/10110311/

Horn, I. S. (2017). Motivated: Designing math classrooms where students want to join in . Heinemann.

Hoyles, C., Noss, R., & Pozzi, S. (2001). Proportional reasoning in nursing practice. Journal for Research in Mathematics Education , 32 (1), 4–27. https://doi.org/10.2307/749619

Ito, M., Martin, C., Pfister, R. C., Rafalow, M. H., Salen, K., & Wortman, A. (2018). Affinity online: How connection and shared interest fuel learning . NYU Press.

Jackson, K. (2011). Approaching participation in school-based mathematics as a cross-setting phenomenon. The Journal of the Learning Sciences , 20 (1), 111–150. https://doi.org/10.1080/10508406.2011.528319

Jansen, A., Herbel-Eisenmann, B., & Smith III, J. P. (2012). Detecting students’ experiences of discontinuities between middle school and high school mathematics programs: Learning during boundary crossing. Mathematical Thinking and Learning , 14 (4), 285–309. https://doi.org/10.1080/10986065.2012.717379

Johnson, L. F., Smith, R. S., Smythe, J. T., & Varon, R. K. (2009). Challenge-based learning: An approach for our time (pp. 1–38). The New Media Consortium https://www.learntechlib.org/p/182083

Jullien, F. (2018). Living off landscape: Or the unthought-of in reason . Rowman & Littlefield.

Kazima, M. (2019). What is proven to work in successful countries should be implemented in other countries: The case of Malawi and Zambia. In M. Graven, H. Venkat, A. A. Essien, & P. Vale (Eds.), Proceedings of the 43rd conference of the international group for the Psychology of Mathematics Education (Vol. 1, pp. 73–78). PME.

Kim, H. (2019). Ask again, “why should we implement what works in successful countries?” In M. Graven, H. Venkat, A. A. Essien, & P. Vale (Eds.), Proceedings of the 43rd conference of the international group for the Psychology of Mathematics Education (Vol. 1, pp. 79–82). PME.

Kolovou, A., Van Den Heuvel-Panhuizen, M., & Bakker, A. (2009). Non-routine problem solving tasks in primary school mathematics textbooks—a needle in a haystack. Mediterranean Journal for Research in Mathematics Education , 8 (2), 29–66.

Kwon, O. N., Han, C., Lee, C., Lee, K., Kim, K., Jo, G., & Yoon, G. (2021). Graphs in the COVID-19 news: A mathematics audit of newspapers in Korea. Educational Studies in Mathematics . https://doi.org/10.1007/s10649-021-10029-0

Lefebvre, H. (2004). Rhythmanalysis: Space, time and everyday life (Original 1992; Translation by S. Elden & G. Moore) . Bloomsbury Academic. https://doi.org/10.5040/9781472547385 .

Li, Y. (2019). Should what works in successful countries be implemented in other countries? In M. Graven, H. Venkat, A. A. Essien, & P. Vale (Eds.), Proceedings of the 43rd conference of the international group for the Psychology of Mathematics Education (Vol. 1, pp. 67–72). PME.

Martin, D., Gholson, M., & Leonard, J. (2010). Mathematics as gatekeeper: Power and privilege in the production of power. Journal of Urban Mathematics Education , 3 (2), 12–24.

Masschelein, J., & Simons, M. (2019). Bringing more ‘school’ into our educational institutions. Reclaiming school as pedagogic form. In A. Bikner-Ahsbahs & M. Peters (Eds.), Unterrichtsentwicklung macht Schule (pp. 11–26) . Springer. https://doi.org/10.1007/978-3-658-20487-7_2

Meeter, M., Bele, T., den Hartogh, C., Bakker, T., de Vries, R. E., & Plak, S. (2020). College students’ motivation and study results after COVID-19 stay-at-home orders. https://psyarxiv.com .

Nemirovsky, R., Kelton, M. L., & Civil, M. (2017). Toward a vibrant and socially significant informal mathematics education. In J. Cai (Ed.), Compendium for Research in Mathematics Education (pp. 968–979). National Council of Teachers of Mathematics.

Niss, M. (2019). The very multi-faceted nature of mathematics education research. For the Learning of Mathematics , 39 (2), 2–7.

OECD. (2020). Back to the Future of Education: Four OECD Scenarios for Schooling. Educational Research and Innovation . OECD Publishing. https://doi.org/10.1787/20769679

Potari, D., Psycharis, G., Sakonidis, C., & Zachariades, T. (2019). Collaborative design of a reform-oriented mathematics curriculum: Contradictions and boundaries across teaching, research, and policy. Educational Studies in Mathematics , 102 (3), 417–434. https://doi.org/10.1007/s10649-018-9834-3

Proulx, J., & Maheux, J. F. (2019). Effect sizes, epistemological issues, and identity of mathematics education research: A commentary on editorial 102(1). Educational Studies in Mathematics , 102 (2), 299–302. https://doi.org/10.1007/s10649-019-09913-7

Roos, H. (2019). Inclusion in mathematics education: An ideology, A way of teaching, or both? Educational Studies in Mathematics , 100 (1), 25–41. https://doi.org/10.1007/s10649-018-9854-z

Saenz, M., Medina, A., & Urbine Holguin, B. (2020). Colombia: La prender al onda (to turn on the wave). Education continuity stories series . OECD Publishing https://oecdedutoday.com/wp-content/uploads/2020/12/Colombia-a-prender-la-onda.pdf

Schindler, M., & Bakker, A. (2020). Affective field during collaborative problem posing and problem solving: A case study. Educational Studies in Mathematics , 105 , 303–324. https://doi.org/10.1007/s10649-020-09973-0

Schoenfeld, A. H. (1999). Looking toward the 21st century: Challenges of educational theory and practice. Educational Researcher , 28 (7), 4–14. https://doi.org/10.3102/0013189x028007004

Schukajlow, S., Rakoczy, K., & Pekrun, R. (2017). Emotions and motivation in mathematics education: Theoretical considerations and empirical contributions. ZDM-Mathematics Education , 49 (3), 307–322. https://doi.org/10.1007/s11858-017-0864-6

Sfard, A. (2005). What could be more practical than good research? Educational Studies in Mathematics , 58 (3), 393–413. https://doi.org/10.1007/s10649-005-4818-5

Shimizu, Y., & Vithal, R. (Eds.). (2019). ICMI Study 24 Conference Proceedings. School mathematics curriculum reforms: Challenges, changes and opportunities . ICMI: University of Tsukuba & ICMI http://www.human.tsukuba.ac.jp/~icmi24/

Sierpinska, A. (1990). Some remarks on understanding in mathematics. For the Learning of Mathematics , 10 (3), 24–41.

Stephan, M. L., Chval, K. B., Wanko, J. J., Civil, M., Fish, M. C., Herbel-Eisenmann, B., … Wilkerson, T. L. (2015). Grand challenges and opportunities in mathematics education research. Journal for Research in Mathematics Education , 46 (2), 134–146. https://doi.org/10.5951/jresematheduc.46.2.0134

Suazo-Flores, E., Alyami, H., Walker, W. S., Aqazade, M., & Kastberg, S. E. (2021). A call for exploring mathematics education researchers’ interdisciplinary research practices. Mathematics Education Research Journal , 1–10. https://doi.org/10.1007/s13394-021-00371-0

Svensson, P., Meaney, T., & Norén, E. (2014). Immigrant students’ perceptions of their possibilities to learn mathematics: The case of homework. For the Learning of Mathematics , 34 (3), 32–37.

UNESCO. (2015). Teacher policy development guide . UNESCO, International Task Force on Teachers for Education 2030. https://teachertaskforce.org/sites/default/files/2020-09/370966eng_0_1.pdf .

Van den Heuvel-Panhuizen, M. (2005). Can scientific research answer the ‘what’ question of mathematics education? Cambridge Journal of Education , 35 (1), 35–53. https://doi.org/10.1080/0305764042000332489

Wittmann, E. C. (1995). Mathematics education as a ‘design science’. Educational Studies in Mathematics , 29 (4), 355–374.

Yoon, H., Byerley, C. O. N., Joshua, S., Moore, K., Park, M. S., Musgrave, S., Valaas, L., & Drimalla, J. (2021). United States and South Korean citizens’ interpretation and assessment of COVID-19 quantitative data. The Journal of Mathematical Behavior . https://doi.org/10.1016/j.jmathb.2021.100865 .

Download references

Acknowledgments

We thank Anna Sfard for her advice on the survey, based on her own survey published in Sfard ( 2005 ). We are grateful for Stephen Hwang’s careful copyediting for an earlier version of the manuscript. Thanks also to Elisabeth Angerer, Elske de Waal, Paul Ernest, Vilma Mesa, Michelle Stephan, David Wagner, and anonymous reviewers for their feedback on earlier drafts.

Author information

Authors and affiliations.

Utrecht University, Utrecht, Netherlands

Arthur Bakker & Linda Zenger

University of Delaware, Newark, DE, USA

You can also search for this author in PubMed   Google Scholar

Corresponding author

Correspondence to Arthur Bakker .

Ethics declarations

In line with the guidelines of the Code of Publication Ethics (COPE), we note that the review process of this article was blinded to the authors.

Additional information

Publisher’s note.

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix 1: Explanation of Fig. 1

We have divided Fig. 1 in 12 rectangles called A1 (bottom left) up to C4 (top right) to explain the details (for image annotation go to https://www.fisme.science.uu.nl/toepassingen/28937 )

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/ .

Reprints and permissions

About this article

Bakker, A., Cai, J. & Zenger, L. Future themes of mathematics education research: an international survey before and during the pandemic. Educ Stud Math 107 , 1–24 (2021). https://doi.org/10.1007/s10649-021-10049-w

Download citation

Accepted : 04 March 2021

Published : 06 April 2021

Issue Date : May 2021

DOI : https://doi.org/10.1007/s10649-021-10049-w

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

• Grand challenges
• Mathematics education research
• Research agenda
• Find a journal
• Publish with us
• Track your research

Recent Master's Theses - Applied Mathematics

2007 | 2008 | 2009 | 2010 | 2011 | 2012 | 2013 | 2014 | 2015  | 2016  | 2017  | 2018  | 2019  | 2022  | 2023  | 2024

Master's Theses 2024

Master's Theses 2023

Master's Theses 2022

Master's Theses 2019

Master's Theses 2018

Master's Theses 2017

Master's Theses 2016

Master's Theses 2015

Master's Theses 2014

Master's Theses 2013

Master's Theses 2012

Master's Theses 2011

Master's Theses 2010

Master's Theses 2009

Master's Theses 2008

Master's Theses 2007

• Departments
• Giving to SSE

Information for

Sse math dept., account login.

• GIBSON ONLINE

Graduate Studies in Mathematics

Our department offers Masters degrees in Mathematics, Applied Mathematics, and Statistics as well as a Ph.D. Degree in Mathematics, which can have an emphasis in any of the three areas mentioned. The Masters degrees emphasize different aspects of theory and applications in order to prepare the students for either continuing studies at the Ph.D. Level or employment. The Ph.D. Program prepares the students for a career in research at a university, a government agency or in industry. Our faculty members are actively engaged in research and mentoring so that doctoral students can choose a faculty advisor according to the students’ interests.

Tulane is a privately endowed university located in New Orleans, Louisiana. At present it has an enrollment of about 10,000 students from almost every state and from 60 foreign countries.

The Mathematics program has, if anything, been strengthened by the reorganization of Tulane University in the aftermath of Hurricane Katrina.

Tulane's mathematical tradition can be traced back to the late nineteenth century, when Florian Cajori, later an expert in the history of mathematical notation, and the most famous translator of Isaac Newton's Principia, was the recipient of the first doctorate in mathematics from Tulane University (1894). Four undergraduates from the period up to the end of World War II (as well as Cajori) later became presidents of the Mathematical Association of America (Duren, McShane, Moise, Young); one (McShane) was a member of the National Academy of Sciences. In the 1950's Tulane became a major center in mathematical research. From 1970 to 2021, 207 Ph.D.'s were awarded.

The mathematics life at Tulane is enlivened by the distinguished mathematicians who visit each year for long or short periods, and by the international symposia which are held here from time to time. The department holds the annual Clifford Lectures, a week-long series of talks by a distinguished mathematician. A mini-conference supported by the National Science Foundation is held in conjunction with each of the Clifford lecture series. The first Clifford lecturer, in 1984, was Fields Medal recipient Charles Fefferman of Princeton University. In 1985 and 1986, the Clifford lecturers were Fields Medal winners, S. T. Yau of UC San Diego and William Thurston of Princeton University. The Clifford lecturers from 1987 through 1990 were Saharon Shelah of Hebrew University in Jerusalem, Clifford Taubes of Harvard University, Charles Peskin of Courant Institute and Haim Brezis of Université de Paris and Rutgers. From 1991 to 1996 they were Sylvain Cappell of Courant Institute of Mathematical Sciences, Nigel Hitchin of the University of Warwick and Persi Diaconis of Harvard University, Peter Sarnak of Princeton University and Dan Voiculescu of UC Berkeley. In 1994, a special conference on semigroups was held in honor of Alfred H. Clifford. In 1997 the Clifford lecturers were Paul Fife of University of Utah (Spring) and Peter Kronheimer of Harvard University (Fall). The speakers from 1998 to the present were Peter Bickel and Alexander Chorin of UC Berkeley, Robert Friedman of Columbia University, Sergei N. Artemov of City University of New York, T. J. Pedley of Cambridge University and Yakov Eliashberg of Stanford University.

The Mathematics Department at Tulane University offers a Ph.D. degree in Mathematics as well as Master of Science degrees in Mathematics, Applied Mathematics and in Statistics. These programs are described below. Undergraduate students majoring in mathematics or other sciences (like engineering, physics or computer science) with a strong interest in mathematics are encouraged to apply for admission to any one of the graduate programs. People who already hold undergraduate degrees in mathematics or other sciences are also encouraged to apply.

Requirements for admission into the Tulane Graduate School include:

• The GRE Test is not required as part of the graduate application for any program.

The way to apply is to fill out and submit a Web-based application form.

Web-based application form

If you have any problems receiving the application, you may inquire at:

Director of Graduate Studies

Mathematics Department Tulane University 6823 St. Charles Ave. New Orleans, LA 70118 phone: (504) 865-5727 fax: (504) 865-5063 [email protected]

Most graduate students receive tuition waivers and teaching assistantships , which carry a stipend of $26,500 approximately. Teaching Assistants typically teach two laboratories (each meets weekly), although more advanced students may teach one section of an undergraduate course. All Ph.D. students are required to teach an undergraduate course, or to serve as teaching assistants in problem sessions attached to undergraduate courses, for at least two semesters during their residence.

Ph.D. students (U.S. Citizens or Legally Permanent Residents) are strongly recommended to apply the external funding resources such as NSF GRFP (Graduate Research Fellowship Program). More details on here https://www.nsfgrfp.org . For non-U.S. Citizens, there is a bountiful funding resources from several fields. Please find the information on https://immigrantsrising.org .

The Tulane Mathematics Department is known for its friendly atmosphere and its practice of fostering close contact and cordial relations between faculty and graduate students. To us, this is a very important aspect of life here, and we strive to maintain it. The ratio of graduate students to faculty members is kept between 1.0 and 1.5. This is important to us because it allows all new graduate students to soon become familiar with everyone and feel at home. It also allows the faculty to get to know the students during their first semester.

The incoming graduate students are advised by the Director of Graduate Studies of the Mathematics Department. The Director, in consultation with the students, determines appropriate first-year courses for each student, according to their preparation and interests. Throughout the program, the Director of Graduate Studies continues to help the students plan their studies and realize their mathematical interests.

All graduate (both Master and Ph.D.) students are given a cubicle in either 6 rooms at Gibson or Stanley Thomas 4th floor equipped with desks, desktop, bookcases and chalkboards. Students also have unlimited access to the lounge and the Mathematics Library, all located within the 4th floor at Gibson Hall. The lounge, or "commons room", is a place where people gather after seminars and colloquia for refreshments and discussion.

Ph.D. in Mathematics

Completing this degree takes about 5 years, depending on the student's preparation and progress satisfying the requirements. For advanced incoming students, limited transfer credit is possible. The Ph.D. prepares the students for a research career in mathematics in industry or academia.

Mathematicians with a Ph.D. from Tulane University have been successful getting jobs in a variety of colleges, research universities, government research laboratories and industries, including biotechnology, e-commerce and financial institutions.

More details on Ph.D. Requirements

Master of Science in Mathematics

This program is designed to provide students with the opportunity to broaden and deepen their knowledge of core areas of mathematics. The course work is designed to provide both breadth of knowledge and depth in an area of interest to the student. This experience will prepare the student for further studies leading to a Ph.D. degree in mathematics.

More details on MS in Mathematics

Master of Science in Applied Mathematics

This program is designed to provide students with the opportunity to broaden and deepen their knowledge of mathematics with an emphasis on those areas that have been most important in science and engineering. Students will also examine, through seminars and case studies, examples of significant applications of mathematics to other areas. This expanded base of knowledge, together with extensive experience in problem solving, is excellent preparation for further studies leading to the Ph.D. degree or for immediate employment in many areas of industry and government.

The program is open to students who have a Bachelor's degree in mathematics or a related field, and have completed undergraduate courses equivalent to Linear Algebra, Numerical Methods, and Analytical Methods. Proficiency in a programming language is essential. Students who have not completed all of these courses may be admitted and are required to take them during the first year.

More details on MS in Applied Mathematics

Master of Science in Statistics

The Master of Science degree in Statistics combines theory and application. Students in statistics will be trained in data collection, the editing and presentation of large data sets, the analyses of these sets and the mathematical foundations upon which all of these areas are based. The training has the two-fold purpose of preparing the student to enter commercial, governmental and other work areas which extensively rely on statistical information and to prepare the student to continue in pursuit of a more advanced degree. Students with appropriate background (three semesters of Calculus and some knowledge of elementary statistics) usually complete the program in one or two academic years.

More details on MS in Statistics

Master of Data Science

The M.S. in Data Science (MSDS) program is a professional, non-thesis degree that is jointly offered by the Mathematics and Computer Science departments. The burst of data in the modern world has fundamentally changed many fields of human activity, including healthcare, energy, manufacturing and scientific research. It has also generated an ever-increasing demand for a new type of professional: the data scientist. The MSDS program aims at providing the next generation of practitioners with cutting-edge data-driven problem-solving skills. These are based on rigorous mathematical foundations, and include data management, advanced statistical modeling, as well as the practical implementation and use of state-of-the-art algorithms.

More details on M.S. in Data Science

Math 6030: Stochastic Processes

Math 6050-6060: Real Analysis I & II

Math 6070: Introduction to Probability

Math 6080: Introduction to Statistical Inference

Math 6090: Linear Algebra

Math 6110-6120: Abstract Algebra I & II

Math 6210: Differential Geometry

Math 6240: Ordinary Differential Equations

Math 6250: Mathematical Foundation of Computer Security

Math 6280: Information Theory

Math 6300: Complex Analysis

Math 6310: Scientific Computing

Math 6350: Numerical Optimization

Math 6370: Time Series Analysis

Math 6470: Analytic Methods of Applied Mathematics

Math 7010-7020: Topology I & II

Math7150: Probability Theory I

Math 7110-7120: Algebra I & II

Math 7210-7220: Analysis I & II

Math 7240: Mathematical Statistics

Math 7260: Linear Models

Math 7291-7292: Algebraic Geometry I & II

Math 7310-7320: Applied Math I & II

Math 7360: Data Analysis

Math 7510-7520: Differential Geometry I& II

Math 7530-7540: Partial Differential Equations I & II

Math 7550: Probability Theory II

Math 7560: Stochastic Processes II

Math 7570-7580: Scientific Computation II & III

Math 7710-7790: Special Topics Courses

More information on the courses

The Mathematics Department consists of 24 regular faculty members, several postdoctoral researchers and frequent visiting faculty in many areas of mathematics.

Its faculty enjoys national and international recognition in Algebra, Analysis, Differential Geometry, Mathematical Physics, Probability and Statistics, Scientific Computation, Theoretical Computer Science, and Topology. The researchers in Scientific Computation and in Statistics, and an increasing number of faculty in other areas, collaborate actively with colleagues in other units of the university such as the Schools of Engineering, Liberal Arts and Sciences, Medicine, and Public Health.

During the past five years our regular faculty have published over 100 research articles and several books. The regular faculty direct theses in very diverse areas which range through all of Pure Mathematics, Applied Mathematics, and Statistics. Detailed information can be found on the faculty page.

The Mathematics Department is housed in the upper floors of Gibson Hall, a stone structure built in 1894. Here are located faculty, graduate students, and staff offices, as well as classrooms, seminar rooms and computers linked to Tulane's main computing system. The department also contains the A. H. Clifford Mathematics Research Library, housing some 28,000 bound volumes and subscribing to 243 journals devoted to all areas of mathematics.

The department has a Microsoft Network with Windows and Mac workstations. The Math Department is connected by a cloud based network and the ability to store your files on Box (a cloud base storage).  Graduate students are provided with adequate computing resources, Ethernet connections, Matlab, Mathematica, Microsoft office and other essential programs that are aimed to help aid in your success.

Tulane University is located in America's most exciting and most visited city. Our department is on St. Charles Avenue, across from Audubon Park, in a quiet residential area full of majestic oak trees and fine old antebellum homes. Often-photographed streetcars provide an easy ride to the picturesque French Quarter. New Orleans has a rich cultural life, with a symphony orchestra, operas, ballets, plays, a noted art museum, many art galleries, excellent jazz, a major jazz festival and many other events. During Mardi Gras (40 days before Easter) the town fills with parades and revelry. New Orleans is also famous for its cuisine; it boasts a number of great restaurants, and many more with good inexpensive meals.

Remember Me

• Forgot your password?
• Forgot your username?
• Create an account
• Math & Stat Office Automation
• Class and Seminar Room Booking
• New Core Lab (Classroom Booking)
• Office Automation Portal
• OARS Port 6060
• OARS Port 4040
• OARS Port 2020
• Departmental Committees
• Annual Reports
• Plan Your Visit
• Faculty Opening
• Post Doc Opening

Intranet | Webmail | Forms | IITK Facility

• Regular Faculty
• Visiting Faculty
• Inspire faculty
• Former Faculty
• Former Heads
• Ph.D. Students
• BS (Stat. & Data Sc.)
• BS-MS (Stat. & Data Sc.)
• Double Major (Stat. & Data Sc.)
• BS (Math. & Sc. Comp.)
• BS-MS (Math. & Sc. Comp.)
• Double Major (Math. & Sc. Comp.)
• BSH (Math & Sc. Comp.)
• BSM (Math & Sc. Comp.)
• Minor in Stat. & Data Sc.
• M.Sc. Mathematics (2 Year)
• M.Sc. Statistics (2 Year)
• Mathematics & Scientific Computing
• Statistics and Data Science
• Project Guidelines
• BS(SDS) Internship Courses Guidelines
• Courses in the Summer Term
• Analysis, Topology, Geometry
• Algebra, Number Theory and Mathematical Logic
• Numerical Analysis & Scientific Computing, ODE, PDE, Fluid Mechanics
• Probability , Statistics
• Research Areas in Mathematics and Statistics
• Publication
• UG/ PG Admission
• PhD Admission
• Financial Aid
• International Students
• Research scholars day
• National Mathematics Day
• Prof. U B Tewari Distinguished Lecture Series
• Conferences
• Student Seminars

Here are the areas of Mathematics in which research is being done currently. + - Algebraic geometry Click to collapse Geometric Invariant Theory : Faculty: Santosha Pattanayak The geometry of algebraic varieties: : The geometry of algebraic varieties with reductive group actions, including flag varieties, toric varieties, torus actions on algebraic varieties, and spherical varieties. I am also interested in studying the structure of algebraic groups and GIT (Geometric Invariant Theory) quotients of algebraic varieties and exploring toric degenerations of algebraic varieties. Moreover, my interest extends to areas such as the Seshadri constants of algebraic varieties and the symplectic invariants of smooth projective algebraic varieties, including the Gromov width. Faculty: Narasimha Chary Bonala + - Commutative Algebra Click to collapse The research areas are Derivations, Higher Derivations, Differential Ideals, Multiplication Modules and the Radical Formula. Faculty: A. K. Maloo + - Complex Analysis & Operator Theory Click to collapse I mainly consider various analytic function spaces defined on the unit disk or on some half plane of the complex plane and various operators on these spaces such as multiplication operators, composition operators, Cesaro operators. Also, I work on similar operators on some discrete function spaces defined on an infinite rooted tree (graph), in particular, on the discrete analogue of Hardy spaces. I deal with number of other problems which connects geometric function theory with function spaces and operator theory. Faculty: P. Muthukumar + - Computational Acoustics and Electromagnetics Click to collapse The study of interaction of electromagnetic fields with physical objects and the environment constitutes the main subject matter of Computational Electromagnetics. One of the major challenges in this area of research is in the development of efficient, accurate and rapidly-convergent algorithms for the simulation of propagation and scattering of acoustic and electromagnetic fields within and around structures that possess complex geometrical characteristics. These problems are of fundamental importance in diverse fields, with applications ranging from space exploration, medical imaging and oil exploration on the civilian side to aircraft design and decoy detection on the military side - just to name a few. Computational modeling of electromagnetic scattering problems has typically been attempted on the basis of classical, low-order Finite-Difference-Time-Domain (FDTD) or Finite-Element-Method (FEM) approaches. An important computational alternative to these approaches is provided by boundary integral-equation formulations that we have adopted owing to a number of excellent properties that they enjoy. Listed below are some of the key areas of interest in related research: 1. Design of high-order integrators for boundary integral equations arising from surface and volumetric scattering of acoustic and electromagnetic waves from complex engineering structures including from open surfaces and from geometries with singular features like edges and corners. 2. Accurate representation of complex surfaces in three dimensions with applications to enhancement of low quality CAD models and in the development of direct CAD-to-EM tools. 3. High frequency scattering methods in three dimensions with frequency independent cost in the context of multiple scattering configurations. A related field of interest in this regard includes high-order geometrical optics simulator for inverse ray tracing. 4. High performance computing. Faculty: Akash Anand , B. V. Rathish Kumar + - Computational Fluid Dynamics Click to collapse Development of Numerical Schemes for Incompressible Newtonian and Non-Newtonian Fluid Flows based on FDM, FEM, FVM, Wavelets, SEM, BEM etc. Development of Parallel Numerical Methods for Heat & Fluid Flow Analysis on Large Scale Parallel Computing systems based on MPI-OpenMP-Cuda programming concepts, ANN/ML methods for Flow Analysis. Global Climate Modelling on Very Large Scale Parallel Systems. Faculty: B. V. Rathish Kumar , Saktipada Ghorai + - Differential Equations Click to collapse Semigroups of linear operators and their applications, Functional differential equations, Galerkin approximations Many unsteady state physical problems are governed by partial differential equations of parabolic or hyperbolic types. These problems are mostly prototypes since they represent as members of large classes of such similar problems. So, to make a useful study of these problems we concentrate on their invariant properties which are satisfied by each member of the class. We reformulate these problems as evolution equations in abstract spaces such as Hilbert or more generally Banach spaces. The operators appearing in these equations have the property that they are the generators of semigroups. The theory of semigroups then plays an important role of establishing the well-posedness of these evolution equations. The analysis of functional differential equations enhances the applicability of evolution equations as these include the equations involving finite as well as infinite delays. Equations involving integrals can also be tackled using the techniques of functional differential equations. The Galerkin method and its nonlinear variants are fundamental tools to obtain the approximate solutions of the evolution and functional differential equations.   Faculty: D. Bahuguna Homogenization and Variational methods for partial differential equation The main interest is on Aysmptotic Analysis of partial differential equations. This is a technique to understand the macroscopic behaviour of a composite medium through its microscopic properties. The technique is commonly used for PDE with highly oscillating coefficients. The idea is to replace a given heterogeneous medium by a fictitious homogeneous one (the `homogenized' material) for numerical computations. The technique is also known as ``Multi scale analysis''. The known and unknown quantities in the study of physical or mechanical processes in a medium with micro structure depend on a small parameter $\varepsilon$. The study of the limit as $ \varepsilon \rightarrow0 $, is the aim of the mathematical theory of homogenization. The notion of $G$-convergence, $H$-convergence, two-scale convergence are some examples of the techniques employed for specific cases. The variational characterization of the technique for problems in calculus of variations is given by $\Gamma$- convergence.   Faculty: T. Muthukumar , B.V. Rathish Kumar Functional inequalities on Sobolev space Sobolev spaces are the natural spaces where one looks for solutions of Partial differential equations (PDEs). Functional inequalities on this spaces ( for example Moser-Trudinger Inequality, Poincare Inequality, Hardy- Sobolev Inequality and many other) plays a very significant role in establishing existence of solutions for various PDEs. Existence of extremal function for such inequalities is another key aspect that is investigated Asymptotic analysis on changing domains Study of differential equations on long cylinders appears naturally in various branches of Physics, Engineering applications and real life problems. Problems (not necessarily PDEs, can be purely variational in nature) set on cylindrical domains whose length tends to infinity, is analysed. Faculty: Prosenjit Roy , Kaushik Bal , Indranil Chowdhury Analysis of Nonlinear PDEs involving fractional/nonlocal operator: Fully nonlinear elliptic and parabolic equation involving nonlocal operators. Equations motivated from stochastic control/ Game problems including Hamilton Jacobi Bellman Equations, Isaacs Equations, Mean Field Games problems - Viscosity Solution theory, Comparison Principle, Wellposedness theory, Stability, Continuous Dependence. Numerical Analysis –Wellposedness, Convergence, Error estimates of Finite difference method, Semi-lagrangian Method.   Faculty: Indranil Chowdhury Control Theory and its applications: We study several aspects of controllability, say exact controllability, null controllability, approximate controllability, controllability to the trajectories of a given system of ordinary and partial differential equations (both linear and nonlinear). We study stabilizability (exponential, asymptotic) of a system of differential equations and construct feedback control for that system. Currently we are studying controllability of reaction diffusion systems of partial equations using Carleman inequalities and fixed point technique. Multiplier techniques are also used to show controllability of system of hyperbolic partial differential equations. Several mixed systems (hyperbolic and parabolic) are also been studied.   Faculty: Mrinmay Biswas + - Functional Analysis & Operator Theory Click to collapse Banach space theory Geometric and proximinality aspects in Banach spaces. Faculty: P. Shunmugaraj Function-theoretic and graph-theoretic operator theory The primary goal is to implement methods from the complex function theory and the graph theory into the multivariable operator theory. The topics of interests include de Branges-Rovnyak spaces and weighted shifts on directed graphs. Faculty: Sameer Chavan Non-commutative geometry The main emphasis is on the metric aspect of noncommutative geometry. Faculty: Satyajit Guin Bounded linear operators A central theme in operator theory is the study of B(H), the algebra of bounded linear operators on a separable complex Hilbert space. We focus on operator ideals, subideals and commutators of compact operators in B(H). There is also a continuing interest in semigroups of operators in B(H) from different perspectives. We work in operator semigroups involve characterization of special classes of semigroups which relate to solving certain operator equations. Faculty: Sasmita Patnaik + - Harmonic Analysis Click to collapse Operator spaces The main emphasis is on operator space techniques in abstract Harmonic Analysis. In the Euclidean setting Analysis, boundedness and weighted boundedness of singular integral operators are major thrust areas in the department. In abstract Harmonic analysis we do work in studying Lacunary sets in the noncommutative Lp spaces.   Faculty: Parasar Mohanty   On Lie groups Problems related to integral geometry on Lie groups are being studied.   Faculty: Rama Rawat     + - Homological Algebra Click to collapse Cohomology and Deformation theory of algebraic structures Research work in this area encompasses cohomology and deformation theory of algebraic structures, mainly focusing on Lie and Leibniz algebras arising out of topology and geometry. In particular, one is interested in the cohomology and Versal deformation for Lie and Leibniz brackets on the space of sections of vector bundles e.g. Lie algebroids and Courant algebroids. This study naturally relate questions about other algebraic structures which include Lie-Rinehart algebras, hom-Lie-Rinehart algebras, Hom-Gerstenhaber algebras, homotopy algebras associated to Courant algebras, higher categories and related fields.   Faculty: Ashis Mandal + - Image Processing Click to collapse TPDE based Image processing for Denoising, Inpainting, Classification, Compression, Registration, Optical flow analysis etc. Bio-Medical Image Analysis based on CT/MRI/US clinical data, ANN/ML methods in Image Analysis, Wavelet methods for Image processing.   Faculty: B. V. Rathish Kumar + - Mathematical Biology Click to collapse There is an active group working in the area of Mathematical Biology. The research is carried out in the following directions. Mathematical ecology 1. Research in this area is focused on the local and global stability analysis, detection of possible bifurcation scenario and derivation of normal form, chaotic dynamics for the ordinary as well as delay differential equation models, stochastic stability analysis for stochastic differential equation model systems and analysis of noise induced phenomena. Also the possible spatio-temporal pattern formation is studied for the models of interacting populations dispersing over two dimensional landscape. 2. Mathematical Modeling of the survival of species in polluted water bodies; depletion of dissolved oxygen in water bodies due to organic pollutants. Mathematical epidemiology 1. Mathematical Modeling of epidemics using stability analysis; effects of environmental, demographic and ecological factors. 2. Mathematical Modeling of HIV Dynamics in vivo Bioconvection Bioconvection is the process of spontaneous pattern formation in a suspension of swimming microorganisms. These patterns are associated with up- and down-welling of the fluid. Bioconvection is due to the individual and collective behaviours of the micro-organisms suspended in a fluid. The physical and biological mechanisms of bioconvection are investigated by developing mathematical models and analysing them using a variety of linear, nonlinear and computational techniques. Bio-fluid dynamics Mathematical Models for blood flow in cardiovascular system; renal flows; Peristaltic transport; mucus transport; synovial joint lubrication.   Faculty: Malay Banerjee , Saktipada Ghorai , B.V. Rathish Kumar   Cardiac electrophysiology Theory, Modeling & Simulation of Cardiac Electrical Activity (CEA) in Human Cardiac Tissue based on PDEODE models such as Monodomain Model, Biodomain model, Cardiac Arrhythmia, pace makers etc   Faculty: B.V. Rathish Kumar + - Number Theory & Arithmetic Geometry Click to collapse Algebraic number theory and Arithmetic geometry Iwasawa Theory of elliptic curves and modular forms, Galois representations, Congruences between special values of L-functions.   Faculty: Sudhanshu Shekhar Analytic number theory L-functions, sub-convexity problems, Sieve method   Faculty: Saurabh Kumar Singh Number theory and Arithmetic geometry Iwasawa Theory of elliptic curves and modular forms, Selmer groups   Faculty: Somnath Jha Number theory, Dynamical systems, Random walks on groups During the last four decades, it has been realized that some problems in number theory and, in particular, in Diophantine approximation, can be solved using techniques from the theory of homogeneous dynamics, random walks on homogeneous spaces etc. Indeed, one translates such problems into a problem on the behavior of certain trajectories in homogeneous spaces of Lie groups under flows or random walks; and subsequently resolves using very powerful techniques from the theory of dynamics on homogeneous spaces, random walk etc. I undertake this theme.   Faculty: Arijit Ganguly + - Numerical Analysis and Scientific Computing Click to collapse The faculty group in the area of Numerical Analysis & Scientific Computing are very actively engaged in high-quality research in the areas that include (but are not limited to): Singular Perturbation problems, Multiscale Phenomena, Hyperbolic Conservation Laws, Elliptic and Parabolic PDEs, Integral Equations, Computational Acoustics and Electromagnetics, Computational Fluid Dynamics, Computer-Aided Tomography and Parallel Computing. The faculty group is involved in the development, analysis, and application of efficient and robust algorithms for solving challenging problems arising in several applied areas. There is expertise in several discretization methods that include: Finite Difference Methods, Finite Element Methods, Spectral Element Methods, Boundary Element Methods, Nyström Method, Spline and Wavelet approximations, etc. This encompasses a very high level of computation that requires software skills of the highest order and parallel computing as well.   Faculty: B. V. Rathish Kumar , Akash Anand , Indranil Chowdhury + - Operator Algebra Click to collapse Broadly speaking, I work with topics in C*-algebras and von Neumann algebras. More precisely, my work involves Jones theory of subfactors and planar algebras.   Faculty: Keshab Chandra Bakshi + - Representation Theory Click to collapse Representation of Lie and linear algebraic groups over local fields, Representation-theoretic methods, automorphic representations over local and global fields, Linear algebraic groups and related topics MSC classification (22E50, 11F70, 20Gxx:)   Faculty: Santosh Nadimpalli Representations of finite and arithmetic groups Current research interests: Representations of Linear groups over local rings, Projective representations of finite and arithmetic groups, Applications of representation theory.   Faculty: Pooja Singla Representation theory of Lie algebras and algebraic groups  Faculty: Santosha Pattanayak Representation theory of infinite dimensional Lie algebras Current research interest: Representation theory of Kac-Moody algebras; Toroidal Lie algebras and extended affine Lie algebras.   Faculty: Sachin S. Sharma Representation theory and Invariant theory Current research interest: Representation and structure theory of algebraic groups, Classical invariant theory of reductive algebraic groups and associated Weyl groups.   Faculty: Preena Samuel Combinatorial representation theory String algebras form a class of tame representation type algebras that are presented combinatorially using quivers and relations. Currently I am interested in studying the combinatorics of strings to understand the Auslander-Reiten quiver that encodes the generators for the category of finite length R-modules as well as the Ziegler spectrum associated with string algebras whose topology is described model-theoretically   Faculty: Amit Kuber Representation Theory of Algebraic groups: Representation theory of Algebraic groups and Lie algebras, and its applications to Invariant theory and Algebraic geometry.   Faculty: Narasimha Chary Bonala + - Set Theory and Logic Click to collapse Set theory (MSC Classification 03Exx) We apply tools from set theory to problems from other areas of mathematics like measure theory and topology. Most of these applications involve the use of forcing to establish independence results. For examples of such results see https://home.iitk.ac.in/~krashu/   Faculty: Ashutosh Kumar Rough set theory and Modal logic Algebraic studies of structures and corresponding logics that have arisen in the course of investigations in Rough Set Theory (RST) constitute a primary part of my research. Currently, we are working on algebras and logics stemming from a combination of formal concept analysis and RST, and also from different approaches to paraconsistency.   Faculty: Mohua Banerjee + - Several Complex Variables Click to collapse Broadly speaking, my work lies in the theory of functions of several complex variables. Two major themes of my work till now are related to _Pick-Nevanlinna interpolation problem_ and on the _Kobayashi geometry of bounded domains_. I am also interested in complex potential theory and complex dynamics in one variable setting.   Faculty: Vikramjeet Singh Chandel + - Topology and Geometry Click to collapse Algebraic topology and Homotopy theory The primary interest is in studying equivariant algebraic topology and homotopy theory with emphasis on unstable homotopy. Specific topics include higher operations such as Toda bracket, pi-algebras, Bredon cohomology, simplicial/ cosimplicial methods, homotopical algebra.   Faculty: Debasis Sen Algebraic topology, Combinatorial topology I apply tools from algebraic topology and combinatorics to address problems in topology and graph theory.   Faculty: Nandini Nilakantan Differential geometry Geometric Analysis and Geometric PDEs. Interested in geometry of the eigenvalues of Laplace operator, Geometry of geodesics.   Faculty: G. Santhanam Low dimensional topology The main interest is in Knot Theory and its Applications. This includes the study of amphicheirality, the study of closed braids, and the knot polynomials, specially the Jones polynomial.   Faculty: Aparna Dar Geometric group theory and Hyperbolic geometry Work in this area involves relatively hyperbolic groups and Cannon-Thurston maps between relatively hyperbolic boundaries. Mapping Class Groups are also explored.   Faculty: Abhijit Pal Manifolds and Characteristic classes We are interested in the construction of new examples of non-Kahler complex manifolds. We aim also at answering the question of existence of almost-complex structures on certain even dimension real manifolds. Characteristic classes of vector bundles over certain spaces are also studied.   Faculty: Ajay Singh Thakur Moduli spaces of hyperbolic surfaces The central question we study here to find combinatorial descriptions of moduli spaces of closed and oriented hyperbolic surfaces. Also, we study isometric embedding of metric graphs on surfaces of following types: (a) quasi-essential on closed and oriented hyperbolic surfaces (b) non-compact surfaces, where complementary regions are punctured discs, (c) on half-translation surfaces etc.   Faculty: Bidyut Sanki Systolic topology and Geometry We are interested to study the configuration of systolic geodesics (i.e., shortest closed geodesics) on oriented hyperbolic surfaces. Also, we are interested in studying the maximal surfaces and deformations on hyperbolic surfaces of finite type to increase systolic lengths. Topological graph theory We study configuration of graphs, curves, arcs on surfaces, fillings, action of mapping class groups on graphs on surfaces, minimal graphs of higher genera.   Faculty: Bidyut Sanki + - Tribology Click to collapse Active work has been going on in the area of "Tribology". Tribology deals with the issues related to lubrication, friction and wear in moving machine parts. Work is going in the direction of hydrodynamic and elastohydrodynamic lubrication, including thermal, roughness and non-newtonian effects. The work is purely theoretical in nature leading to a system on non-linear partial differential equations, which are solved using high speed computers.   Faculty: B. V. Rathish Kumar Research Areas in Statistics and Probability Theory Here are the areas of Statistics in which research is being done currently. + - Bayesian Nonparametric Methods Click to collapse Exponential growth in computing power in the past few decades has made Bayesian methods for infinitedimensional models possible, which is termed as the Bayesian nonparametric (BN) methods. BN is a vast area dealing with modelling and making inference in various fields of Statistics, including, and not restricted to density estimation, regression, variable selection, classification, clustering. Irrespective of the field of execution, a BN method deals with prior construction on an infinite-dimensional parameter space, posterior computation and thereby making posterior predictive inference. Finally, the method is validated by supportive asymptotic properties to show the closeness of the proposed method to the true underlying data generating process. Faculty: Minerva Mukhopadhyay + - Data Mining in Finance Click to collapse Economic globalization and evolution of information technology has in recent times accounted for huge volume of financial data being generated and accumulated at an unprecedented pace. Effective and efficient utilization of massive amount of financial data using automated data driven analysis and modelling to help in strategic planning, investment, risk management and other decision-making goals is of critical importance. Data mining techniques have been used to extract hidden patterns and predict future trends and behaviours in financial markets. Data mining is an interdisciplinary field bringing together techniques from machine learning, pattern recognition, statistics, databases and visualization to address the issue of information extraction from such large databases. Advanced statistical, mathematical and artificial intelligence techniques are typically required for mining such data, especially the high frequency financial data. Solving complex financial problems using wavelets, neural networks, genetic algorithms and statistical computational techniques is thus an active area of research for researchers and practitioners. Faculty: Amit Mitra , Sharmishtha Mitra + - Econometric Modelling Click to collapse Econometric modelling involves analytical study of complex economic phenomena with the help of sophisticated mathematical and statistical tools. The size of a model typically varies with the number of relationships and variables it is applying to replicate and simulate in a regional, national or international level economic system. On the other hand, the methodologies and techniques address the issues of its basic purpose – understanding the relationship, forecasting the future horizon and/or building "what-if" type scenarios. Econometric modelling techniques are not only confined to macro-economic theory, but also are widely applied to model building in micro-economics, finance and various other basic and social sciences. The successful estimation and validation part of the model-building relies heavily on the proper understanding of the asymptotic theory of statistical inference. A challenging area of econometric Faculty: Shalabh , Sharmishtha Mitra + - Entropy Estimation and Applications Click to collapse Estimation of entropies of molecules is an important problem in molecular sciences. A commonly used method by molecular scientist is based on the assumption of a multivariate normal distribution for the internal molecular coordinates. For the multivariate normal distribution, we have proposed various estimators of entropy and established their optimum properties. The assumption of a multivariate normal distribution for the internal coordinates of molecules is adequate when the temperature at which the molecule is studied is low, and thus the fluctuations in internal coordinates are small. However, at higher temperatures, the multivariate normal distribution is inadequate as the dihedral angles at higher temperatures exhibit multimodes and skewness in their distribution. Moreover the internal coordinates of molecules are circular variables and thus the assumption of multivariate normality is inappropriate. Therefore a nonparametric and circular statistic approach to the problem of estimation of entropy is desirable. We have adopted a circular nonparametric approach for estimating entropy of a molecule. This approach is getting a lot of attention among molecular scientists. Faculty: Neeraj Misra + - Environmental Statistics Click to collapse The main goal of environmental statistics is to build sophisticated modelling techniques that are necessary for analysing temperature, precipitation, ozone concentration in air, salinity in seawater, fire weather index, etc. There are multiple sources of such observations, like weather stations, satellites, ships, and buoys, as well as climate models. While station-based data are generally available for long time periods, the geographical coverage of such stations is mostly sparse. On the other hand, satellite-derived data are available only for the last few decades, but they are generally of much higher spatial resolution. While the current statistical literature has already explored various techniques for station-based data, methods available for modelling high-resolution satellite-based datasets are relatively scarce and there is ample opportunity for building statistical methods to handle such datasets. Here, the data are not only huge in volume, but they are also spatially dependent. Modelling such complex dependencies is challenging also due to the high nonstationary often present in the data. The sophisticated methods also need suitable computational tools and thus provide scopes for novel research directions in computational statistics. Apart from real datasets, statistical modelling of climate model outputs is a new area of research, particularly keeping in mind the issue of climate change. Under different representative concentration pathways (RCPs) of the Intergovernmental Panel for Climate Change (IPCC), different carbon emission Faculty: Arnab Hazra + - Estimation in Restricted Parameter Space Click to collapse In many practical situations, it is natural to restrict the parameter space. This additional information of restricted parameter space can be intelligently used to derive estimators that improve upon the standard (natural) estimators, meant for the case of unrestricted parameter space. We deal with the problems of estimation parameters of one or more populations when it is known apriori that some or all of them satisfy certain restrictions, leading to the consideration of restricted parameter space. The goal is to find estimators that improve upon the standard (natural) estimators, meant for the case of unrestricted parameter space. We also deal with the decision theoretic aspects of this problem. + - Game Theory Click to collapse The mathematical discipline of Game theory models and analyses interactions between competing and cooperative players. Some research areas in game theory are choice theory, mechanism design, differential games, stochastic games, graphon games, combinatorial games, evolutionary games, cooperative games, Bayesian games, algorithmic games - and this list is certainly not exhaustive. Gametheoretic models are used in many real-life problems such as decision making, voting, matching, auctioning, bargaining/negotiating, queuing, distributing/dividing wealth, dealing with cheap talks, the evolution of living organisms, disease propagation, cancer treatment, and many more. Game Theory is also a popular research area in computer science where equilibrium structures are explored using computer algorithms. Mathematical topics such as combinatorics, graph theory, probability (discrete and measure-theoretic), analysis (real and functional), algebra (linear and abstract), etc., are used in solving game-theoretic problems. Faculty: Soumyarup Sadhukhan + - Machine Learning and Statistical Pattern Recognition Click to collapse Build machine learning algorithms based on statistical modeling of data. With a statistical model in hand, we apply probability theory to get a sound understanding of the algorithms. Faculty: Subhajit Dutta + - Markov chain Monte Carlo Click to collapse Markov chain Monte Carlo (MCMC) algorithms produce correlated samples from a desired target distribution, using an ergodic Markov chain. Due to the lack of independence of the samples, and the challenges of working with Markov chains, many theoretical and practical questions arise. Much of the research in this area can be divided into three broad topics: (1) development of new sampling algorithms for complicated target distributions, (2) studying rates of convergence of the Markov chains employed in various applications like variable selection, regression, survival analysis etc, and (3) measuring the quality of MCMC samples in an effort to quantify the variability in the final estimators of the features of the target. Faculty: Dootika Vats + - Non-Parametric and Robust Statistical methods Click to collapse Detection of different features (in terms of shape) of non-parametric regression functions are studied; asymptotic distributions of the proposed estimators (along with their robustness properties) of the shaperestricted regression function are also investigated. Apart from this, work on the test of independence for more than two random variables is pursued. Statistical Signal Processing and Statistical Pattern Recognition are the other areas of interest. Faculty: Subhra Sankar Dhar + - Optimal Experimental Design Click to collapse The area of optimal experimental design has been an integral part of many scientific investigation including agriculture and animal husbandry, biology, medicine, physical and chemical sciences, and industrial research. A well-designed experiment utilizes the limited recourse (cost, time, experimental units, etc) optimally to answer the underlying scientific question. For example, optimal cluster/crossover designs may be applied to cluster/cross randomized trials to efficiently estimates the treatment effects. Optimal standard ANOVA designs can be utilized to test the equality of several experimental groups. Most popular categories of optimal designs include Bayesian designs, longitudinal designs, designs for ordered experiments and factorial designs to name a few. Faculty: Satya Prakash Singh + - Ranking and Selection Problems Click to collapse About fifty years ago statistical inference problems were first formulated in the now-familiar "Ranking and Selection" framework. Ranking and selection problems broadly deal with the goal of ordering of different populations in terms of unknown parameters associated with them. We deal with the following aspects of Ranking and Selection Problems:1. Obtaining optimal ranking and selection procedures using decision theoretic approach;2. Obtaining optimal ranking and selection procedures under heteroscedasticity;3. Simultaneous confidence intervals for all distances from the best and/or worst populations, where the best (worst) population is the one corresponding to the largest (smallest) value of the parameter;4. Estimation of ranked parameters when the ranking between parameters is not known apriori;5. Estimation of (random) parameters of the populations selected using a given decision rule for ranking and selection problems. + - Regression Modelling Click to collapse The outcome of any experiment depends on several variables and such dependence involves some randomness which can be characterized by a statistical model. The statistical tools in regression analysis help in determining such relationships based on the sample experimental data. This helps further in describing the behaviour of the process involved in experiment. The tools in regression analysis can be applied in social sciences, basic sciences, engineering sciences, medical sciences etc. The unknown and unspecified form of relationship among the variables can be linear as well as nonlinear which is to be determined on the basis of a sample of experimental data only. The tools in regression analysis help in the determination of such relationships under some standard statistical assumptions. In many experimental situations, the data do not satisfy the standard assumptions of statistical tools, e.g. the input variables may be linearly related leading to the problem of multicollinearity, the output data may not have constant variance giving rise to the hetroskedasticity problem, parameters of the model may have some restrictions, the output data may be auto correlated, some data on input and/or output variables may be missing, the data on input and output variables may not be correctly observable but contaminated with measurement errors etc. Different types of models including the econometric models, e.g., multiple regression models, restricted regression models, missing data models, panel data models, time series models, measurement error models, simultaneous equation models, seemingly unrelated regression equation models etc. are employed in such situations. So the need of development of new statistical tools arises for the detection of problem, analysis of such non-standard data in different models and to find the relationship among different variables under nonstandard statistical conditions. The development of such tools and the study of their theoretical statistical properties using finite sample theory and asymptotic theory supplemented with numerical studies based on simulation and real data are the objectives of the research work in this area. Faculty: Shalabh + - Robust Estimation in Nonlinear Models Click to collapse Efficient estimation of parameters of nonlinear regression models is a fundamental problem in applied statistics. Isolated large values in the random noise associated with model, which is referred to as an outliers or an atypical observation, while of interest, should ideally not influence estimation of the regular pattern exhibited by the model and the statistical method of estimation should be robust against outliers. The nonlinear least squares estimators are sensitive to presence of outliers in the data and other departures from the underlying distributional assumptions. The natural choice of estimation technique in such a scenario is the robust M-estimation approach. Study of the asymptotic theoretical properties of Mestimators under different possibilities of the M-estimation function and noise distribution assumptions is an interesting problem. It is further observed that a number of important nonlinear models used to model real life phenomena have a nested superimposed structure. It is thus desirable also to have robust order estimation techniques and study the corresponding theoretical asymptotic properties. Theoretical asymptotic properties of robust model selection techniques for linear regression models are well established in the literature, it is an important and challenging problem to design robust order estimation techniques for nonlinear nested models and establish their asymptotic optimality properties. Furthermore, study of the asymptotic properties of robust M-estimators as the number of nested superimposing terms increase is also an important problem. Huber and Portnoy established asymptotic behavior of the M-estimators when the number of components in a linear regression model is large and established conditions under which consistency and asymptotic normality results are valid. It is possible to derive conditions under which similar results hold for different nested nonlinear models. Faculty: Debasis Kundu , Amit Mitra + - Rough Paths and Regularity structures Click to collapse The seminal works of Terry Lyons on extensions of Young integration, the latter being an extension of Riemann integration, to functions with Holder regular paths (or those with finite p-variation for some 0 < p < 1) lead to the study of Rough Paths and Rough Differential Equations. Martin Hairer, Massimiliano Gubinelli and their collaborators developed fundamental results in this area of research. Extensions of these ideas to functions with negative regularity (read as "distributions") opened up the area of Regularity structures. Important applications of these topics include constructions of `pathwise' solutions of stochastic differential equations and stochastic partial differential equations. Faculty: Suprio Bhar Numerical analysis of differential equation driven by rough noise: Developing numerical scheme for differential equations driven by rough noise and studying its convergence, rate of convergence etc. Faculty: Mrinmay Biswas and Suprio Bhar + - Spatial statistics Click to collapse The branch of statistics that focuses on the methods for analysing data observed across some spatial locations in 2-D or 3-D (most common), is called spatial statistics. The spatial datasets can be broadly divided into three types: point-referenced data, areal data, and point patterns. Temperature data collected by a few monitoring stations spread across a city on some specific day is an example of the first type. When data are obtained as summaries of some geographical regions, they are of the second type, crime rate dataset from the different states of India on a specific year is an example. An example of the third type is the IED attack locations in Afghanistan during a year, where the geographical coordinates are themselves the data. Because of the natural dependence among the observations obtained from two close locations, the data cannot be assumed to be independent. When the study domain is large, often we have a large number of observational sites and at the same time, those sites are possibly distributed across a nonhomogeneous area. This leads to the necessity of models that can handle a large number of sites as well as the nonstationary dependence structure and this is a very active area of research. Apart from common geostatistical models, a very active area of research is focused on spatial extreme value theory where max-stable stochastic processes are the natural models to explain the tail-dependence. While the available methods for such spatial extremes are highly scarce, specifically for moderately highdimensional problems, different future research directions are being explored currently in the literature. For better uncertainty quantification and computational flexibility using hierarchically defined models, the Bayesian paradigm is often a natural choice. + - Statistical Signal Processing Click to collapse Signal processing may broadly be considered to involve the recovery of information from physical observations. The received signals are usually disturbed by thermal, electrical, atmospheric or intentional interferences. Due to the random nature of the signal, statistical techniques play an important role in signal processing. Statistics is used in the formulation of appropriate models to describe the behaviour of the system, the development of appropriate techniques for estimation of model parameters, and the assessment of model performances. Statistical Signal Processing basically refers to the analysis of random signals using appropriate statistical techniques. Different one and multidimensional models have been used in analyzing various one and multidimensional signals. For example ECG and EEG signals, or different grey and white or colour textures can be modelled quite effectively, using different non-linear models. Effective modelling are very important for compression as well as for prediction purposes. The important issues are to develop efficient estimation procedures and to study their properties. Due to non-linearity, finite sample properties of the estimators cannot be derived; most of the results are asymptotic in nature. Extensive Monte Carlo simulations are generally used to study the finite sample behaviour of the different estimators. + - Step-Stress Modelling Click to collapse Traditionally, life-data analysis involves analysing the time-to-failure data obtained under normal operating conditions. However, such data are difficult to obtain due to long durability of modern days. products, lack of time-gap in designing, manufacturing and actually releasing such products in market, etc. Given these difficulties as well as the ever-increasing need to observe failures of products to better understand their failure modes and their life characteristics in today's competitive scenario, attempts have been made to devise methods to force these products to fail more quickly than they would under normal use conditions. Various methods have been developed to study this type of "accelerated life testing" (ALT) models. Step-stress modelling is a special case of ALT, where one or more stress factors are applied in a life-testing experiment, which are changed according to pre-decided design. The failure data observed as order statistics are used to estimate parameters of the distribution of failure times under normal operating conditions. The process requires a model relating the level of stress and the parameters of the failure distribution at that stress level. The difficulty level of estimation procedure depends on several factors like, the lifetime distribution and number of parameters thereof, the uncensored or various censoring (Type I, Type II, Hybrid, Progressive, etc.) schemes adopted, the application of non-Bayesian or Bayesian estimation procedures, etc. Faculty: Debasis Kundu , Sharmishtha Mitra + - Stochastic Partial Differential Equations Click to collapse The study of Stochastic calculus, more specifically, that of stochastic differential equations and stochastic partial differential equations, has a broad range of applications across various disciplines or branches of Mathematics, such as Partial Differential Equations, Evolution systems, Interacting particle systems, Finance, Mathematical Biology. Theoretical understanding for such equations was first obtained in finite dimensional Euclidean spaces. Later on, to describe various natural phenomena, models were constructed (and analyzed) with values in Banach spaces, Hilbert spaces and in the duals of nuclear spaces. Important topics/questions in this area of research include existence and uniqueness of solutions, Stability, Stationarity, Stochastic flows, Stochastic Filtering theory and Stochastic Control Theory, to name a few. + - Theory of Stochastic Orders and Aging and Applications Click to collapse The manner in which a component (or system) improves or deteriorates with time can be described by concepts of aging. Various aging notions have been proposed in the literature. Similarly lifetimes of two different systems can be compared using the concepts of stochastic orders between the probability distributions of corresponding (random) lifetimes. Various stochastic orders between probability distributions have been defined in the literature. We study the concepts of aging and stochastic orders for various coherent systems. In many situations, the performance of a system can be improved by introducing some kind of redundancy into the system. The problem of allocating redundant components to the components of a coherent system, in order to optimize its reliability or some other system performance characteristic, is of considerable interest in reliability engineering. These problems often lead to interesting theoretical results in Probability Theory. We study the problem of optimally allocating spares to the components of various coherent systems, in order to optimize their reliability or some other system performance characteristic. Performances of systems arising out of different allocations are studied using concepts of aging and stochastic orders. DEPARTMENT OF Mathematics & Statistics INDIAN INSTITUTE OF TECHNOLOGY KANPUR Kanpur, UP 208016 | Phone: 0512-259-xxxx | Fax: 0512-259-xxxx Suo-Moto Disclosure Areas of Study Course Catalog Academic Calendar Graduate Programs Winter Study Experiential Learning & Community Engagement Pathways for Inclusive Excellence Admission & Aid Affordability Financial Aid Life at Williams Health Services Integrative Wellbeing Services Religion & Spirituality Students    ▶ Alumni    ▶ Parents & Families    ▶ Employment    ▶ Faculty & Staff    ▶ Quick Links: Diversity, Equity & Inclusion Sustainability News & Stories Williams Email Math/Stats Thesis and Colloquium Topics Updated: April 2024 Math/Stats Thesis and Colloquium Topics 2024- 2025 The degree with honors in Mathematics or Statistics is awarded to the student who has demonstrated outstanding intellectual achievement in a program of study which extends beyond the requirements of the major. The principal considerations for recommending a student for the degree with honors will be: Mastery of core material and skills, breadth and, particularly, depth of knowledge beyond the core material, ability to pursue independent study of mathematics or statistics, originality in methods of investigation, and, where appropriate, creativity in research. An honors program normally consists of two semesters (MATH/STAT 493 and 494) and a winter study (WSP 031) of independent research, culminating in a thesis and a presentation. Under certain circumstances, the honors work can consist of coordinated study involving a one semester (MATH/STAT 493 or 494) and a winter study (WSP 030) of independent research, culminating in a “minithesis” and a presentation. At least one semester should be in addition to the major requirements, and thesis courses do not count as 400-level senior seminars. An honors program in actuarial studies requires significant achievement on four appropriate examinations of the Society of Actuaries. Highest honors will be reserved for the rare student who has displayed exceptional ability, achievement or originality. Such a student usually will have written a thesis, or pursued actuarial honors and written a mini-thesis. An outstanding student who writes a mini-thesis, or pursues actuarial honors and writes a paper, might also be considered. In all cases, the award of honors and highest honors is the decision of the Department. Here is a list of possible colloquium topics that different faculty are willing and eager to advise. You can talk to several faculty about any colloquium topic, the sooner the better, at least a month or two before your talk. For various reasons faculty may or may not be willing or able to advise your colloquium, which is another reason to start early. RESEARCH INTERESTS OF MATHEMATICS AND STATISTICS FACULTY Here is a list of faculty interests and possible thesis topics.  You may use this list to select a thesis topic or you can use the list below to get a general idea of the mathematical interests of our faculty. Colin Adams (On Leave 2024 – 2025) Research interests:   Topology and tiling theory.  I work in low-dimensional topology.  Specifically, I work in the two fields of knot theory and hyperbolic 3-manifold theory and develop the connections between the two. Knot theory is the study of knotted circles in 3-space, and it has applications to chemistry, biology and physics.  I am also interested in tiling theory and have been working with students in this area as well. Hyperbolic 3-manifold theory utilizes hyperbolic geometry to understand 3-manifolds, which can be thought of as possible models of the spatial universe. Possible thesis topics: Investigate various aspects of virtual knots, a generalization of knots. Consider hyperbolicity of virtual knots, building on previous SMALL work. For which virtual knots can you prove hyperbolicity? Investigate why certain virtual knots have the same hyperbolic volume. Consider the minimal Turaev volume of virtual knots, building on previous SMALL work. Investigate which knots have totally geodesic Seifert surfaces. In particular, figure out how to interpret this question for virtual knots. Investigate n-crossing number of knots. An n-crossing is a crossing with n strands of the knot passing through it. Every knot can be drawn in a picture with only n-crossings in it. The least number of n-crossings is called the n-crossing number. Determine the n-crossing number for various n and various families of knots. An übercrossing projection of a knot is a projection with just one n-crossing. The übercrossing number of a knot is the least n for which there is such an übercrossing projection. Determine the übercrossing number for various knots, and see how it relates to other traditional knot invariants. A petal projection of a knot is a projection with just one n-crossing such that none of the loops coming out of the crossing are nested. In other words, the projection looks like a daisy. The petal number of a knot is the least n for such a projection. Determine petal number for various knots, and see how it relates to other traditional knot invariants. In a recent paper, we extended petal number to virtual knots. Show that the virtual petal number of a classical knot is equal to the classical petal number of the knot (This is a GOOD question!) Similarly, show that the virtual n-crossing number of a classical knot is equal to the classical n-crossing number. (This is known for n = 2.) Find tilings of the branched sphere by regular polygons. This would extend work of previous research students. There are lots of interesting open problems about something as simple as tilings of the sphere. Other related topics. Possible colloquium topics : Particularly interested in topology, knot theory, graph theory, tiling theory and geometry but will consider other topics. Christina Athanasouli Research Interests:   Differential equations, dynamical systems (both smooth and non-smooth), mathematical modeling with applications in biological and mechanical systems My research focuses on analyzing mathematical models that describe various phenomena in Mathematical Neuroscience and Engineering. In particular, I work on understanding 1) the underlying mechanisms of human sleep (e.g. how sleep patterns change with development or due to perturbations), and 2) potential design or physical factors that may influence the dynamics in vibro-impact mechanical systems for the purpose of harvesting energy. Mathematically, I use various techniques from dynamical systems and incorporate both numerical and analytical tools in my work.  Possible colloquium topics:   Topics in applied mathematics, such as: Mathematical modeling of sleep-wake regulation Mathematical modeling vibro-impact systems Bifurcations/dynamics of mathematical models in Mathematical Neuroscience and Engineering Bifurcations in piecewise-smooth dynamical systems Julie Blackwood Research Interests:   Mathematical modeling, theoretical ecology, population biology, differential equations, dynamical systems. My research uses mathematical models to uncover the complex mechanisms generating ecological dynamics, and when applicable emphasis is placed on evaluating intervention programs. My research is in various ecological areas including ( I ) invasive species management by using mathematical and economic models to evaluate the costs and benefits of control strategies, and ( II ) disease ecology by evaluating competing mathematical models of the transmission dynamics for both human and wildlife diseases. Mathematical modeling of invasive species Mathematical modeling of vector-borne or directly transmitted diseases Developing mathematical models to manage vector-borne diseases through vector control Other relevant topics of interest in mathematical biology Each topic (1-3) can focus on a case study of a particular invasive species or disease, and/or can investigate the effects of ecological properties (spatial structure, resource availability, contact structure, etc.) of the system. Possible colloquium topics:   Any topics in applied mathematics, such as: Research Interest :  Statistical methodology and applications.  One of my research topics is variable selection for high-dimensional data.  I am interested in traditional and modern approaches for selecting variables from a large candidate set in different settings and studying the corresponding theoretical properties. The settings include linear model, partial linear model, survival analysis, dynamic networks, etc.  Another part of my research studies the mediation model, which examines the underlying mechanism of how variables relate to each other.  My research also involves applying existing methods and developing new procedures to model the correlated observations and capture the time-varying effect.  I am also interested in applications of data mining and statistical learning methods, e.g., their applications in analyzing the rhetorical styles in English text data. Variable selection uses modern techniques such as penalization and screening methods for several different parametric and semi-parametric models. Extension of the classic mediation models to settings with correlated, longitudinal, or high-dimensional mediators. We could also explore ways to reduce the dimensionality and simplify the structure of mediators to have a stable model that is also easier to interpret. We shall analyze the English text dataset processed by the Docuscope environment with tools for corpus-based rhetorical analysis. The data have a hierarchical structure and contain rich information about the rhetorical styles used. We could apply statistical models and statistical learning algorithms to reduce dimensions and gain a more insightful understanding of the text. Possible colloquium topics:  I am open to any problems in statistical methodology and applications, not limited to my research interests and the possible thesis topics above. Richard De Veaux  Research interests: Statistics. My research interests are in both statistical methodology and in statistical applications.  For the first, I look at different methods and try to understand why some methods work well in particular settings, or more creatively, to try to come up with new methods.  For the second, I work in collaboration with an investigator (e.g. scientist, doctor, marketing analyst) on a particular statistical application.  I have been especially interested in problems dealing with large data sets and the associated modeling tools that work for these problems. Human Performance and Aging.I have been working on models for assessing the effect of age on performance in running and swimming events. There is still much work to do. So far I’ve looked at masters’ freestyle swimming and running data and a handicapped race in California, but there are world records for each age group and other events in running and swimming that I’ve not incorporated. There are also many other types of events. Variable Selection.  How do we choose variables when we have dozens, hundreds or even thousands of potential predictors? Various model selection strategies exist, but there is still a lot of work to be done to find out which ones work under what assumptions and conditions. Problems at the interface.In this era of Big Data, not all methods of classical statistics can be applied in practice. What methods scale up well, and what advances in computer science give insights into the statistical methods that are best suited to large data sets? Applying statistical methods to problems in science or social science.In collaboration with a scientist or social scientist, find a problem for which statistical analysis plays a key role. Possible colloquium topics: Almost any topic in statistics that extends things you’ve learned in courses —  specifically topics in Experimental design, regression techniques or machine learning Model selection problems Thomas Garrity (On Leave 2024 – 2025) Research interest:   Number Theory and Dynamics. My area of research is officially called “multi-dimensional continued fraction algorithms,” an area that touches many different branches of mathematics (which is one reason it is both interesting and rich).  In recent years, students writing theses with me have used serious tools from geometry, dynamics, ergodic theory, functional analysis, linear algebra, differentiability conditions, and combinatorics.  (No single person has used all of these tools.)  It is an area to see how mathematics is truly interrelated, forming one coherent whole. While my original interest in this area stemmed from trying to find interesting methods for expressing real numbers as sequences of integers (the Hermite problem), over the years this has led to me interacting with many different mathematicians, and to me learning a whole lot of math.  My theses students have had much the same experiences, including the emotional rush of discovery and the occasional despair of frustration.  The whole experience of writing a thesis should be intense, and ultimately rewarding.   Also, since this area of math has so many facets and has so many entrance points, I have had thesis students from wildly different mathematical backgrounds do wonderful work; hence all welcome. Generalizations of continued fractions. Using algebraic geometry to study real submanifolds of complex spaces. Possible colloquium topics:   Any interesting topic in mathematics. Leo Goldmakher Research interests:   Number theory and arithmetic combinatorics. I’m interested in quantifying structure and randomness within naturally occurring sets or sequences, such as the prime numbers, or the sequence of coefficients of a continued fraction, or a subset of a vector space. Doing so typically involves using ideas from analysis, probability, algebra, and combinatorics. Possible thesis topics:   Anything in number theory or arithmetic combinatorics. Possible colloquium topics:   I’m happy to advise a colloquium in any area of math. Susan Loepp Research interests: Commutative Algebra.  I study algebraic structures called commutative rings.  Specifically, I have been investigating the relationship between local rings and their completion.  One defines the completion of a ring by first defining a metric on the ring and then completing the ring with respect to that metric.  I am interested in what kinds of algebraic properties a ring and its completion share.  This relationship has proven to be intricate and quite surprising.  I am also interested in the theory of tight closure, and Homological Algebra. Topics in Commutative Algebra including: Using completions to construct Noetherian rings with unusual prime ideal structures. What prime ideals of C[[ x 1 ,…, x n ]] can be maximal in the generic formal fiber of a ring? More generally, characterize what sets of prime ideals of a complete local ring can occur in the generic formal fiber. Characterize what sets of prime ideals of a complete local ring can occur in formal fibers of ideals with height n where n ≥1. Characterize which complete local rings are the completion of an excellent unique factorization domain. Explore the relationship between the formal fibers of R and S where S is a flat extension of R . Determine which complete local rings are the completion of a catenary integral domain. Determine which complete local rings are the completion of a catenary unique factorization domain. Possible colloquium topics:   Any topics in mathematics and especially commutative algebra/ring theory. Steven Miller For more information and references, see http://www.williams.edu/Mathematics/sjmiller/public_html/index.htm Research interests :  Analytic number theory, random matrix theory, probability and statistics, graph theory. My main research interest is in the distribution of zeros of L-functions.  The most studied of these is the Riemann zeta function, Sum_{n=1 to oo} 1/n^s.  The importance of this function becomes apparent when we notice that it can also be written as Prod_{p prime} 1 / (1 – 1/p^s); this function relates properties of the primes to those of the integers (and we know where the integers are!).  It turns out that the properties of zeros of L-functions are extremely useful in attacking questions in number theory.  Interestingly, a terrific model for these zeros is given by random matrix theory: choose a large matrix at random and study its eigenvalues.  This model also does a terrific job describing behavior ranging from heavy nuclei like Uranium to bus routes in Mexico!  I’m studying several problems in random matrix theory, which also have applications to graph theory (building efficient networks).  I am also working on several problems in probability and statistics, especially (but not limited to) sabermetrics (applying mathematical statistics to baseball) and Benford’s law of digit bias (which is often connected to fascinating questions about equidistribution).  Many data sets have a preponderance of first digits equal to 1 (look at the first million Fibonacci numbers, and you’ll see a leading digit of 1 about 30% of the time).  In addition to being of theoretical interest, applications range from the IRS (which uses it to detect tax fraud) to computer science (building more efficient computers).  I’m exploring the subject with several colleagues in fields ranging from accounting to engineering to the social sciences. Possible thesis topics:  Theoretical models for zeros of elliptic curve L-functions (in the number field and function field cases). Studying lower order term behavior in zeros of L-functions. Studying the distribution of eigenvalues of sets of random matrices. Exploring Benford’s law of digit bias (both its theory and applications, such as image, voter and tax fraud). Propagation of viruses in networks (a graph theory / dynamical systems problem). Sabermetrics. Additive number theory (questions on sum and difference sets). Possible colloquium topics:  Plus anything you find interesting.  I’m also interested in applications, and have worked on subjects ranging from accounting to computer science to geology to marketing…. Ralph Morrison Research interests:   I work in algebraic geometry, tropical geometry, graph theory (especially chip-firing games on graphs), and discrete geometry, as well as computer implementations that study these topics. Algebraic geometry is the study of solution sets to polynomial equations.  Such a solution set is called a variety.  Tropical geometry is a “skeletonized” version of algebraic geometry. We can take a classical variety and “tropicalize” it, giving us a tropical variety, which is a piecewise-linear subset of Euclidean space.  Tropical geometry combines combinatorics, discrete geometry, and graph theory with classical algebraic geometry, and allows for developing theory and computations that tell us about the classical varieties.  One flavor of this area of math is to study chip-firing games on graphs, which are motivated by (and applied to) questions about algebraic curves. Possible thesis topics : Anything related to tropical geometry, algebraic geometry, chip-firing games (or other graph theory topics), and discrete geometry.  Here are a few specific topics/questions: Study the geometry of tropical plane curves, perhaps motivated by results from algebraic geometry.  For instance:  given 5 (algebraic) conics, there are 3264 conics that are tangent to all 5 of them.  What if we look at tropical conics–is there still a fixed number of tropical conics tangent to all of them?  If so, what is that number?  How does this tropical count relate to the algebraic count? What can tropical plane curves “look like”?  There are a few ways to make this question precise.  One common way is to look at the “skeleton” of a tropical curve, a graph that lives inside of the curve and contains most of the interesting data.  Which graphs can appear, and what can the lengths of its edges be?  I’ve done lots of work with students on these sorts of questions, but there are many open questions! What can tropical surfaces in three-dimensional space look like?  What is the version of a skeleton here?  (For instance, a tropical surface of degree 4 contains a distinguished polyhedron with at most 63 facets. Which polyhedra are possible?) Study the geometry of tropical curves obtained by intersecting two tropical surfaces.  For instance, if we intersect a tropical plane with a tropical surface of degree 4, we obtain a tropical curve whose skeleton has three loops.  How can those loops be arranged?  Or we could intersect degree 2 and degree 3 tropical surfaces, to get a tropical curve with 4 loops; which skeletons are possible there? One way to study tropical geometry is to replace the usual rules of arithmetic (plus and times) with new rules (min and plus).  How do topics like linear algebra work in these fields?  (It turns out they’re related to optimization, scheduling, and job assignment problems.) Chip-firing games on graphs model questions from algebraic geometry.  One of the most important comes in the “gonality” of a graph, which is the smallest number of chips on a graph that could eliminate (via a series of “chip-firing moves”) an added debt of -1 anywhere on the graph.  There are lots of open questions for studying the gonality of graphs; this include general questions, like “What are good lower bounds on gonality?” and specific ones, like “What’s the gonality of the n-dimensional hypercube graph?” We can also study versions of gonality where we place -r chips instead of just -1; this gives us the r^th gonality of a graph.  Together, the first, second, third, etc. gonalities form the “gonality sequence” of a graph.  What sequences of integers can be the gonality sequence of some graph?  Is there a graph whose gonality sequence starts 3, 5, 8? There are many computational and algorithmic questions to ask about chip-firing games.  It’s known that computing the gonality of a general graph is NP-hard; what if we restrict to planar graphs?  Or graphs that are 3-regular? And can we implement relatively efficient ways of computing these numbers, at least for small graphs? What if we changed our rules for chip-firing games, for instance by working with chips modulo N?  How can we “win” a chip-firing game in that context, since there’s no more notion of debt? Study a “graph throttling” version of gonality.  For instance, instead of minimizing the number of chips we place on the graph, maybe we can also try to decrease the number of chip-firing moves we need to eliminate debt. Chip-firing games lead to interesting questions on other topics in graph theory.  For instance, there’s a conjectured upper bound of (|E|-|V|+4)/2 on the gonality of a graph; and any graph is known to have gonality at least its tree-width.  Can we prove the (weaker) result that (|E|-|V|+4)/2 is an upper bound on tree-width?  (Such a result would be of interest to graph theorists, even the idea behind it comes from algebraic geometry!) Topics coming from discrete geometry.  For example:  suppose you want to make “string art”, where you have one shape inside of another with string weaving between the inside and the outside shapes.  For which pairs of shapes is this possible? Possible Colloquium topics:   I’m happy to advise a talk in any area of math, but would be especially excited about talks related to algebra, geometry, graph theory, or discrete mathematics. Shaoyang Ning (On Leave 2024 – 2025) Research Interest :  Statistical methodologies and applications. My research focuses on the study and design of statistical methods for integrative data analysis, in particular, to address the challenges of increasing complexity and connectivity arising from “Big Data”. I’m interested in innovating statistical methods that efficiently integrate multi-source, multi-resolution information to solve real-life problems. Instances include tracking localized influenza with Google search data and predicting cancer-targeting drugs with high-throughput genetic profiling data. Other interests include Bayesian methods, copula modeling, and nonparametric methods. Digital (disease) tracking: Using Internet search data to track and predict influenza activities at different resolutions (nation, region, state, city); Integrating other sources of digital data (e.g. Twitter, Facebook) and/or extending to track other epidemics and social/economic events, such as dengue, presidential approval rates, employment rates, and etc. Predicting cancer drugs with multi-source profiling data: Developing new methods to aggregate genetic profiling data of different sources (e.g., mutations, expression levels, CRISPR knockouts, drug experiments) in cancer cell lines to identify potential cancer-targeting drugs, their modes of actions and genetic targets. Social media text mining: Developing new methods to analyze and extract information from social media data (e.g. Reddit, Twitter). What are the challenges in analyzing the large-volume but short-length social media data? Can classic methods still apply? How should we innovate to address these difficulties? Copula modeling: How do we model and estimate associations between different variables when they are beyond multivariate Normal? What if the data are heavily dependent in the tails of their distributions (commonly observed in stock prices)? What if dependence between data are non-symmetric and complex? When the size of data is limited but the dimension is large, can we still recover their correlation structures? Copula model enables to “link” the marginals of a multivariate random variable to its joint distribution with great flexibility and can just be the key to the questions above. Other cross-disciplinary, data-driven projects: Applying/developing statistical methodology to answer an interesting scientific question in collaboration with a scientist or social scientist. Possible colloquium topics:   Any topics in statistical methodology and application, including but not limited to: topics in applied statistics, Bayesian methods, computational biology, statistical learning, “Big Data” mining, and other cross-disciplinary projects. Anna Neufeld Research interests:  My research is motivated by the gap between classical statistical tools and practical data analysis. Classic statistical tools are designed for testing a single hypothesis about a single, pre-specified model. However, modern data analysis is an adaptive process that involves exploring the data, fitting several models, evaluating these models, and then testing a potentially large number of hypotheses about one or more selected models. With this in mind, I am interested in topics such as (1) methods for model validation and selection, (2) methods for testing data-driven hypotheses (post-selection inference), and (3) methods for testing a large number of hypotheses. I am also interested in any applied project where I can help a scientist rigorously answer an important question using data.  Cross-validation for unsupervised learning. Cross-validation is one of the most widely-used tools for model validation, but, in its typical form, it cannot be used for unsupervised learning problems. Numerous ad-hoc proposals exist for validating unsupervised learning models, but there is a need to compare and contrast these proposals and work towards a unified approach. Identifying the number of cell types in single-cell genomics datasets. This is an application of the topic above, since the cell types are typically estimated via unsupervised learning. There is growing interest in “post-prediction inference”, which is the task of doing valid statistical inference when some inputs to your statistical model are the outputs of other statistical models (i.e. predictions). Frameworks have recently been proposed for post-prediction inference in the setting where you have access to a gold-standard dataset where the true inputs, rather than the predicted inputs, have been observed. A thesis could explore the possibility of post-prediction inference in the absence of this gold-standard dataset. Any other topic of student interest related to selective inference, multiple testing, or post-prediction inference. Any collaborative project in which we work with a scientist to identify an interesting question in need of non-standard statistics. I am open to advising colloquia in almost any area of statistical methodology or applications, including but not limited to: multiple testing, post-selection inference, post-prediction inference, model selection, model validation, statistical machine learning, unsupervised learning, or genomics. Allison Pacelli Research interests:   Math Education, Math & Politics, and Algebraic Number Theory. Math Education.  Math education is the study of the practice of teaching and learning mathematics, at all levels. For example, do high school calculus students learn best from lecture or inquiry-based learning? What mathematical content knowledge is critical for elementary school math teachers? Is a flipped classroom more effective than a traditional learning format? Many fascinating questions remain, at all levels of education. We can talk further to narrow down project ideas. Math & Politics.  The mathematics of voting and the mathematics of fair division are two fascinating topics in the field of mathematics and politics. Research questions look at types of voting systems, and the properties that we would want a voting system to satisfy, as well as the idea of fairness when splitting up a single object, like cake, or a collection of objects, such as after a divorce or a death. Algebraic Number Theory.  The Fundamental Theorem of Arithmetic states that the ring of integers is a unique factorization domain, that is, every integer can be uniquely factored into a product of primes. In other rings, there are analogues of prime numbers, but factorization into primes is not necessarily unique! In order to determine whether factorization into primes is unique in the ring of integers of a number field or function field, it is useful to study the associated class group – the group of equivalence classes of ideals. The class group is trivial if and only if the ring is a unique factorization domain. Although the study of class groups dates back to Gauss and played a key role in the history of Fermat’s Last Theorem, many basic questions remain open.   Possible thesis topics: Topics in math education, including projects at the elementary school level all the way through college level. Topics in voting and fair division. Investigating the divisibility of class numbers or the structure of the class group of quadratic fields and higher degree extensions. Exploring polynomial analogues of theorems from number theory concerning sums of powers, primes, divisibility, and arithmetic functions. Possible colloquium topics:   Anything in number theory, algebra, or math & politics. Anna Plantinga Research interests:   I am interested in both applied and methodological statistics. My research primarily involves problems related to statistical analysis within genetics, genomics, and in particular the human microbiome (the set of bacteria that live in and on a person).  Current areas of interest include longitudinal data, distance-based analysis methods such as kernel machine regression, high-dimensional data, and structured data. Impacts of microbiome volatility. Sometimes the variability of a microbial community is more indicative of an unhealthy community than the actual bacteria present. We have developed an approach to quantifying microbiome variability (“volatility”). This project will use extensive simulations to explore the impact of between-group differences in volatility on a variety of standard tests for association between the microbiome and a health outcome. Accounting for excess zeros (sparse feature matrices). Often in a data matrix with many zeros, some of the zeros are “true” or “structural” zeros, whereas others are simply there because we have fewer observations for some subjects. How we account for these zeros affects analysis results. Which methods to account for excess zeros perform best for different analyses? Longitudinal methods for compositional data. When we have longitudinal data, we assume the same variables are measured at every time point. For high-dimensional compositions, this may not be the case. We would generally assume that the missing component was absent at any time points for which it was not measured. This project will explore alternatives to making that assumption. Applied statistics research. In collaboration with a scientist or social scientist, use appropriate statistical methodology (or variations on existing methods) to answer an interesting scientific question. Any topics in statistical application, education, or methodology, including but not restricted to: Topics in applied statistics. Methods for microbiome data analysis. Statistical genetics. Electronic health records. Variable selection and statistical learning. Longitudinal methods. Cesar Silva Research interests :  Ergodic theory and measurable dynamics; in particular mixing properties and rank one examples, and infinite measure-preserving and nonsingular transformations and group actions.  Measurable dynamics of transformations defined on the p-adic field.  Measurable sensitivity.  Fractals.  Fractal Geometry. Possible thesis topics:    Ergodic Theory.   Ergodic theory studies the probabilistic behavior of abstract dynamical systems.  Dynamical systems are systems that change with time, such as the motion of the planets or of a pendulum.  Abstract dynamical systems represent the state of a dynamical system by a point in a mathematical space (phase space).  In many cases this space is assumed to be the unit interval [0,1) with Lebesgue measure.  One usually assumes that time is measured at discrete intervals and so the law of motion of the system is represented by a single map (or transformation) of the phase space [0,1).  In this case one studies various dynamical behaviors of these maps, such as ergodicity, weak mixing, and mixing.  I am also interested in studying the measurable dynamics of systems defined on the p-adics numbers.  The prerequisite is a first course in real analysis.  Topological Dynamics.  Dynamics on compact or locally compact spaces. Topics in mathematics and in particular: Any topic in measure theory.  See for example any of the first few chapters in “Measure and Category” by J. Oxtoby. Possible topics include the Banach-Tarski paradox, the Banach-Mazur game, Liouville numbers and s-Hausdorff measure zero. Topics in applied linear algebra and functional analysis. Fractal sets, fractal generation, image compression, and fractal dimension. Dynamics on the p-adic numbers. Banach-Tarski paradox, space filling curves. Mihai Stoiciu Research interests: Mathematical Physics and Functional Analysis. I am interested in the study of the spectral properties of various operators arising from mathematical physics – especially the Schrodinger operator. In particular, I am investigating the distribution of the eigenvalues for special classes of self-adjoint and unitary random matrices. Topics in mathematical physics, functional analysis and probability including: Investigate the spectrum of the Schrodinger operator. Possible research topics: Find good estimates for the number of bound states; Analyze the asymptotic growth of the number of bound states of the discrete Schrodinger operator at large coupling constants. Study particular classes of orthogonal polynomials on the unit circle. Investigate numerically the statistical distribution of the eigenvalues for various classes of random CMV matrices. Study the general theory of point processes and its applications to problems in mathematical physics. Possible colloquium topics:   Any topics in mathematics, mathematical physics, functional analysis, or probability, such as: The Schrodinger operator. Orthogonal polynomials on the unit circle. Statistical distribution of the eigenvalues of random matrices. The general theory of point processes and its applications to problems in mathematical physics. Elizabeth Upton Research Interests: My research interests center around network science, with a focus on regression methods for network-indexed data. Networks are used to capture the relationships between elements within a system. Examples include social networks, transportation networks, and biological networks. I also enjoy tackling problems with pragmatic applications and am therefore interested in applied interdisciplinary research. Regression models for network data: how can we incorporate network structure (and dependence) in our regression framework when modeling a vertex-indexed response? Identify effects shaping network structure. For example, in social networks, the phrase “birds of a feather flock together” is often used to describe homophily. That is, those who have similar interests are more likely to become friends. How can we capture or test this effect, and others, in a regression framework when modeling edge-indexed responses? Extending models for multilayer networks. Current methodologies combine edges from multiple networks in some sort of weighted averaging scheme. Could a penalized multivariate approach yield a more informative model? Developing algorithms to make inference on large networks more efficient. Any topic in linear or generalized linear modeling (including mixed-effects regression models, zero-inflated regressions, etc.). Applied statistics research. In collaboration with a scientist or social scientist, use appropriate statistical methodology to answer an interesting scientific question. Any applied statistics research project/paper Topics in linear or generalized linear modeling Network visualizations and statistics Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow , the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Q&A for work Connect and share knowledge within a single location that is structured and easy to search. How do mathematicians conduct research? I am curious as to how mathematicians conduct research. I hope some of you can help me solve this little mystery. To me, mathematics is a branch where you either get it or you don't. If you see the solution, then you've solved the problem, otherwise you will have to tackle it bit by bit. Exactly how this is done is elusive to me. Unlike physicists, chemists, engineers or even sociologists, I can't see where a mathematician (other than statisticians) gather their data from. Also, unlike the other professions mentioned above, it is not apparent that mathematicians perform any experiments. Additionally, a huge amount of work has already been laid down by other mathematicians, I wonder if there is a lot of "copy and pasting" as we see in software engineering (think of using other people's code) So my question is, where do mathematicians get their research topics from and how do they go about conducting research? What is considered acceptable progress in mathematics? research-process mathematics 29 I like the question just fine, but: you do realize that mathematics as an academic field is not uniquely characterized by a lack of data and experiments, right? In other words, you correctly point out that theoretical mathematics is not a science . There are other non-sciences too... –  Pete L. Clark Commented Dec 11, 2014 at 6:14 28 ff524: The NSF disagrees with me on the science thing, sometimes to the extent of putting money in my pocket. Nevertheless I think that everyone agrees that there is a sense in which (traditional, theoretical) mathematics is not a science: deductive versus inductive reasoning and all that. My point is that the OP seems to express wonderment about an academic field which lies largely outside of the scientific method. I agree and say: more amazing still, there are multiple fields like that. –  Pete L. Clark Commented Dec 11, 2014 at 6:27 12 I have the opposite problem: I don't understand how you can call collecting some numbers from nowhere and deducing some non-sense from them without giving proper evidence (proof) as conducting research :D </sarcasm> –  yo' Commented Dec 11, 2014 at 7:57 21 Legend has it that mathematics research consists of the following iterations coffee -> think -> coffee -> theorem -> coffee -> paper . Rinse and repeat. There may be more coffee steps involved but the general idea boils down to this (pun intended). –  Marc Claesen Commented Dec 11, 2014 at 11:21 14 You can find a video clip on YouTube of two characters from the show "The Big Bang Theory" acting like they are "doing research", set to the song "Eye of the Tiger". The characters are playing physicists, but the clip is frighteningly accurate for what much mathematical research looks like. –  Oswald Veblen Commented Dec 11, 2014 at 23:46 4 Answers 4 As far as pure mathematics, you are quite right: there are neither data nor experiments. Drastically oversimplified, a mathematics research project goes like this: Develop, or select from the existing literature, a mathematical statement ("conjecture") that you think will be of interest to other mathematicians, and whose truth or falsity is not known. (For example, "There are infinitely many pairs of prime numbers that differ by 2.") This is your problem . Construct a mathematical proof (or disproof) of this statement. See below. This is the solution of the problem. Write a paper explaining your proof, and submit it to a journal. Peer reviewers will decide whether your problem is interesting and whether your solution is logically correct. If so, it can be published, and the conjecture is now a theorem. The following discussion will make much more sense to anyone who has tried to write mathematical proofs at any level, but I'll try an analogy. A mathematical proof is often described as a chain of logical deductions, starting from something that is known (or generally agreed) to be true, and ending with the statement you are trying to prove. Each link must be a logical consequence of the one before it. For a very simple problem, a proof might have only one link: in that case one can often see the solution immediately. This would normally not be interesting enough to publish on its own, though mathematics papers typically contain several such results ("lemmas") used as intermediate steps on the way to something more interesting. So one is left to, as you say, "tackle it bit by bit". You construct the chain a link at a time. Maybe you start at the beginning (something that is already known to be true) and try to build toward the statement you want to prove. Maybe you go the other way: from the desired statement, work backward toward something that is known. Maybe you try to build free-standing lengths of chain in the middle and hope that you will later manage to link them together. You need a certain amount of experience and intuition to guess which direction you should direct your chain to eventually get it where it needs to go. There are generally lots of false starts and dead ends before you complete the chain. (If, indeed, you ever do. Maybe you just get completely stuck, abandon the project, and find a new one to work on. I suspect this happens to the vast majority of mathematics research projects that are ever started.) Of course, you want to take advantage of work already done by other people: using their theorems to justify steps in your proof. In an abstract sense, you are taking their chain and splicing it into your own. But in mathematics, as in software design, copy-and-paste is a poor methodology for code reuse. You don't repeat their proof; you just cite their paper and use their theorem. In the software analogy, you link your program against their library. You might also find a published theorem that doesn't prove exactly the piece you need, but whose proof can be adapted. So this sometimes turns into the equivalent of copying and pasting someone else's code (giving them due credit, of course) but changing a few lines where needed. More often the changes are more extensive and your version ends up looking like a reimplementation from scratch, which now supports the necessary extra features. "Acceptable progress" is quite subjective and usually based on how interesting or useful your theorem is, compared to the existing body of knowledge. In some cases, a theorem that looks like a very slight improvement on something previously known can be a huge breakthrough. In other cases, a theorem could have all sorts of new results, but maybe they are not useful for proving further theorems that anyone finds interesting, and so nobody cares. Now, through this whole process, here is what an outside observer actually sees you doing: Search for books and papers. Stare into space for a while. Scribble inscrutable symbols on a chalkboard. (The symbols themselves are usually meaningful to other mathematicians, but at any given moment, the context in which they make sense may exist only in your head.) Scribble similar inscrutable symbols on paper. Use LaTeX to produce beautifully-typeset inscrutable symbols interspersed with incomprehensible technical terms, connected by lots of "therefore"s and "hence"s. Loop until done. Submit said beautifully-typeset gibberish to a journal. Apply for funding. Attend a conference, where you speak unintelligibly about your gibberish, and listen to others do the same about theirs. Loop until emeritus, or perhaps until dead ( in the sense of Erdős ). 6 What's interesting is that sometimes in the course of proving something, you might invent an entirely new kind of mathematics, which in turn winds up being useful for other purposes. This is very loosely analogous to inventing new programming languages for the purpose of more efficiently expressing your intention and hence developing things more quickly. Many of the names of our everyday mathematical abstractions come from the names of the living, breathing people who spent their lives constructing and refining them. –  Dan Bryant Commented Dec 11, 2014 at 16:21 15 In my own experience, it's not that common to begin with a specific problem. More often, I begin with a feeling that something I've read or heard about could be done more elegantly or more clearly. My initial goal is then just to understand better what someone else has done, but if I can really achieve a better understanding, then that often suggests improvements or generalizations of that work. Indeed, it sometimes makes such improvements obvious. If the improvement is big enough, it can constitute a paper; if not, it can sometimes become part of a paper, or of a talk. –  Andreas Blass Commented Dec 11, 2014 at 21:23 4 Rather than starting with a conjecture (although I sometimes do that), I more often start with an idea: some specific thing that I'd like to understand. This is based on my intuition about what problems seem likely to have interesting results. As I work through the thing I am studying, I come up with specific conjectures and theorems. But the beginning of the project rarely has specific conjectures, just goals. –  Oswald Veblen Commented Dec 11, 2014 at 23:31 6 You forgot "meet with a colleague, stare at a blackboard together and argue passionately on which definition looks the most beautiful". Pretty accurate nevertheless. –  Federico Poloni Commented Dec 13, 2014 at 17:04 2 @Jack: The goal of pure mathematics research at any level is as I described: to be able to prove or disprove statements whose truth or falsity was not previously known. At the undergraduate level, it often begins with computations (by hand or computer) to try to evaluate whether a conjecture is plausible, and sometimes it doesn't get any further than that. There will also be a lot more interaction with an advisor. –  Nate Eldredge Commented Apr 4, 2015 at 15:10 Actually, even in pure mathematics, it very often is possible to do experiments of a sort. It's very common to come up with a hypothesis that seems plausible but you're not sure if it's true or not. If it's true, proving that is probably quite a lot of work; if it's false, proving that could be quite a lot of work, too. But, if it's true, trying to prove that it's false is a huge amount of work! Before you invest a lot of effort into trying to prove the wrong direction, it's good to gain some intuition about the situation and whether the statement seems more likely to be true or to be false. Computers can be very useful for this kind of thing: you can generate lots of examples and see if they satisfy your hypothesis. If they do, you might try to prove your hypothesis is true; if they don't, you might try to refine your hypothesis by adding more conditions to it. See also Oswald Veblen's answer which talks about doing similar "experiments" by hand. 8 I "do experiments" by working out conjectures in the context of specific examples. If the conjecture works out in several examples, that makes me more confident that it may be true in general. –  Oswald Veblen Commented Dec 11, 2014 at 23:42 I "gather data" and perform experiments" by working out my conjectures in the context of specific examples. If the conjecture works out in several examples, that makes me more confident that it may be true in general. For example, suppose that I think that every topological space of a certain form has a particular property. I will start by looking at some "simple" spaces, like the real line, and see if they have the property. If they do, I may look at some more complicated space. Often, when I look at what specific attributes of the examples were necessary to show they had the property in question, it tells me what hypotheses I need to add to make my conjecture into a theorem. This is not the same as scientific experimentation, nor the same as computer experimentation, which is also important in various areas of mathematics. But it is its own form of experimentation, nevertheless. 15 I think this is an important answer (especially in light of my comments above). From a philosophy of science standpoint, one must be clear that theoretical mathematics does not follow the scientific method. However, an important part of what mathematicians do in practice bears a lot of similarity to scientific experimentation. As a result, mathematical research has a similar flavor to scientific research in many respects. (There are other academic fields in which one really doesn't do experiments in any sense: philosophy, literature, law...) –  Pete L. Clark Commented Dec 12, 2014 at 3:40 One point to note is that, for some questions, it is possible to do experiments to get data. Certain questions are things we now have computer programs to generate, and previously they could have been done on a far more limited scale by hand. So in some cases mathematicians do work more like experimental scientists. On the other hand, once they've found what seems to be a pattern, they change approach. Gathering further examples isn't much use (unless you then find a counter-example, but it can be encouraging) - you need to find an actual proof. More generally, nearly every big result will come from some 'experiments': you try special cases, cases with more hypotheses, extreme cases that might result in failures... On the 'copy-and-paste' point, mathematicians do use a lot of what other people have done (generally they must), but whereas you might copy someone's code to use it, when you cite a theorem you don't need to copy out the proof. So in terms of written space in a paper, the 'copied' section is very small. There are (fairly large) exceptions to this: fairly often a proof someone has given is very close to what you need, but not quite good enough, because you want to use it for something different to what they did. So you may end up writing out something very similar, but with your own subtle tweaks. I guess you could see this as like adjusting someone else's machine (we call things machines too, but here I mean a physical one). The difference is that generally in order to do this sort of thing you must completely understand what the machine does. Another big reason for 'copying' is that you may need (for actual theoretical reasons or for expositional ones) to build on the actual workings of the machine, not just on the output it gives. More to the point of the question: As a mathematician, you generally read, and aim to understand, what other people have done. That gives you a bank of tools you can use - results (which you may or may or may not be completely able to prove yourself), and methods that have worked in the past. You build up an idea of things that tend to work, and how to adapt things slightly to work in similar situations. You do a fair amount of trial and error - you try something, but realise you get stuck at some point. Then you try and understand why you are stuck, and if there's a way round. You try proving the opposite to what you want, and see where you get stuck (or don't!). Once you have a working proof, you see whether there are closely related things you can/can't prove. What happens if you remove/change a hypothesis? Also, does the reverse statement hold? If not entirely, are there some cases in which it does? Can you give examples to show your result is as good as possible? Can you combine it with other things you know about? Another source of questions is what other people are interested in. Sometimes you know how to do something they want doing, but you didn't think of it until they asked. One more point I'd like to make in the 'methods of proof category' is that, for me at least, there's a degree to which I work by 'feel'. You know those puzzles where all the pieces seem to be jammed in place but you're meant to take them apart (and put them back together again)? You sort of play around until you feel a bit that's looser than the rest, right? Sometimes proofs are a bit like that. When you understand something well, you can 'feel' where things are wedged tight and where they are looser. Sometimes you also hope that lightning (inspiration) will strike. Occasionally it does. (All of this may not exactly answer the question, but hopefully it gives some insight.) 3 "whereas you might copy someone's code to use it, when you cite a theorem you don't need to copy out the proof" - and when you call someone else's function, you don't need to copy the source. If you're copying the source, that's a bad sign. –  user2357112 Commented Dec 11, 2014 at 8:57 2 @user2357112 or a sign that they don't provide a library, just an integrated implementation; or that the full library has too many requirements, or does not compile on your system. Seriously, in academic code you can usually find truly horrific things, and just copying the body of a function is one of the least abhorrent things. –  Davidmh Commented Dec 11, 2014 at 9:34 You must log in to answer this question. Not the answer you're looking for browse other questions tagged research-process mathematics .. Featured on Meta Bringing clarity to status tag usage on meta sites We've made changes to our Terms of Service & Privacy Policy - July 2024 Announcing a change to the data-dump process Hot Network Questions When was this photo taken? Why are AAVs not a replacement for through-roof vent pipe? How can these humans cross the ocean(s) at the first possible chance? Is the theory of ordinals in Cantor normal form with just addition decidable? What is it called when perception of a thing is replaced by an pre-existing abstraction of that thing? Is it possible for a fuse to blow at extremely low temperatures? Purpose of burn permit? How to justify our beliefs so that it is not circular? Do metal objects attract lightning? Is く in 驚く here not just devoiced but completely removed? Post apocalyptic horror/thriller movie set in a building with puzzle rooms and killer dogs The last person in Daniel 11. Who is it? What does "garb" mean in this passage? Submitting a paper as a nonacademic practitioner in a field How much missing data is too much (part 2)? statistical power, effective sample size How to assess whether it is imposter syndrome or actual low quality work during Ph.D.? Can I use "historically" to mean "for a long time" in "Historically, the Japanese were almost vegetarian"? Can a rope thrower act as a propulsion method for land based craft? What is the spiritual difference between hungering and thirsting? (Matthew 5:6) Is there any video of an air-to-air missile shooting down an aircraft? Which cards use −5 V and −12 V in IBM PC compatible systems? On a 3D Gagliardo-Nirenberg inequality `Drop` for list of elements of different dimensions What are the risks of a compromised top tube and of attempts to repair it? Applied Mathematics Research In applied mathematics, we look for important connections with other disciplines that may inspire interesting and useful mathematics, and where innovative mathematical reasoning may lead to new insights and applications. Applied Mathematics Fields Combinatorics Computational Biology Physical Applied Mathematics Computational Science & Numerical Analysis Theoretical Computer Science Mathematics of Data Applied Math Committee An official website of the United States government Here's how you know Official websites use .gov A .gov website belongs to an official government organization in the United States. Secure .gov websites use HTTPS. A lock ( Lock Locked padlock ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites. 'Quantum coherence' survives in ultracold molecules Scientists are hot on the trail of a process called quantum coherence, the ability of particles to maintain phase relationships and exist in multiple states simultaneously. It's akin to the parts of a wave being synchronized. But whether quantum coherence can persist through a chemical reaction, where bonds break and form, has been questioned.   Scientists have demonstrated that quantum coherence can survive in a chemical reaction with ultracold molecules. Researchers' findings highlight the potential of harnessing chemical reactions for applications in quantum information science. The work was published in the journal Science and funded by the U.S. National Science Foundation .   Scientists are using sophisticated laser techniques to research quantum entanglement between the states of a chemical reaction. Quantum entanglement is a key concept at the heart of quantum information science, whereby two particles can occupy a shared quantum state.    "Our findings precede the study of coherence in reactions under wet and warm conditions, which may be of interest for a wide range of chemical phenomena, including in the brain," write Kang-Kuen Ni of Harvard University and co-authors.   This story is based on an article published by Harvard University. Read the full story: Finding quantum order in chaos Research areas Information Author Services Initiatives You are accessing a machine-readable page. In order to be human-readable, please install an RSS reader. All articles published by MDPI are made immediately available worldwide under an open access license. No special permission is required to reuse all or part of the article published by MDPI, including figures and tables. For articles published under an open access Creative Common CC BY license, any part of the article may be reused without permission provided that the original article is clearly cited. For more information, please refer to https://www.mdpi.com/openaccess . Feature papers represent the most advanced research with significant potential for high impact in the field. A Feature Paper should be a substantial original Article that involves several techniques or approaches, provides an outlook for future research directions and describes possible research applications. Feature papers are submitted upon individual invitation or recommendation by the scientific editors and must receive positive feedback from the reviewers. Editor’s Choice articles are based on recommendations by the scientific editors of MDPI journals from around the world. Editors select a small number of articles recently published in the journal that they believe will be particularly interesting to readers, or important in the respective research area. The aim is to provide a snapshot of some of the most exciting work published in the various research areas of the journal. Original Submission Date Received: . Active Journals Find a Journal Proceedings Series For Authors For Reviewers For Editors For Librarians For Publishers For Societies For Conference Organizers Open Access Policy Institutional Open Access Program Special Issues Guidelines Editorial Process Research and Publication Ethics Article Processing Charges Testimonials Preprints.org SciProfiles Encyclopedia Article Menu Subscribe SciFeed Recommended Articles Google Scholar on Google Scholar Table of Contents Find support for a specific problem in the support section of our website. Please let us know what you think of our products and services. Visit our dedicated information section to learn more about MDPI. JSmol Viewer Empirical research on ai technology-supported precision teaching in high school science subjects, 1. introduction, 1.1. development and application of precision teaching, 1.2. the present study, 2. precision teaching model supported by ai technology. Click here to enlarge figure 2.1. Teachers and Parents: Precision Teaching and Precision Intervention Supported by Formative Assessment 2.1.1. learning preview, 2.1.2. classroom interaction, 2.1.3. learning report, 2.1.4. stage report, 2.2. students: personalized learning and individual development supported by intelligent technology systems, 2.2.1. pre-class study, 2.2.2. homework, 2.2.3. practice, 2.2.4. exams, 2.2.5. error logbook, 2.3. examples of pedagogical models in use, 3.1. procedure and sample, 3.2. measures, 3.2.1. midterm examination papers, 3.2.2. self-directed learning report, 3.2.3. teacher emotional attitude survey questionnaire ( questionnaire s1 ), 3.3. data analysis, 4.1. results of t-test, 4.2. results of regression analysis, 4.3. results of correlation analysis, 4.4. results of descriptive analysis, 5. discussion, 5.1. measures 1 and 2, 5.2. measure 3, 5.3. limitations for research, 6. conclusions, supplementary materials, author contributions, institutional review board statement, informed consent statement, data availability statement, conflicts of interest. Schmitz, M.-L.; Antonietti, C.; Cattaneo, A.; Gonon, P.; Petko, D. When barriers are not an issue: Tracing the relationship between hindering factors and technology use in secondary schools across Europe. Comput. Educ. 2022 , 179 , 104411. [ Google Scholar ] [ CrossRef ] European Commission. European Commission 2nd Survey of Schools–ICT in Education–Objective 1–Benchmark Progress in ICT in Schools, Final Report ; Publications Office: Luxembourg, 2019. [ Google Scholar ] European Commission. EU European Commission Survey of Schools–ICT in Education–Benchmarking Access, Use and Attitudes to Technology in Europe’s Schools ; Publications Office of the European Union: Luxembourg, 2013. [ Google Scholar ] Zhan, Z.; Tong, Y.; Lan, X.; Zhong, B. A systematic literature review of game-based learning in Artificial Intelligence education. Interact. Learn. Environ. 2024 , 32 , 1137–1158. [ Google Scholar ] [ CrossRef ] Park, W.; Kwon, H. Implementing artificial intelligence education for middle school technology education in Republic of Korea. Int. J. Technol. Des. Educ. 2024 , 34 , 109–135. [ Google Scholar ] [ CrossRef ] Cook, C.R.; Kilgus, S.P.; Burns, M.K. Advancing the science and practice of precision education to enhance student outcomes. J. Sch. Psychol. 2018 , 66 , 4–10. [ Google Scholar ] [ CrossRef ] Hwang, G.-J.; Xie, H.; Wah, B.W.; Gašević, D. Vision, challenges, roles and research issues of Artificial Intelligence in Education. Comput. Educ. Artif. Intell. 2020 , 1 , 100001. [ Google Scholar ] [ CrossRef ] Guan, C.; Mou, J.; Jiang, Z. Artificial intelligence innovation in education: A twenty-year data-driven historical analysis. Int. J. Innov. Stud. 2020 , 4 , 134–147. [ Google Scholar ] [ CrossRef ] Tsai, S.-C.; Chen, C.-H.; Shiao, Y.-T.; Ciou, J.-S.; Wu, T.-N. Precision education with statistical learning and deep learning: A case study in Taiwan. Int. J. Educ. Technol. High. Educ. 2020 , 17 , 12. [ Google Scholar ] [ CrossRef ] Lu, O.H.; Huang, A.Y.; Huang, J.C.; Lin, A.J.; Ogata, H.; Yang, S.J. Applying Learning Analytics for the Early Prediction of Students’ Academic Performance in Blended Learning. J. Educ. Technol. Soc. 2018 , 21 , 220–232. [ Google Scholar ] Forero-Corba, W.; Bennasar, F.N. Techniques and Applications of Machine Learning and Artificial Intelligence in Education: A Systematic Review. RIED-Rev. Iberoam. Educ. Distancia 2024 , 27 , 1–19. [ Google Scholar ] Deepika, A.; Kandakatla, R.; Saida, A.; Reddy, V.B. Implementation of ICAP Principles through Technology Tools: Exploring the Alignment between Pedagogy and Technology. J. Eng. Educ. Transform. 2021 , 34 , 542. [ Google Scholar ] [ CrossRef ] Hew, K.F.; Lan, M.; Tang, Y.; Jia, C.; Lo, C.K. Where is the “theory” within the field of educational technology research? Br. J. Educ. Technol. 2019 , 50 , 956–971. [ Google Scholar ] [ CrossRef ] Chen, X.; Xie, H.; Zou, D.; Hwang, G.-J. Application and theory gaps during the rise of Artificial Intelligence in Education. Comput. Educ. Artif. Intell. 2020 , 1 , 100002. [ Google Scholar ] [ CrossRef ] Schunk, D.H. Learning Theories an Educational Perspective , 8th ed.; Pearson Education, Inc.: London, UK, 2020. [ Google Scholar ] Sønderlund, A.L.; Hughes, E.; Smith, J. The efficacy of learning analytics interventions in higher education: A systematic review. Br. J. Educ. Technol. 2019 , 50 , 2594–2618. [ Google Scholar ] [ CrossRef ] Viberg, O.; Hatakka, M.; Bälter, O.; Mavroudi, A. The Current Landscape of Learning Analytics in Higher Education. Comput. Hum. Behav. 2018 , 89 , 98–110. [ Google Scholar ] [ CrossRef ] Luan, H.; Tsai, C.-C. A Review of Using Machine Learning Approaches for Precision Education. Educ. Technol. Soc. 2021 , 24 , 250–266. [ Google Scholar ] Shan, S.; Liu, Y. Blended Teaching Design of College Students’ Mental Health Education Course Based on Artificial Intelligence Flipped Class. Math. Probl. Eng. 2021 , 2021 , 1–10. [ Google Scholar ] [ CrossRef ] Dong, X. Application of Precision Teaching Under the Guidance of Big Data in The Course of Internal Medicine Nursing. Front. Bus. Econ. Manag. 2022 , 5 , 37–39. [ Google Scholar ] [ CrossRef ] Wei, X.; Jiang, J.; Zhang, L.; Feng, H. Research on Precision Teaching Management Methods in Universities in the Era of Big Data Based on Entropy Weight Method. In Frontiers in Artificial Intelligence and Applications ; Grigoras, G., Lorenz, P., Eds.; IOS Press: Amsterdam, The Netherlands, 2023; ISBN 978-1-64368-444-4. [ Google Scholar ] Yanfei, M. Online and Offline Mixed Intelligent Teaching Assistant Mode of English Based on Mobile Information System. Mob. Inf. Syst. 2021 , 2021 , 7074629. [ Google Scholar ] [ CrossRef ] Wang, Y.; Xiao, L.; Mo, S.; Shen, Y.; Tong, G. Research on the Effectiveness of Precision Teaching Model Empowered by e-Schoolbag—A Case Study of Mathematics Review Lessons in Junior High School. China Educ. Technol. 2019 , 5 , 106–113+119. Available online: https://qikan.cqvip.com/Qikan/Article/Detail?id=7002036138 (accessed on 20 August 2024). Lindsley, O.R. Precision teaching: Discoveries and effects. J. Appl. Behav. Anal. 1992 , 25 , 51–57. [ Google Scholar ] [ CrossRef ] Kubina, R.M.; Yurich, K.K. Precision Teaching Book ; Greatness Achieved Publishing Company Lemont: Pittsburgh, PA, USA, 2012; ISBN 0-615-55420-2. [ Google Scholar ] Yin, B.; Yuan, C.-H. Precision Teaching and Learning Performance in a Blended Learning Environment. Front. Psychol. 2021 , 12 , 631125. [ Google Scholar ] [ CrossRef ] Binder, C.; Watkins, C.L. Precision Teaching and Direct Instruction: Measurably Superior Instructional Technology in Schools. Perform. Improv. Q. 1990 , 3 , 74–96. [ Google Scholar ] [ CrossRef ] Hughes, J.C.; Beverley, M.; Whitehead, J. Using precision teaching to increase the fluency of word reading with problem readers. Eur. J. Behav. Anal. 2007 , 8 , 221–238. [ Google Scholar ] [ CrossRef ] Liu, C.; Zhang, L. Research Focuses and Future Directions of Precision Teaching in China: A Visualized Analysis Based on CiteSpace. J. Suzhou Vocat. Univ. 2023 , 34 , 72–78. [ Google Scholar ] Guo, L.; Yang, X.; Zhang, Y. Analysis on New Development and Value Orientation of Precision Teaching in the Era of Big Data. E-Educ. Res. 2019 , 40 , 76–81+88. [ Google Scholar ] Yang, X.; Luo, J.; Liu, Y.; Chen, S. Data-Driven Instruction: A New Trend of Teaching Paradigm in Big Data Era. E-Educ. Res. 2017 , 38 , 13–20+26. [ Google Scholar ] Zhang, X.; Mou, Z. The Research on the Design of Precise Instruction Model Facing Personalized Learning under the Data Learning Environment. Mod. Distance Educ. 2018 , 5 , 65–72. Available online: https://qikan.cqvip.com/Qikan/Article/Detail?id=676261576 (accessed on 20 August 2024). Yang, Z.; Wang, J.; Wu, D.; Wang, M. Developing Intelligent Education to Promote Sustainable Development of Education. E-Educ. Res. 2022 , 43 , 5–10+17. [ Google Scholar ] Shemshack, A.; Spector, J.M. A systematic literature review of personalized learning terms. Smart Learn. Environ. 2020 , 7 , 1–20. [ Google Scholar ] [ CrossRef ] Gallagher, E. Improving a mathematical key skill using precision teaching. Ir. Educ. Stud. 2006 , 25 , 303–319. [ Google Scholar ] [ CrossRef ] Strømgren, B.; Berg-Mortensen, C.; Tangen, L. The Use of Precision Teaching to Teach Basic Math Facts. Eur. J. Behav. Anal. 2014 , 15 , 225–240. [ Google Scholar ] [ CrossRef ] Gist, C.; Bulla, A.J. A Systematic Review of Frequency Building and Precision Teaching with School-Aged Children. J. Behav. Educ. 2022 , 31 , 43–68. [ Google Scholar ] [ CrossRef ] Yang, S.J.H. Precision Education: New Challenges for AI in Education [Conference Keynote]. In Proceedings of the 27th International Conference on Computers in Education (ICCE), Kenting, Taiwan, 2–6 December 2019; Asia-Pacific Society for Computers in Education (APSCE): Taoyuan City, Taiwan, 2019; pp. XXVII–XXVIII. [ Google Scholar ] Peng, X.; Wu, B. How Is Data-Driven Precision Teaching Possible?From the Perspective of Cultivating Teacher’s Data Wisdom. J. East China Norm. Univ. Sci. 2021 , 39 , 45–56. [ Google Scholar ] Taber, K.S. Mediated Learning Leading Development—The Social Development Theory of Lev Vygotsky. In Science Education in Theory and Practice: An Introductory Guide to Learning Theory ; Springer: Cham, Switzerland, 2020; pp. 277–291. [ Google Scholar ] Ness, I.J. Zone of Proximal Development. In The Palgrave Encyclopedia of the Possible ; Springer: Berlin/Heidelberg, Germany, 2023; pp. 1781–1786. [ Google Scholar ] Liu, N.; Yu, S. Research on Precision Teaching Based on Zone of Proximal Development. E-Educ. Res. 2020 , 41 , 77–85. [ Google Scholar ] Liu, H.; Sun, J.; Chen, J.; Zhang, Y. Persona Model and Its Application in Library. Libr. Theory Pract. 2018 , 92 , 97. Available online: https://qikan.cqvip.com/Qikan/Article/Detail?id=7000905917 (accessed on 20 August 2024). Liu, H.; Sun, J.; Su, Y.; Zhang, Y. A Multi Contextual Interest Recommender Method for Library Big Data Knowledge Service. J. Mod. Inf. 2018 , 38 , 62–67,156. [ Google Scholar ] Liu, H.; Sun, J.; Su, Y.; Zhang, Y. Research on the Tourism Situational Recommendation Service Based on Persona. Inf. Stud. Theory Appl. 2018 , 41 , 87–92. [ Google Scholar ] Liu, H. Contextual Recommendation for the Big Data Knowledge Service Oriented the Cloud Computing. Libr. Dev. 2014 , 31–35. Available online: https://qikan.cqvip.com/Qikan/Article/Detail?id=661733950 (accessed on 20 August 2024). Liu, H.; Liu, X.; Yao, S.; Xie, S. Statistical Analysis of Information Behavior Characteristics of Online Social Users Based on Public Opinion Portrait. J. Mod. Inf. 2019 , 39 , 64–73. [ Google Scholar ] Liu, H.; Sun, J.; Zhang, Y.; Zhao, P. Research on User Portrayal and Information Dissemination Behavior in Online Social Activities. Inf. Sci. 2018 , 36 , 17–21. [ Google Scholar ] Liu, H.; Huang, W.; Xie, S. Research on the Situational Recommendation-Oriented Library User Profiles. Res. Libr. Sci. 2018 , 62–68. Available online: https://qikan.cqvip.com/Qikan/Article/Detail?id=676852789 (accessed on 20 August 2024). Erümit, A.K.; Çetin, I. Design framework of adaptive intelligent tutoring systems. Educ. Inf. Technol. 2020 , 25 , 4477–4500. [ Google Scholar ] [ CrossRef ] U.S. Department of Education, Office of Educational Technology. Transforming American Education: Learning Powered by Technology ; U.S. Department of Education, Office of Educational Technology: Washington, DC, USA, 2010. [ Google Scholar ] Fei, L.; Ma, Y. Developing Personalized Learning to Promote Educational Equity: An Exploration of the Basic Theory and Practical Experience of Personalized Learning in the UK. Glob. Educ. 2010 , 39 , 42–46. Available online: https://qikan.cqvip.com/Qikan/Article/Detail?id=34931923 (accessed on 20 August 2024). Yu, S. Internet Plus Education: Future Schools ; Publishing House of Electronics Industry: Beijing, China, 2019; ISBN 978-7-121-36043-5. [ Google Scholar ] Luan, H.; Geczy, P.; Lai, H.; Gobert, J.; Yang, S.J.; Ogata, H.; Baltes, J.; Guerra, R.; Li, P.; Tsai, C.-C. Challenges and Future Directions of Big Data and Artificial Intelligence in Education. Front. Psychol. 2020 , 11 , 580820. [ Google Scholar ] [ CrossRef ] Bray, B.; McClaskey, K. Personalization vs. Differentiation vs Individualization. Dostopno Na Httpeducation Ky Govschool-innovDocumentsBB-KM-Pers.-2012 Pdf Pridobljeno 12 10 2013 2012. Available online: https://www.marshfieldschools.org/cms/lib/WI01919828/Centricity/Domain/82/PL_Diff_Indiv.pdf (accessed on 20 August 2024). Li, y.; Zhang, S. Self-study and Adaptive Adjusting of Exam-question Difficulty Coefficient. Comput. Eng. 2005 , 31 , 181–182. [ Google Scholar ] Lourdusamy, R.; Magendiran, P. A systematic analysis of difficulty level of the question paper using student’s marks: A case study. Int. J. Inf. Technol. 2021 , 13 , 1127–1143. [ Google Scholar ] [ CrossRef ] Peng, J.; Sun, M.; Yuan, B.; Lim, C.P.; van Merriënboer, J.J.G.; Wang, M. Visible thinking to support online project-based learning: Narrowing the achievement gap between high- and low-achieving students. Educ. Inf. Technol. 2024 , 29 , 2329–2363. [ Google Scholar ] [ CrossRef ] Chiesa, M.; Robertson, A. Precision Teaching and Fluency Training: Making maths easier for pupils and teachers. Educ. Psychol. Pract. 2000 , 16 , 297–310. [ Google Scholar ] [ CrossRef ] Yang, Z. Empowering Teaching and Learning with Artificial Intelligence. Front. Digit. Educ. 2024 , 1 , 1–3. [ Google Scholar ] Subject Exam Type Total Number of Participants Full Score Maximum Value Minimum Value Mean Value Standard Deviation Test Difficulty M Pre-test 545 150 122 5 55.77 20.78 0.37 Post-test 530 150 148 10 69.86 27.58 0.46 P Pre-test 545 100 97 6 51.52 20.47 0.52 Post-test 531 100 100 4 48.20 21.68 0.48 C Pre-test 547 100 96 9 53.48 19.31 0.53 Post-test 530 100 98 5 58.82 24.04 0.58 B Pre-test 547 100 94 16 64.13 16.53 0.64 Post-test 531 100 93 14 55.95 14.49 0.56 Class Pre-Test M Post-Test M Pre-Test P Post-Test P Pre-Test C Post-Test C Pre-Test B Post-Test B Pre-Test Total Score Post-Test Total Score Difference from Grade Average Total Score (Pre-Test) Difference from Grade Average Total Score (Post-Test) 1 80.46 103.00 78.00 73.11 76.13 80.32 80.89 70.08 315.48 326.51 86.08 94.57 2 80.79 103.41 77.62 72.59 77.31 78.80 82.62 73.00 318.34 327.8 88.94 95.86 3 51.19 81.67 50.95 54.17 57.79 59.03 67.26 60.13 227.19 255 −2.21 23.06 4 53.63 79.03 44.48 50.23 50.63 56.28 61.84 56.95 210.58 242.49 −18.82 10.55 5 63.86 80.97 60.74 60.97 63.41 67.95 73.66 69.61 261.67 279.5 32.27 42.00 6 65.09 78.32 59.6 51.63 62.38 64.75 66.02 67.62 253.09 262.32 24.58 24.82 7 39.61 50.46 34.76 26.02 36.05 29.23 47.39 41.21 157.81 146.92 −71.59 −85.02 8 37.93 44.02 31.19 27.51 32.82 24.37 49.98 43.61 151.92 139.51 −77.84 −92.43 9 38.5 52.97 36.25 31.48 35.22 28.97 50.58 44 160.55 157.42 −68.85 −74.52 Grade Level 56.78 74.87 52.62 49.75 54.64 54.41 64.47 58.47 228.51 237.50 0 0 Subject Homework Completion Rate Similar Questions Completed Count Personalized Exercises Completed Count M Y = 0.0031 × X + 9.663 Y = −0.5400 × X + 30.81 Y = 0.0167 × X + 4.917 P Y = −0.1662 × X + 95.13 Y = 1.277 × X + 21.50 Y = −0.0298 × X + 3.857 C Y = 0.4216 × X + 95.66 Y = 4.283 × X + 5.579 Y = 2.174 × X−2.325 B Y = −0.2373 × X + 94.84 Y = 1.306 × X + 35.34 Y = −0.4493 × X + 21.37 Subject Similar Questions Completed Count Personalized Exercises Completed Count M Y = 0.084 × X + 24.28 Y = 0.047 × X + 17.85 P Y = 0.020 × X + 131.6 Y = 0.007 × X + 4.82 C Y = 0.2111 × X + 27.84 Y = 0.0190 × X + 35.92 B Y = 0.024 × X + 43.20 Y = −0.124 × X + 176.2 Questions Options and Answers Based on your teaching needs, do you think the pre-class study report is helpful for your teaching? Yes: 15 (78.95%) No: 0 (0%) Not very helpful: 4 (21.05%) Are you satisfied with the types of homework provided by the AI learning system, or do you have any suggestions? Satisfied: 7 (36.84%) Dissatisfied: 0 (0%) It is okay: 11 (57.89%) Other Suggestions: 1 (5.26%) Are you satisfied with the difficulty level of the homework provided by the AI learning system, or do you have any suggestions? Satisfied: 7 (36.84%) Dissatisfied: 0 (0%) It is okay: 12 (63.16%) Other Suggestions: 0 (0%) Are you satisfied with the homework grading provided by the AI learning system, or do you have any suggestions? Satisfied: 9 (47.37%) Dissatisfied: 0 (0%) It is okay: 10 (52.63%) Other Suggestions: 0 (0%) Does the collection period and source of incorrect questions in the AI teaching class meet the teaching requirements? Satisfied: 5 (26.32%) Not Satisfied: 0 (0%) It is okay: 13 (68.42%) Other Suggestions: 1 (5.26%) The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. Share and Cite Hao, M.; Wang, Y.; Peng, J. Empirical Research on AI Technology-Supported Precision Teaching in High School Science Subjects. Appl. Sci. 2024 , 14 , 7544. https://doi.org/10.3390/app14177544 Hao M, Wang Y, Peng J. Empirical Research on AI Technology-Supported Precision Teaching in High School Science Subjects. Applied Sciences . 2024; 14(17):7544. https://doi.org/10.3390/app14177544 Hao, Miaomiao, Yi Wang, and Jun Peng. 2024. "Empirical Research on AI Technology-Supported Precision Teaching in High School Science Subjects" Applied Sciences 14, no. 17: 7544. https://doi.org/10.3390/app14177544 Article Metrics Article access statistics, supplementary material. ZIP-Document (ZIP, 71 KiB) Further Information Mdpi initiatives, follow mdpi. Subscribe to receive issue release notifications and newsletters from MDPI journals 211 COMMENTS 251+ Math Research Topics [2024 Updated] 251+ Math Research Topics [2024 Updated] General / By admin / 2nd March 2024. Mathematics, often dubbed as the language of the universe, holds immense significance in shaping our understanding of the world around us. It's not just about crunching numbers or solving equations; it's about unraveling mysteries, making predictions, and creating ... 181 Math Research Topics If you are an undergraduate looking for some research topics for your next math paper, you will surely appreciate our list of interesting undergraduate math research topics: Methods to count discrete objects. The origins of Greek symbols in mathematics. Methods to solve simultaneous equations. Real-world applications of the theorem of Pythagoras. Research Areas Applied Math. Applied mathematics at the Stanford Department of Mathematics focuses, very broadly, on the areas of scientific computing, stochastic modeling, and applied analysis. Research Areas Department members engage in cutting-edge research on a wide variety of topics in mathematics and its applications. Topics continually evolve to reflect emerging interests and developments, but can roughly grouped into the following areas. Algebra, Combinatorics, and Geometry Algebra, combinatorics, and geometry are areas of very active research at the University of Pittsburgh. Research Areas Research Areas. Ranked among the top 20 math graduate programs by U.S. News & World Report, our faculty conduct more than $3.7 million in research each year for industry, the Department of Defense, the National Science Foundation, and the National Institutes of Health. Our faculty of 35 includes three National Academy of Science members and two ... Mathematics Theses, Projects, and Dissertations bio-mathematics: introduction to the mathematical model of the hepatitis c virus, lucille j. durfee. pdf. analysis and synthesis of the literature regarding active and direct instruction and their promotion of flexible thinking in mathematics, genelle elizabeth gonzalez. pdf. life expectancy, ali r. hassanzadah. pdf Guide To Graduate Study Guide to Graduate Studies. The PhD Program. The Ph.D. program of the Harvard Department of Mathematics is designed to help motivated students develop their understanding and enjoyment of mathematics. Enjoyment and understanding of the subject, as well as enthusiasm in teaching it, are greater when one is actively thinking about mathematics in ... Applied Mathematics Applied Mathematics. Faculty and students interested in the applications of mathematics are an integral part of the Department of Mathematics; there is no formal separation between pure and applied mathematics, and the Department takes pride in the many ways in which they enrich each other. We also benefit tremendously from close collaborations ... Pure Mathematics Research Department of Mathematics. Headquarters Office. Simons Building (Building 2), Room 106. 77 Massachusetts Avenue. Cambridge, MA 02139-4307. Campus Map. (617) 253-4381. Website Questions: [email protected]. Undergraduate Admissions: [email protected]. Frontiers in Applied Mathematics and Statistics Explores how the application of mathematics and statistics can drive scientific developments across data science, engineering, finance, physics, biology, ecology, business, medicine, and beyond ... 119 Research Topics Guest edit your own article collection Suggest a topic. Submission. null. Submission MSc by Research in Mathematics About the course. The MSc by Research is an advanced research degree which provides the opportunity to investigate a project in depth and write a thesis which makes a significant contribution in the field. The research project is however designed to take less time than a Doctorate degree (normally two years, though it is possible to complete ... Research Areas Major research areas in this department include computational fluid dynamics (CFD), interface and front tracking methods, iterative methods in numerical linear algebra, and algorithms for parallel computers.Current research topics in CFD include: high resolution methods for solving nonlinear conservation laws with shock wave solutions. Research projects Research field projects. In addition to individual projects listed on FindAPhD, we are also looking for postgraduate researchers for potential projects within a number of other research fields. Browse these fields below and get in contact with the named supervisor to find out more. Applied Mathematics and Numerical Analysis. Continuum mechanics. Computational Mathematics Duke's Mathematics Department has a large group of mathematicians whose research involves scientific computing, numerical analysis, machine learning, computational topology, and algorithmic algebraic geometry. The computational mathematics research of our faculty has applications in data analysis and signal processing, fluid and solid mechanics ... Future themes of mathematics education research: an international Before the pandemic (2019), we asked: On what themes should research in mathematics education focus in the coming decade? The 229 responses from 44 countries led to eight themes plus considerations about mathematics education research itself. The themes can be summarized as teaching approaches, goals, relations to practices outside mathematics education, teacher professional development ... Research areas The department has strong research programs in: Control and Dynamical Systems (including differential equations) Fluid Mechanics. Mathematical Medicine and Biology. Mathematical Physics. Mathematics of Data Science and Machine Learning. Scientific Computing. Researchers in our department are at the forefront of a number of exciting research areas. Recent Master's Theses Research topics Continuum and Fluid Mechanics students AMATH 361 ... Recent Master's research papers Conference research posters Graduate Student Profiles ... Department of Applied Mathematics University of Waterloo Waterloo, Ontario Canada N2L 3G1 Phone: 519-888-4567, ext. 45098 ... Graduate Studies in Mathematics Our department offers Masters degrees in Mathematics, Applied Mathematics, and Statistics as well as a Ph.D. Degree in Mathematics, which can have an emphasis in any of the three areas mentioned. ... Math 7710-7790: Special Topics Courses. More information on the courses . Faculty and Areas of Research. ... The department also contains the A. H ... Research Areas in Mathematics Game Theory is also a popular research area in computer science where equilibrium structures are explored using computer algorithms. Mathematical topics such as combinatorics, graph theory, probability (discrete and measure-theoretic), analysis (real and functional), algebra (linear and abstract), etc., are used in solving game-theoretic problems. Lists of mathematics topics Basic mathematics. This branch is typically taught in secondary education or in the first year of university. Outline of arithmetic. Outline of discrete mathematics. List of calculus topics. List of geometry topics. Outline of geometry. List of trigonometry topics. Outline of trigonometry. Math/Stats Thesis and Colloquium Topics Updated: April 2024 Math/Stats Thesis and Colloquium Topics 2024- 2025 The degree with honors in Mathematics or Statistics is awarded to the student who has demonstrated outstanding intellectual achievement in a program of study which extends beyond the requirements of the major. The principal considerations for recommending a student for the degree with honors will be: Mastery of core ... (PDF) A Systematic Review on Graduate Mathematics Theses and Research Mapping of Conducted Mathematics Graduate Researches in Region I (2007-2016) Figure 3 also presents a research map of every research agenda conducted in the region along mathematics ... mathematics Drastically oversimplified, a mathematics research project goes like this: Develop, or select from the existing literature, a mathematical statement ("conjecture") that you think will be of interest to other mathematicians, and whose truth or falsity is not known. (For example, "There are infinitely many pairs of prime numbers that differ by 2.") Applied Mathematics Research Department of Mathematics. Headquarters Office. Simons Building (Building 2), Room 106. 77 Massachusetts Avenue. Cambridge, MA 02139-4307. Campus Map. (617) 253-4381. Website Questions: [email protected]. Undergraduate Admissions: [email protected]. 'Quantum coherence' survives in ultracold molecules Scientists are using sophisticated laser techniques to research quantum entanglement between the states of a chemical reaction. Quantum entanglement is a key concept at the heart of quantum information science, whereby two particles can occupy a shared quantum state. Applied Sciences The empowerment of educational reform and innovation through AI technology has become a topic of increasing interest in the field of education. The advent of AI technology has made comprehensive and in-depth teaching evaluation possible, serving as a significant driving force for efficient and precise teaching. There were few empirical studies on the application of high-quality precision ... assignment essay homework paper phd report review speech thesis