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Department of Mathematics

The Department of Mathematics offers training at the undergraduate, graduate, and postgraduate levels. Its expertise covers a broad spectrum of fields ranging from the traditional areas of "pure" mathematics, such as analysis, algebra, geometry, and topology, to applied mathematics areas such as combinatorics, computational biology, fluid dynamics, theoretical computer science, and theoretical physics.

Course 18 includes two undergraduate degrees: a Bachelor of Science in Mathematics and a Bachelor of Science in Mathematics with Computer Science. Undergraduate students may choose one of three options leading to the Bachelor of Science in Mathematics: applied mathematics, pure mathematics, or general mathematics. The general mathematics option provides a great deal of flexibility and allows students to design their own programs in conjunction with their advisors. The Mathematics with Computer Science degree is offered for students who want to pursue interests in mathematics and theoretical computer science within a single undergraduate program.

At the graduate level, the Mathematics Department offers the PhD in Mathematics, which culminates in the exposition of original research in a dissertation. Graduate students also receive training and gain experience in the teaching of mathematics.

The CLE Moore instructorships and Applied Mathematics instructorships bring mathematicians at the postdoctoral level to MIT and provide them with training in research and teaching.

Bachelor of Science in Mathematics (Course 18)

Bachelor of science in mathematics with computer science (course 18-c), minor in mathematics, undergraduate study.

An undergraduate degree in mathematics provides an excellent basis for graduate work in mathematics or computer science, or for employment in such fields as finance, business, or consulting. Students' programs are arranged through consultation with their faculty advisors.

Undergraduates in mathematics are encouraged to elect an undergraduate seminar during their junior or senior year. The experience gained from active participation in a seminar conducted by a research mathematician has proven to be valuable for students planning to pursue graduate work as well as for those going on to other careers. These seminars also provide training in the verbal and written communication of mathematics and may be used to fulfill the Communication Requirement.

Many mathematics majors take 18.821 Project Laboratory in Mathematics , which fulfills the Institute's Laboratory Requirement and counts toward the Communication Requirement.

General Mathematics Option

In addition to the General Institute Requirements, the requirements consist of Differential Equations, plus eight additional 12-unit subjects in Course 18 of essentially different content, including at least six advanced subjects (first decimal digit one or higher) that are distributed over at least three distinct areas (at least three distinct first decimal digits). One of these eight subjects must be Linear Algebra. This leaves available 84 units of unrestricted electives. The requirements are flexible in order to accommodate students who pursue programs that combine mathematics with a related field (such as physics, economics, or management) as well as students who are interested in both pure and applied mathematics. More details can be found on the degree chart .

Applied Mathematics Option

Applied mathematics focuses on the mathematical concepts and techniques applied in science, engineering, and computer science. Particular attention is given to the following principles and their mathematical formulations: propagation, equilibrium, stability, optimization, computation, statistics, and random processes.

Sophomores interested in applied mathematics typically enroll in 18.200 Principles of Discrete Applied Mathematics and 18.300 Principles of Continuum Applied Mathematics . Subject 18.200 is devoted to the discrete aspects of applied mathematics and may be taken concurrently with 18.03 Differential Equations . Subject 18.300 , offered in the spring term, is devoted to continuous aspects and makes considerable use of differential equations.

The subjects in Group I of the program correspond roughly to those areas of applied mathematics that make heavy use of discrete mathematics, while Group II emphasizes those subjects that deal mainly with continuous processes. Some subjects, such as probability or numerical analysis, have both discrete and continuous aspects.

Students planning to go on to graduate work in applied mathematics should also take some basic subjects in analysis and algebra.

More detail on the Applied Mathematics option can be found on the degree chart .

Pure Mathematics Option

Pure (or "theoretical") mathematics is the study of the basic concepts and structure of mathematics. Its goal is to arrive at a deeper understanding and an expanded knowledge of mathematics itself.

Traditionally, pure mathematics has been classified into three general fields: analysis, which deals with continuous aspects of mathematics; algebra, which deals with discrete aspects; and geometry. The undergraduate program is designed so that students become familiar with each of these areas. Students also may wish to explore other topics such as logic, number theory, complex analysis, and subjects within applied mathematics.

The subjects 18.701 Algebra I and 18.901 Introduction to Topology are more advanced and should not be elected until a student has had experience with proofs, as in Real Analysis ( 18.100A , 18.100B , 18.100P or 18.100Q ) or 18.700 Linear Algebra .

For more details, see the degree chart .

Mathematics and computer science are closely related fields. Problems in computer science are often formalized and solved with mathematical methods. It is likely that many important problems currently facing computer scientists will be solved by researchers skilled in algebra, analysis, combinatorics, logic and/or probability theory, as well as computer science.

The purpose of this program is to allow students to study a combination of these mathematical areas and potential areas of application in computer science. Required subjects include linear algebra ( 18.06 , 18.C06[J] , or 18.700 ) because it is so broadly used, and discrete mathematics ( 18.062[J] or 18.200 ) to give experience with proofs and the necessary tools for analyzing algorithms. The required subjects covering complexity ( 18.404 Theory of Computation or 18.400[J] Computability and Complexity Theory ) and algorithms ( 18.410[J] Design and Analysis of Algorithms ) provide an introduction to the most theoretical aspects of computer science. We also require exposure to other areas of computer science ( 6.1020 , 6.1800 , 6.4100 , or 6.3900 ) where mathematical issues may also arise. More details can be found on the degree chart .

Some flexibility is allowed in this program. In particular, students may substitute the more advanced subject 18.701 Algebra I for 18.06 Linear Algebra , and, if they already have strong theorem-proving skills, may substitute 18.211 Combinatorial Analysis or 18.212 Algebraic Combinatorics for 18.062[J] Mathematics for Computer Science or 18.200 Principles of Discrete Applied Mathematics .

The requirements for a Minor in Mathematics are as follows: six 12-unit subjects in mathematics, beyond the Institute's Mathematics Requirement, of essentially different content, including at least three advanced subjects (first decimal digit one or higher).

See the Undergraduate Section for a general description of the minor program .

For further information, see the department's website or contact Math Academic Services, 617-253-2416.

Graduate Study

The Mathematics Department offers programs covering a broad range of topics leading to the Doctor of Philosophy or Doctor of Science degree. Candidates are admitted to either the Pure or Applied Mathematics programs but are free to pursue interests in both groups. Of the roughly 120 doctoral students, about two thirds are in Pure Mathematics, one third in Applied Mathematics.

The programs in Pure and Applied Mathematics offer basic and advanced classes in analysis, algebra, geometry, Lie theory, logic, number theory, probability, statistics, topology, astrophysics, combinatorics, fluid dynamics, numerical analysis, theoretical physics, and the theory of computation. In addition, many mathematically oriented subjects are offered by other departments. Students in Applied Mathematics are especially encouraged to take subjects in engineering and scientific subjects related to their research.

All students pursue research under the supervision of the faculty and are encouraged to take advantage of the many seminars and colloquia at MIT and in the Boston area.

Doctor of Philosophy or Doctor of Science

The requirements for these degrees are described on the department's website . In outline, they consist of an oral qualifying examination, a thesis proposal, completion of a minimum of 96 units (8 graduate subjects), experience in classroom teaching, and a thesis containing original research in mathematics.

Interdisciplinary Programs

Students with primary interest in computational science may also consider applying to the interdisciplinary Computational Science and Engineering (CSE) program, with which the Mathematics Department is affiliated. For more information, see the CSE website .

Mathematics and Statistics

The Interdisciplinary Doctoral Program in Statistics provides training in statistics, including classical statistics and probability as well as computation and data analysis, to students who wish to integrate these valuable skills into their primary academic program. The program is administered jointly by the departments of Aeronautics and Astronautics, Economics, Mathematics, Mechanical Engineering, Physics, and Political Science, and the Statistics and Data Science Center within the Institute for Data, Systems, and Society. It is open to current doctoral students in participating departments. For more information, including department-specific requirements, see the full program description under Interdisciplinary Graduate Programs.

Financial Support

Financial support is guaranteed for up to five years to students making satisfactory academic progress. Financial aid after the first year is usually in the form of a teaching or research assistantship.

For further information, see the department's website or contact Math Academic Services, 617-253-2416.

Faculty and Teaching Staff

Michel X. Goemans, PhD

RSA Professor of Mathematics

Head, Department of Mathematics

William Minicozzi, PhD

Singer Professor of Mathematics

Associate Head, Department of Mathematics

Martin Z. Bazant, PhD

E. G. Roos Professor

Professor of Chemical Engineering

Professor of Mathematics

Bonnie Berger, PhD

Simons Professor of Mathematics

Member, Health Sciences and Technology Faculty

Roman Bezrukavnikov, PhD

Alexei Borodin, PhD

John W. M. Bush, PhD

Hung Cheng, PhD

Tobias Colding, PhD

Cecil and Ida Green Distinguished Professor

Laurent Demanet, PhD

Professor of Earth, Atmospheric and Planetary Sciences

Joern Dunkel, PhD

MathWorks Professor of Mathematics

Alan Edelman, PhD

Pavel I. Etingof, PhD

Lawrence Guth, PhD

Claude E. Shannon (1940) Professor of Mathematics

Anette E. Hosoi, PhD

Neil and Jane Pappalardo Professor

Professor of Mechanical Engineering

Member, Institute for Data, Systems, and Society

David S. Jerison, PhD

Steven G. Johnson, PhD

Professor of Physics

Victor Kac, PhD

Kenneth N. Kamrin, PhD

Jonathan Adam Kelner, PhD

Ju-Lee Kim, PhD

Frank Thomson Leighton, PhD

George Lusztig, PhD

Edward A. Abdun-Nur (1924) Professor of Mathematics

Davesh Maulik, PhD

Richard B. Melrose, PhD

Ankur Moitra, PhD

Norbert Wiener Professor of Mathematics

Associate Director, Institute for Data, Systems, and Society

Elchanan Mossel, PhD

Tomasz S. Mrowka, PhD

Pablo A. Parrilo, PhD

Joseph F. and Nancy P. Keithley Professor in Electrical Engineering

Professor of Electrical Engineering and Computer Science

Bjorn Poonen, PhD

Distinguished Professor in Science

(On leave, spring)

Alexander Postnikov, PhD

Philippe Rigollet, PhD

Rodolfo R. Rosales, PhD

Paul Seidel, PhD

Levinson Professor of Mathematics

Scott Roger Sheffield, PhD

Leighton Family Professor of Mathematics

Peter W. Shor, PhD

Henry Adams Morss and Henry Adams Morss, Jr. (1934) Professor of Mathematics

Michael Sipser, PhD

Donner Professor of Mathematics

Gigliola Staffilani, PhD

Abby Rockefeller Mauzé Professor of Mathematics

Daniel W. Stroock, PhD

Professor Post-Tenure of Mathematics

Martin J. Wainwright, PhD

Cecil H. Green Professor in Electrical Engineering

Zhiwei Yun, PhD

Wei Zhang, PhD

Associate Professors

Tristan Collins, PhD

Class of 1948 Career Development Professor

Associate Professor of Mathematics

Semyon Dyatlov, PhD

Andrew Lawrie, PhD

Andrei Negut, PhD

Nike Sun, PhD

Yufei Zhao, PhD

Assistant Professors

Daniel Alvarez-Gavela, PhD

Assistant Professor of Mathematics

Jeremy Hahn, PhD

Rockwell International Career Development Professor

(On leave, fall)

Dor Minzer, PhD

Tristan Ozuch-Meersseman, PhD

Lisa Piccirillo, PhD

Lisa Sauermann, PhD

John Urschel, PhD

Visiting Associate Professors

Leonid Rybnikov, PhD

Visiting Simons Associate Professor of Mathematics

Adjunct Professors

Henry Cohn, PhD

Adjunct Professor of Mathematics

Jonathan Bloom, PhD

Lecturer in Mathematics

Slava Gerovitch, PhD

Peter J. Kempthorne, PhD

Tanya Khovanova, PhD

CLE Moore Instructors

Qin Deng, PhD

CLE Moore Instructor of Mathematics

Marjorie Drake, PhD

Giada Franz, PhD

Yuchen Fu, PhD

Jimmy He, PhD

Felipe Hernandez, PhD

Malo Pierig Jezequel, PhD

Ruojing Jiang, PhD

Konstantinos Kavvadias, PhD

Aaron Landesman, PhD

Miguel Moreira, PhD

Changkeun Oh, PhD

Jia Shi, PhD

Minh-Tam Trinh, PhD

David Yang, PhD

Jingze Zhu, PhD

Jonathan Zung, PhD

Instructors

Karol Bacik, PhD

Instructor of Applied Mathematics

Mitali Bafna, PhD

Omri Ben-Eliezer, PhD

Elijah Bodish, PhD

Instructor of Mathematics

Pengning Chao, PhD

Ziang Chen, PhD

Nicholas Derr, PhD

Manik Dhar, PhD

Andrew James Horning, PhD

Artem Kalmykov, PhD

Anya Katsevich, PhD

David Milton Kouskoulas, PhD

Dominique Maldague, PhD

Dan Mikulincer, PhD

Keaton Naff, PhD

Alex Pieloch, PhD

Bauyrzhan Primkulov, PhD

Melissa Sherman-Bennett, PhD

Michael Simkin, PhD

Foster Tom, PhD

Kent Vashaw, PhD

Research Staff

Principal research scientists.

Andrew Victor Sutherland II, PhD

Principal Research Scientist of Mathematics

Research Scientists

Shiva Chidambaram, PhD

Research Scientist of Mathematics

Edgar Costa, PhD

David Roe, PhD

Samuel Schiavone, PhD

Raymond van Bommel, PhD

Professors Emeriti

Michael Artin, PhD

Professor Emeritus of Mathematics

Daniel Z. Freedman, PhD

Professor Emeritus of Physics

Harvey P. Greenspan, PhD

Victor W. Guillemin, PhD

Sigurdur Helgason, PhD

Steven L. Kleiman, PhD

Daniel J. Kleitman, PhD

Haynes R. Miller, PhD

James R. Munkres, PhD

Richard P. Stanley, PhD

Harold Stark, PhD

Gilbert Strang, PhD

Alar Toomre, PhD

David A. Vogan, PhD

General Mathematics

18.01 calculus.

Prereq: None U (Fall, Spring) 5-0-7 units. CALC I Credit cannot also be received for 18.01A , CC.1801 , ES.1801 , ES.181A

Differentiation and integration of functions of one variable, with applications. Informal treatment of limits and continuity. Differentiation: definition, rules, application to graphing, rates, approximations, and extremum problems. Indefinite integration; separable first-order differential equations. Definite integral; fundamental theorem of calculus. Applications of integration to geometry and science. Elementary functions. Techniques of integration. Polar coordinates. L'Hopital's rule. Improper integrals. Infinite series: geometric, p-harmonic, simple comparison tests, power series for some elementary functions.

Fall: L. Guth. Spring: Information: W. Minicozzi

18.01A Calculus

Prereq: Knowledge of differentiation and elementary integration U (Fall; first half of term) 5-0-7 units. CALC I Credit cannot also be received for 18.01 , CC.1801 , ES.1801 , ES.181A

Six-week review of one-variable calculus, emphasizing material not on the high-school AB syllabus: integration techniques and applications, improper integrals, infinite series, applications to other topics, such as probability and statistics, as time permits. Prerequisites: one year of high-school calculus or the equivalent, with a score of 5 on the AB Calculus test (or the AB portion of the BC test, or an equivalent score on a standard international exam), or equivalent college transfer credit, or a passing grade on the first half of the 18.01 advanced standing exam.

18.02 Calculus

Prereq: Calculus I (GIR) U (Fall, Spring) 5-0-7 units. CALC II Credit cannot also be received for 18.022 , 18.02A , CC.1802 , ES.1802 , ES.182A

Calculus of several variables. Vector algebra in 3-space, determinants, matrices. Vector-valued functions of one variable, space motion. Scalar functions of several variables: partial differentiation, gradient, optimization techniques. Double integrals and line integrals in the plane; exact differentials and conservative fields; Green's theorem and applications, triple integrals, line and surface integrals in space, Divergence theorem, Stokes' theorem; applications.

Fall: S Dyatlov. Spring: D Jerison

18.02A Calculus

Prereq: Calculus I (GIR) U (Fall, IAP, Spring; second half of term) 5-0-7 units. CALC II Credit cannot also be received for 18.02 , 18.022 , CC.1802 , ES.1802 , ES.182A

First half is taught during the last six weeks of the Fall term; covers material in the first half of 18.02 (through double integrals). Second half of 18.02A can be taken either during IAP (daily lectures) or during the second half of the Spring term; it covers the remaining material in 18.02 .

Fall, IAP: J. W. M. Bush. Spring: D. Jerison

18.022 Calculus

Prereq: Calculus I (GIR) U (Fall) 5-0-7 units. CALC II Credit cannot also be received for 18.02 , 18.02A , CC.1802 , ES.1802 , ES.182A

Calculus of several variables. Topics as in 18.02 but with more focus on mathematical concepts. Vector algebra, dot product, matrices, determinant. Functions of several variables, continuity, differentiability, derivative. Parametrized curves, arc length, curvature, torsion. Vector fields, gradient, curl, divergence. Multiple integrals, change of variables, line integrals, surface integrals. Stokes' theorem in one, two, and three dimensions.

W. Minicozzi

18.03 Differential Equations

Prereq: None. Coreq: Calculus II (GIR) U (Fall, Spring) 5-0-7 units. REST Credit cannot also be received for CC.1803 , ES.1803

Study of differential equations, including modeling physical systems. Solution of first-order ODEs by analytical, graphical, and numerical methods. Linear ODEs with constant coefficients. Complex numbers and exponentials. Inhomogeneous equations: polynomial, sinusoidal, and exponential inputs. Oscillations, damping, resonance. Fourier series. Matrices, eigenvalues, eigenvectors, diagonalization. First order linear systems: normal modes, matrix exponentials, variation of parameters. Heat equation, wave equation. Nonlinear autonomous systems: critical point analysis, phase plane diagrams.

Fall: J. Dunkel. Spring: L. Demanet

18.031 System Functions and the Laplace Transform

Prereq: None. Coreq: 18.03 U (IAP) 1-0-2 units

Studies basic continuous control theory as well as representation of functions in the complex frequency domain. Covers generalized functions, unit impulse response, and convolution; and Laplace transform, system (or transfer) function, and the pole diagram. Includes examples from mechanical and electrical engineering.

Information: H. R. Miller

18.032 Differential Equations

Prereq: None. Coreq: Calculus II (GIR) U (Spring) 5-0-7 units. REST

Covers much of the same material as 18.03 with more emphasis on theory. The point of view is rigorous and results are proven. Local existence and uniqueness of solutions.

18.04 Complex Variables with Applications

Prereq: Calculus II (GIR) and ( 18.03 or 18.032 ) U (Fall) 4-0-8 units Credit cannot also be received for 18.075 , 18.0751

Complex algebra and functions; analyticity; contour integration, Cauchy's theorem; singularities, Taylor and Laurent series; residues, evaluation of integrals; multivalued functions, potential theory in two dimensions; Fourier analysis, Laplace transforms, and partial differential equations.

18.05 Introduction to Probability and Statistics

Prereq: Calculus II (GIR) U (Spring) 4-0-8 units. REST

Elementary introduction with applications. Basic probability models. Combinatorics. Random variables. Discrete and continuous probability distributions. Statistical estimation and testing. Confidence intervals. Introduction to linear regression.

18.06 Linear Algebra

Prereq: Calculus II (GIR) U (Fall, Spring) 4-0-8 units. REST Credit cannot also be received for 6.C06[J] , 18.700 , 18.C06[J]

Basic subject on matrix theory and linear algebra, emphasizing topics useful in other disciplines, including systems of equations, vector spaces, determinants, eigenvalues, singular value decomposition, and positive definite matrices. Applications to least-squares approximations, stability of differential equations, networks, Fourier transforms, and Markov processes. Uses linear algebra software. Compared with 18.700 , more emphasis on matrix algorithms and many applications.

Fall: TBD. Spring: A. Borodin

18.C06[J] Linear Algebra and Optimization

Same subject as 6.C06[J] Prereq: Calculus II (GIR) U (Fall) 5-0-7 units. REST Credit cannot also be received for 18.06 , 18.700

Introductory course in linear algebra and optimization, assuming no prior exposure to linear algebra and starting from the basics, including vectors, matrices, eigenvalues, singular values, and least squares. Covers the basics in optimization including convex optimization, linear/quadratic programming, gradient descent, and regularization, building on insights from linear algebra. Explores a variety of applications in science and engineering, where the tools developed give powerful ways to understand complex systems and also extract structure from data.

A. Moitra, P. Parrilo

18.062[J] Mathematics for Computer Science

Same subject as 6.1200[J] Prereq: Calculus I (GIR) U (Fall, Spring) 5-0-7 units. REST

See description under subject 6.1200[J] .

Z. R. Abel, F. T. Leighton, A. Moitra

18.065 Matrix Methods in Data Analysis, Signal Processing, and Machine Learning

Subject meets with 18.0651 Prereq: 18.06 U (Spring) 3-0-9 units

Reviews linear algebra with applications to life sciences, finance, engineering, and big data. Covers singular value decomposition, weighted least squares, signal and image processing, principal component analysis, covariance and correlation matrices, directed and undirected graphs, matrix factorizations, neural nets, machine learning, and computations with large matrices.

18.0651 Matrix Methods in Data Analysis, Signal Processing, and Machine Learning

Subject meets with 18.065 Prereq: 18.06 G (Spring) 3-0-9 units

18.075 Methods for Scientists and Engineers

Subject meets with 18.0751 Prereq: Calculus II (GIR) and 18.03 U (Spring) 3-0-9 units Credit cannot also be received for 18.04

Covers functions of a complex variable; calculus of residues. Includes ordinary differential equations; Bessel and Legendre functions; Sturm-Liouville theory; partial differential equations; heat equation; and wave equations.

18.0751 Methods for Scientists and Engineers

Subject meets with 18.075 Prereq: Calculus II (GIR) and 18.03 G (Spring) 3-0-9 units Credit cannot also be received for 18.04

18.085 Computational Science and Engineering I

Subject meets with 18.0851 Prereq: Calculus II (GIR) and ( 18.03 or 18.032 ) U (Fall, Spring, Summer) 3-0-9 units

Review of linear algebra, applications to networks, structures, and estimation, finite difference and finite element solution of differential equations, Laplace's equation and potential flow, boundary-value problems, Fourier series, discrete Fourier transform, convolution. Frequent use of MATLAB in a wide range of scientific and engineering applications.

Fall: D. Kouskoulas. Spring: Staff

18.0851 Computational Science and Engineering I

Subject meets with 18.085 Prereq: Calculus II (GIR) and ( 18.03 or 18.032 ) G (Fall, Spring, Summer) 3-0-9 units

Fall: D. Kouskoulas. Spring: Staff

18.086 Computational Science and Engineering II

Subject meets with 18.0861 Prereq: Calculus II (GIR) and ( 18.03 or 18.032 ) U (Spring) Not offered regularly; consult department 3-0-9 units

Initial value problems: finite difference methods, accuracy and stability, heat equation, wave equations, conservation laws and shocks, level sets, Navier-Stokes. Solving large systems: elimination with reordering, iterative methods, preconditioning, multigrid, Krylov subspaces, conjugate gradients. Optimization and minimum principles: weighted least squares, constraints, inverse problems, calculus of variations, saddle point problems, linear programming, duality, adjoint methods.

Information: W. G. Strang

18.0861 Computational Science and Engineering II

Subject meets with 18.086 Prereq: Calculus II (GIR) and ( 18.03 or 18.032 ) G (Spring) Not offered regularly; consult department 3-0-9 units

18.089 Review of Mathematics

Prereq: Permission of instructor G (Summer) 5-0-7 units

One-week review of one-variable calculus ( 18.01 ), followed by concentrated study covering multivariable calculus ( 18.02 ), two hours per day for five weeks. Primarily for graduate students in Course 2N. Degree credit allowed only in special circumstances.

Information: W. Minicozzi

18.090 Introduction to Mathematical Reasoning

Prereq: None. Coreq: Calculus II (GIR) U (Spring) 3-0-9 units. REST

Focuses on understanding and constructing mathematical arguments. Discusses foundational topics (such as infinite sets, quantifiers, and methods of proof) as well as selected concepts from algebra (permutations, vector spaces, fields) and analysis (sequences of real numbers). Particularly suitable for students desiring additional experience with proofs before going on to more advanced mathematics subjects or subjects in related areas with significant mathematical content.

S. Dyatlov, B. Poonen, P. Seidel

18.094[J] Teaching College-Level Science and Engineering

Same subject as 1.95[J] , 5.95[J] , 7.59[J] , 8.395[J] Subject meets with 2.978 Prereq: None G (Fall) 2-0-2 units

See description under subject 5.95[J] .

18.095 Mathematics Lecture Series

Prereq: Calculus I (GIR) U (IAP) 2-0-4 units Can be repeated for credit.

Ten lectures by mathematics faculty members on interesting topics from both classical and modern mathematics. All lectures accessible to students with calculus background and an interest in mathematics. At each lecture, reading and exercises are assigned. Students prepare these for discussion in a weekly problem session.

18.098 Internship in Mathematics

Prereq: Permission of instructor U (Fall, IAP, Spring, Summer) Units arranged [P/D/F] Can be repeated for credit.

Provides academic credit for students pursuing internships to gain practical experience in the applications of mathematical concepts and methods.

18.099 Independent Study

Prereq: Permission of instructor U (Fall, IAP, Spring, Summer) Units arranged Can be repeated for credit.

Studies (during IAP) or special individual reading (during regular terms). Arranged in consultation with individual faculty members and subject to departmental approval. May not be used to satisfy Mathematics major requirements.

18.1001 Real Analysis

Subject meets with 18.100A Prereq: Calculus II (GIR) G (Fall, Spring) 3-0-9 units Credit cannot also be received for 18.1002 , 18.100A , 18.100B , 18.100P , 18.100Q

Covers fundamentals of mathematical analysis: convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, interchange of limit operations. Shows the utility of abstract concepts and teaches understanding and construction of proofs. Proofs and definitions are less abstract than in 18.100B . Gives applications where possible. Concerned primarily with the real line. Students in Course 18 must register for undergraduate version 18.100A .

Fall: Q. Deng. Spring: J. Zhu

18.1002 Real Analysis

Subject meets with 18.100B Prereq: Calculus II (GIR) G (Fall, Spring) 3-0-9 units Credit cannot also be received for 18.1001 , 18.100A , 18.100B , 18.100P , 18.100Q

Covers fundamentals of mathematical analysis: convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, interchange of limit operations. Shows the utility of abstract concepts and teaches understanding and construction of proofs. More demanding than 18.100A , for students with more mathematical maturity. Places more emphasis on point-set topology and n-space. Students in Course 18 must register for undergraduate version 18.100B .

Fall: R. Melrose. Spring: G. Franz

18.100A Real Analysis

Subject meets with 18.1001 Prereq: Calculus II (GIR) U (Fall, Spring) 3-0-9 units Credit cannot also be received for 18.1001 , 18.1002 , 18.100B , 18.100P , 18.100Q

18.100B Real Analysis

Subject meets with 18.1002 Prereq: Calculus II (GIR) U (Fall, Spring) 3-0-9 units Credit cannot also be received for 18.1001 , 18.1002 , 18.100A , 18.100P , 18.100Q

Covers fundamentals of mathematical analysis: convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, interchange of limit operations. Shows the utility of abstract concepts and teaches understanding and construction of proofs. More demanding than 18.100A , for students with more mathematical maturity. Places more emphasis on point-set topology and n-space.

18.100P Real Analysis

Prereq: Calculus II (GIR) U (Spring) 4-0-11 units Credit cannot also be received for 18.1001 , 18.1002 , 18.100A , 18.100B , 18.100Q

Covers fundamentals of mathematical analysis: convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, interchange of limit operations. Shows the utility of abstract concepts and teaches understanding and construction of proofs. Proofs and definitions are less abstract than in 18.100B . Gives applications where possible. Concerned primarily with the real line. Includes instruction and practice in written communication. Enrollment limited.

18.100Q Real Analysis

Prereq: Calculus II (GIR) U (Fall) 4-0-11 units Credit cannot also be received for 18.1001 , 18.1002 , 18.100A , 18.100B , 18.100P

Covers fundamentals of mathematical analysis: convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, interchange of limit operations. Shows the utility of abstract concepts and teaches understanding and construction of proofs. More demanding than 18.100A , for students with more mathematical maturity. Places more emphasis on point-set topology and n-space. Includes instruction and practice in written communication. Enrollment limited.

18.101 Analysis and Manifolds

Subject meets with 18.1011 Prereq: ( 18.06 , 18.700 , or 18.701 ) and ( 18.100A , 18.100B , 18.100P , or 18.100Q ) U (Fall) 3-0-9 units

Introduction to the theory of manifolds: vector fields and densities on manifolds, integral calculus in the manifold setting and the manifold version of the divergence theorem. 18.901 helpful but not required.

M. Jezequel

18.1011 Analysis and Manifolds

Subject meets with 18.101 Prereq: ( 18.06 , 18.700 , or 18.701 ) and ( 18.100A , 18.100B , 18.100P , or 18.100Q ) G (Fall) 3-0-9 units

18.102 Introduction to Functional Analysis

Subject meets with 18.1021 Prereq: ( 18.06 , 18.700 , or 18.701 ) and ( 18.100A , 18.100B , 18.100P , or 18.100Q ) U (Spring) 3-0-9 units

Normed spaces, completeness, functionals, Hahn-Banach theorem, duality, operators. Lebesgue measure, measurable functions, integrability, completeness of L-p spaces. Hilbert space. Compact, Hilbert-Schmidt and trace class operators. Spectral theorem.

18.1021 Introduction to Functional Analysis

Subject meets with 18.102 Prereq: ( 18.06 , 18.700 , or 18.701 ) and ( 18.100A , 18.100B , 18.100P , or 18.100Q ) G (Spring) 3-0-9 units

18.103 Fourier Analysis: Theory and Applications

Subject meets with 18.1031 Prereq: ( 18.06 , 18.700 , or 18.701 ) and ( 18.100A , 18.100B , 18.100P , or 18.100Q ) U (Fall) 3-0-9 units

Roughly half the subject devoted to the theory of the Lebesgue integral with applications to probability, and half to Fourier series and Fourier integrals.

18.1031 Fourier Analysis: Theory and Applications

Subject meets with 18.103 Prereq: ( 18.06 , 18.700 , or 18.701 ) and ( 18.100A , 18.100B , 18.100P , or 18.100Q ) G (Fall) 3-0-9 units

Roughly half the subject devoted to the theory of the Lebesgue integral with applications to probability, and half to Fourier series and Fourier integrals. Students in Course 18 must register for the undergraduate version, 18.103 .

18.104 Seminar in Analysis

Prereq: 18.100A , 18.100B , 18.100P , or 18.100Q U (Fall, Spring) 3-0-9 units

Students present and discuss material from books or journals. Topics vary from year to year. Instruction and practice in written and oral communication provided. Enrollment limited.

Fall: T. Ozuch-Meersseman. Spring: G. Staffilani

18.112 Functions of a Complex Variable

Subject meets with 18.1121 Prereq: ( 18.06 , 18.700 , or 18.701 ) and ( 18.100A , 18.100B , 18.100P , or 18.100Q ) U (Fall) 3-0-9 units

Studies the basic properties of analytic functions of one complex variable. Conformal mappings and the Poincare model of non-Euclidean geometry. Cauchy-Goursat theorem and Cauchy integral formula. Taylor and Laurent decompositions. Singularities, residues and computation of integrals. Harmonic functions and Dirichlet's problem for the Laplace equation. The partial fractions decomposition. Infinite series and infinite product expansions. The Gamma function. The Riemann mapping theorem. Elliptic functions.

18.1121 Functions of a Complex Variable

Subject meets with 18.112 Prereq: ( 18.06 , 18.700 , or 18.701 ) and ( 18.100A , 18.100B , 18.100P , or 18.100Q ) G (Fall) 3-0-9 units

18.116 Riemann Surfaces

Prereq: 18.112 Acad Year 2023-2024: Not offered Acad Year 2024-2025: G (Fall) 3-0-9 units

Riemann surfaces, uniformization, Riemann-Roch Theorem. Theory of elliptic functions and modular forms. Some applications, such as to number theory.

P. I. Etingof

18.117 Topics in Several Complex Variables

Prereq: 18.112 and 18.965 Acad Year 2023-2024: Not offered Acad Year 2024-2025: G (Spring) 3-0-9 units Can be repeated for credit.

Harmonic theory on complex manifolds, Hodge decomposition theorem, Hard Lefschetz theorem. Vanishing theorems. Theory of Stein manifolds. As time permits students also study holomorphic vector bundles on Kahler manifolds.

18.118 Topics in Analysis

Prereq: Permission of instructor Acad Year 2023-2024: G (Spring) Acad Year 2024-2025: Not offered 3-0-9 units Can be repeated for credit.

Topics vary from year to year.

18.125 Measure Theory and Analysis

Prereq: 18.100A , 18.100B , 18.100P , or 18.100Q Acad Year 2023-2024: Not offered Acad Year 2024-2025: G (Spring) 3-0-9 units

Provides a rigorous introduction to Lebesgue's theory of measure and integration. Covers material that is essential in analysis, probability theory, and differential geometry.

18.137 Topics in Geometric Partial Differential Equations

Prereq: Permission of instructor Acad Year 2023-2024: Not offered Acad Year 2024-2025: G (Fall) 3-0-9 units Can be repeated for credit.

18.152 Introduction to Partial Differential Equations

Subject meets with 18.1521 Prereq: ( 18.06 , 18.700 , or 18.701 ) and ( 18.100A , 18.100B , 18.100P , or 18.100Q ) U (Spring) 3-0-9 units

Introduces three main types of partial differential equations: diffusion, elliptic, and hyperbolic. Includes mathematical tools, real-world examples and applications, such as the Black-Scholes equation, the European options problem, water waves, scalar conservation laws, first order equations and traffic problems.

18.1521 Introduction to Partial Differential Equations

Subject meets with 18.152 Prereq: ( 18.06 , 18.700 , or 18.701 ) and ( 18.100A , 18.100B , 18.100P , or 18.100Q ) G (Spring) 3-0-9 units

18.155 Differential Analysis I

Prereq: 18.102 or 18.103 G (Fall) 3-0-9 units

First part of a two-subject sequence. Review of Lebesgue integration. Lp spaces. Distributions. Fourier transform. Sobolev spaces. Spectral theorem, discrete and continuous spectrum. Homogeneous distributions. Fundamental solutions for elliptic, hyperbolic and parabolic differential operators. Recommended prerequisite: 18.112 .

18.156 Differential Analysis II

Prereq: 18.155 G (Spring) 3-0-9 units

Second part of a two-subject sequence. Covers variable coefficient elliptic, parabolic and hyperbolic partial differential equations.

18.157 Introduction to Microlocal Analysis

Prereq: 18.155 Acad Year 2023-2024: Not offered Acad Year 2024-2025: G (Spring) 3-0-9 units

The semi-classical theory of partial differential equations. Discussion of Pseudodifferential operators, Fourier integral operators, asymptotic solutions of partial differential equations, and the spectral theory of Schroedinger operators from the semi-classical perspective. Heavy emphasis placed on the symplectic geometric underpinnings of this subject.

R. B. Melrose

18.158 Topics in Differential Equations

Prereq: 18.157 Acad Year 2023-2024: G (Spring) Acad Year 2024-2025: Not offered 3-0-9 units Can be repeated for credit.

18.199 Graduate Analysis Seminar

Studies original papers in differential analysis and differential equations. Intended for first- and second-year graduate students. Permission must be secured in advance.

V. W. Guillemin

Discrete Applied Mathematics

18.200 principles of discrete applied mathematics.

Prereq: None. Coreq: 18.06 U (Spring) 4-0-11 units Credit cannot also be received for 18.200A

Study of illustrative topics in discrete applied mathematics, including probability theory, information theory, coding theory, secret codes, generating functions, and linear programming. Instruction and practice in written communication provided. Enrollment limited.

P. W. Shor, A. Moitra

18.200A Principles of Discrete Applied Mathematics

Prereq: None. Coreq: 18.06 Acad Year 2023-2024: Not offered Acad Year 2024-2025: U (Fall) 3-0-9 units Credit cannot also be received for 18.200

Study of illustrative topics in discrete applied mathematics, including probability theory, information theory, coding theory, secret codes, generating functions, and linear programming.

18.204 Undergraduate Seminar in Discrete Mathematics

Prereq: (( 6.1200[J] or 18.200 ) and ( 18.06 , 18.700 , or 18.701 )) or permission of instructor U (Fall, Spring) 3-0-9 units

Seminar in combinatorics, graph theory, and discrete mathematics in general. Participants read and present papers from recent mathematics literature. Instruction and practice in written and oral communication provided. Enrollment limited.

J. He, D. Mikulincer, M. Sherman-Bennett, A. Weigandt

18.211 Combinatorial Analysis

Prereq: Calculus II (GIR) and ( 18.06 , 18.700 , or 18.701 ) U (Fall) 3-0-9 units

Combinatorial problems and methods for their solution. Enumeration, generating functions, recurrence relations, construction of bijections. Introduction to graph theory. Prior experience with abstraction and proofs is helpful.

A. Weigandt

18.212 Algebraic Combinatorics

Prereq: 18.701 or 18.703 U (Spring) 3-0-9 units

Applications of algebra to combinatorics. Topics include walks in graphs, the Radon transform, groups acting on posets, Young tableaux, electrical networks.

A. Postnikov

18.217 Combinatorial Theory

Prereq: Permission of instructor G (Fall) 3-0-9 units Can be repeated for credit.

Content varies from year to year.

18.218 Topics in Combinatorics

Prereq: Permission of instructor G (Spring) 3-0-9 units Can be repeated for credit.

L. Sauermann

18.219 Seminar in Combinatorics

Prereq: Permission of instructor G (Fall) Not offered regularly; consult department 3-0-9 units Can be repeated for credit.

Content varies from year to year. Readings from current research papers in combinatorics. Topics to be chosen and presented by the class.

Information: Y. Zhao

18.225 Graph Theory and Additive Combinatorics

Prereq: (( 18.701 or 18.703 ) and ( 18.100A , 18.100B , 18.100P , or 18.100Q )) or permission of instructor Acad Year 2023-2024: G (Fall) Acad Year 2024-2025: Not offered 3-0-9 units

Introduction to extremal graph theory and additive combinatorics. Highlights common themes, such as the dichotomy between structure versus pseudorandomness. Topics include Turan-type problems, Szemeredi's regularity lemma and applications, pseudorandom graphs, spectral graph theory, graph limits, arithmetic progressions (Roth, Szemeredi, Green-Tao), discrete Fourier analysis, Freiman's theorem on sumsets and structure. Discusses current research topics and open problems.

18.226 Probabilistic Methods in Combinatorics

Prereq: ( 18.211 , 18.600 , and ( 18.100A , 18.100B , 18.100P , or 18.100Q )) or permission of instructor Acad Year 2023-2024: Not offered Acad Year 2024-2025: G (Fall) 3-0-9 units

Introduction to the probabilistic method, a fundamental and powerful technique in combinatorics and theoretical computer science. Focuses on methodology as well as combinatorial applications. Suitable for students with strong interest and background in mathematical problem solving. Topics include linearity of expectations, alteration, second moment, Lovasz local lemma, correlation inequalities, Janson inequalities, concentration inequalities, entropy method.

Continuous Applied Mathematics

18.300 principles of continuum applied mathematics.

Prereq: Calculus II (GIR) and ( 18.03 or 18.032 ) U (Fall) 3-0-9 units

Covers fundamental concepts in continuous applied mathematics. Applications from traffic flow, fluids, elasticity, granular flows, etc. Also covers continuum limit; conservation laws, quasi-equilibrium; kinematic waves; characteristics, simple waves, shocks; diffusion (linear and nonlinear); numerical solution of wave equations; finite differences, consistency, stability; discrete and fast Fourier transforms; spectral methods; transforms and series (Fourier, Laplace). Additional topics may include sonic booms, Mach cone, caustics, lattices, dispersion and group velocity. Uses MATLAB computing environment.

B. Geshkovski

18.303 Linear Partial Differential Equations: Analysis and Numerics

Prereq: 18.06 or 18.700 U (Fall) 3-0-9 units

Provides students with the basic analytical and computational tools of linear partial differential equations (PDEs) for practical applications in science and engineering, including heat/diffusion, wave, and Poisson equations. Analytics emphasize the viewpoint of linear algebra and the analogy with finite matrix problems. Studies operator adjoints and eigenproblems, series solutions, Green's functions, and separation of variables. Numerics focus on finite-difference and finite-element techniques to reduce PDEs to matrix problems, including stability and convergence analysis and implicit/explicit timestepping. Some programming required for homework and final project.

V. Heinonen

18.305 Advanced Analytic Methods in Science and Engineering

Prereq: 18.04 , 18.075 , or 18.112 Acad Year 2023-2024: Not offered Acad Year 2024-2025: G (Fall) 3-0-9 units

Covers expansion around singular points: the WKB method on ordinary and partial differential equations; the method of stationary phase and the saddle point method; the two-scale method and the method of renormalized perturbation; singular perturbation and boundary-layer techniques; WKB method on partial differential equations.

18.306 Advanced Partial Differential Equations with Applications

Prereq: ( 18.03 or 18.032 ) and ( 18.04 , 18.075 , or 18.112 ) Acad Year 2023-2024: G (Spring) Acad Year 2024-2025: Not offered 3-0-9 units

Concepts and techniques for partial differential equations, especially nonlinear. Diffusion, dispersion and other phenomena. Initial and boundary value problems. Normal mode analysis, Green's functions, and transforms. Conservation laws, kinematic waves, hyperbolic equations, characteristics shocks, simple waves. Geometrical optics, caustics. Free-boundary problems. Dimensional analysis. Singular perturbation, boundary layers, homogenization. Variational methods. Solitons. Applications from fluid dynamics, materials science, optics, traffic flow, etc.

R. R. Rosales

18.327 Topics in Applied Mathematics

18.330 introduction to numerical analysis.

Basic techniques for the efficient numerical solution of problems in science and engineering. Root finding, interpolation, approximation of functions, integration, differential equations, direct and iterative methods in linear algebra. Knowledge of programming in a language such as MATLAB, Python, or Julia is helpful.

18.335[J] Introduction to Numerical Methods

Same subject as 6.7310[J] Prereq: 18.06 , 18.700 , or 18.701 G (Spring) 3-0-9 units

Advanced introduction to numerical analysis: accuracy and efficiency of numerical algorithms. In-depth coverage of sparse-matrix/iterative and dense-matrix algorithms in numerical linear algebra (for linear systems and eigenproblems). Floating-point arithmetic, backwards error analysis, conditioning, and stability. Other computational topics (e.g., numerical integration or nonlinear optimization) may also be surveyed. Final project involves some programming.

A. J. Horning

18.336[J] Fast Methods for Partial Differential and Integral Equations

Same subject as 6.7340[J] Prereq: 6.7300[J] , 16.920[J] , 18.085 , 18.335[J] , or permission of instructor G (Fall, Spring) 3-0-9 units

Unified introduction to the theory and practice of modern, near linear-time, numerical methods for large-scale partial-differential and integral equations. Topics include preconditioned iterative methods; generalized Fast Fourier Transform and other butterfly-based methods; multiresolution approaches, such as multigrid algorithms and hierarchical low-rank matrix decompositions; and low and high frequency Fast Multipole Methods. Example applications include aircraft design, cardiovascular system modeling, electronic structure computation, and tomographic imaging.

18.337[J] Parallel Computing and Scientific Machine Learning

Same subject as 6.7320[J] Prereq: 18.06 , 18.700 , or 18.701 Acad Year 2023-2024: Not offered Acad Year 2024-2025: G (Spring) 3-0-9 units

Introduction to scientific machine learning with an emphasis on developing scalable differentiable programs. Covers scientific computing topics (numerical differential equations, dense and sparse linear algebra, Fourier transformations, parallelization of large-scale scientific simulation) simultaneously with modern data science (machine learning, deep neural networks, automatic differentiation), focusing on the emerging techniques at the connection between these areas, such as neural differential equations and physics-informed deep learning. Provides direct experience with the modern realities of optimizing code performance for supercomputers, GPUs, and multicores in a high-level language.

18.338 Eigenvalues of Random Matrices

Prereq: 18.701 or permission of instructor G (Fall) 3-0-9 units

Covers the modern main results of random matrix theory as it is currently applied in engineering and science. Topics include matrix calculus for finite and infinite matrices (e.g., Wigner's semi-circle and Marcenko-Pastur laws), free probability, random graphs, combinatorial methods, matrix statistics, stochastic operators, passage to the continuum limit, moment methods, and compressed sensing. Knowledge of Julia helpful, but not required.

18.352[J] Nonlinear Dynamics: The Natural Environment

Same subject as 12.009[J] Prereq: Calculus II (GIR) and Physics I (GIR) ; Coreq: 18.03 U (Fall) Not offered regularly; consult department 3-0-9 units

See description under subject 12.009[J] .

D. H. Rothman

18.353[J] Nonlinear Dynamics: Chaos

Same subject as 2.050[J] , 12.006[J] Prereq: Physics II (GIR) and ( 18.03 or 18.032 ) U (Fall) 3-0-9 units

See description under subject 12.006[J] .

18.354[J] Nonlinear Dynamics: Continuum Systems

Same subject as 1.062[J] , 12.207[J] Subject meets with 18.3541 Prereq: Physics II (GIR) and ( 18.03 or 18.032 ) U (Spring) 3-0-9 units

General mathematical principles of continuum systems. From microscopic to macroscopic descriptions in the form of linear or nonlinear (partial) differential equations. Exact solutions, dimensional analysis, calculus of variations and singular perturbation methods. Stability, waves and pattern formation in continuum systems. Subject matter illustrated using natural fluid and solid systems found, for example, in geophysics and biology.

B. Primkulov

18.3541 Nonlinear Dynamics: Continuum Systems

Subject meets with 1.062[J] , 12.207[J] , 18.354[J] Prereq: Physics II (GIR) and ( 18.03 or 18.032 ) G (Spring) 3-0-9 units

18.355 Fluid Mechanics

Prereq: 2.25 , 12.800 , or 18.354[J] Acad Year 2023-2024: Not offered Acad Year 2024-2025: G (Spring) 3-0-9 units

Topics include the development of Navier-Stokes equations, inviscid flows, boundary layers, lubrication theory, Stokes flows, and surface tension. Fundamental concepts illustrated through problems drawn from a variety of areas, including geophysics, biology, and the dynamics of sport. Particular emphasis on the interplay between dimensional analysis, scaling arguments, and theory. Includes classroom and laboratory demonstrations.

18.357 Interfacial Phenomena

Prereq: 2.25 , 12.800 , 18.354[J] , 18.355 , or permission of instructor Acad Year 2023-2024: Not offered Acad Year 2024-2025: G (Fall) 3-0-9 units

Fluid systems dominated by the influence of interfacial tension. Elucidates the roles of curvature pressure and Marangoni stress in a variety of hydrodynamic settings. Particular attention to drops and bubbles, soap films and minimal surfaces, wetting phenomena, water-repellency, surfactants, Marangoni flows, capillary origami and contact line dynamics. Theoretical developments are accompanied by classroom demonstrations. Highlights the role of surface tension in biology.

18.358[J] Nonlinear Dynamics and Turbulence

Same subject as 1.686[J] , 2.033[J] Subject meets with 1.068 Prereq: 1.060A Acad Year 2023-2024: Not offered Acad Year 2024-2025: G (Spring) 3-2-7 units

See description under subject 1.686[J] .

L. Bourouiba

18.367 Waves and Imaging

Prereq: Permission of instructor Acad Year 2023-2024: Not offered Acad Year 2024-2025: G (Fall) 3-0-9 units

The mathematics of inverse problems involving waves, with examples taken from reflection seismology, synthetic aperture radar, and computerized tomography. Suitable for graduate students from all departments who have affinities with applied mathematics. Topics include acoustic, elastic, electromagnetic wave equations; geometrical optics; scattering series and inversion; migration and backprojection; adjoint-state methods; Radon and curvilinear Radon transforms; microlocal analysis of imaging; optimization, regularization, and sparse regression.

18.369[J] Mathematical Methods in Nanophotonics

Same subject as 8.315[J] Prereq: 8.07 , 18.303 , or permission of instructor Acad Year 2023-2024: G (Spring) Acad Year 2024-2025: Not offered 3-0-9 units

High-level approaches to understanding complex optical media, structured on the scale of the wavelength, that are not generally analytically soluable. The basis for understanding optical phenomena such as photonic crystals and band gaps, anomalous diffraction, mechanisms for optical confinement, optical fibers (new and old), nonlinearities, and integrated optical devices. Methods covered include linear algebra and eigensystems for Maxwell's equations, symmetry groups and representation theory, Bloch's theorem, numerical eigensolver methods, time and frequency-domain computation, perturbation theory, and coupled-mode theories.

S. G. Johnson

18.376[J] Wave Propagation

Same subject as 1.138[J] , 2.062[J] Prereq: 2.003[J] and 18.075 G (Spring) 3-0-9 units

See description under subject 2.062[J] .

T. R. Akylas, R. R. Rosales

18.377[J] Nonlinear Dynamics and Waves

Same subject as 1.685[J] , 2.034[J] Prereq: Permission of instructor Acad Year 2023-2024: G (Spring) Acad Year 2024-2025: Not offered 3-0-9 units

A unified treatment of nonlinear oscillations and wave phenomena with applications to mechanical, optical, geophysical, fluid, electrical and flow-structure interaction problems. Nonlinear free and forced vibrations; nonlinear resonances; self-excited oscillations; lock-in phenomena. Nonlinear dispersive and nondispersive waves; resonant wave interactions; propagation of wave pulses and nonlinear Schrodinger equation. Nonlinear long waves and breaking; theory of characteristics; the Korteweg-de Vries equation; solitons and solitary wave interactions. Stability of shear flows. Some topics and applications may vary from year to year.

18.384 Undergraduate Seminar in Physical Mathematics

Prereq: 12.006[J] , 18.300 , 18.354[J] , or permission of instructor U (Fall) 3-0-9 units

Covers the mathematical modeling of physical systems, with emphasis on the reading and presentation of papers. Addresses a broad range of topics, with particular focus on macroscopic physics and continuum systems: fluid dynamics, solid mechanics, and biophysics. Instruction and practice in written and oral communication provided. Enrollment limited.

18.385[J] Nonlinear Dynamics and Chaos

Same subject as 2.036[J] Prereq: 18.03 or 18.032 Acad Year 2023-2024: G (Spring) Acad Year 2024-2025: Not offered 3-0-9 units

Introduction to the theory of nonlinear dynamical systems with applications from science and engineering. Local and global existence of solutions, dependence on initial data and parameters. Elementary bifurcations, normal forms. Phase plane, limit cycles, relaxation oscillations, Poincare-Bendixson theory. Floquet theory. Poincare maps. Averaging. Near-equilibrium dynamics. Synchronization. Introduction to chaos. Universality. Strange attractors. Lorenz and Rossler systems. Hamiltonian dynamics and KAM theory. Uses MATLAB computing environment.

18.397 Mathematical Methods in Physics

Prereq: 18.745 or some familiarity with Lie theory G (Fall) Not offered regularly; consult department 3-0-9 units Can be repeated for credit.

Content varies from year to year. Recent developments in quantum field theory require mathematical techniques not usually covered in standard graduate subjects.

Theoretical Computer Science

18.400[j] computability and complexity theory.

Same subject as 6.1400[J] Prereq: ( 6.1200[J] and 6.1210 ) or permission of instructor U (Spring) 4-0-8 units

See description under subject 6.1400[J] .

R. Williams, R. Rubinfeld

18.404 Theory of Computation

Subject meets with 6.5400[J] , 18.4041[J] Prereq: 6.1200[J] or 18.200 U (Fall) 4-0-8 units

A more extensive and theoretical treatment of the material in 6.1400[J] / 18.400[J] , emphasizing computability and computational complexity theory. Regular and context-free languages. Decidable and undecidable problems, reducibility, recursive function theory. Time and space measures on computation, completeness, hierarchy theorems, inherently complex problems, oracles, probabilistic computation, and interactive proof systems.

18.4041[J] Theory of Computation

Same subject as 6.5400[J] Subject meets with 18.404 Prereq: 6.1200[J] or 18.200 G (Fall) 4-0-8 units

18.405[J] Advanced Complexity Theory

Same subject as 6.5410[J] Prereq: 18.404 Acad Year 2023-2024: Not offered Acad Year 2024-2025: G (Spring) 3-0-9 units

Current research topics in computational complexity theory. Nondeterministic, alternating, probabilistic, and parallel computation models. Boolean circuits. Complexity classes and complete sets. The polynomial-time hierarchy. Interactive proof systems. Relativization. Definitions of randomness. Pseudo-randomness and derandomizations. Interactive proof systems and probabilistically checkable proofs.

R. Williams

18.408 Topics in Theoretical Computer Science

Prereq: Permission of instructor Acad Year 2023-2024: Not offered Acad Year 2024-2025: G (Fall, Spring) 3-0-9 units Can be repeated for credit.

Study of areas of current interest in theoretical computer science. Topics vary from term to term.

Fall: D. Minzer. Spring: A. Moitra

18.410[J] Design and Analysis of Algorithms

Same subject as 6.1220[J] Prereq: 6.1200[J] and 6.1210 U (Fall, Spring) 4-0-8 units

See description under subject 6.1220[J] .

E. Demaine, M. Goemans

18.413 Introduction to Computational Molecular Biology

Subject meets with 18.417 Prereq: 6.1210 or permission of instructor U (Spring) Not offered regularly; consult department 3-0-9 units

Introduction to computational molecular biology with a focus on the basic computational algorithms used to solve problems in practice. Covers classical techniques in the field for solving problems such as genome sequencing, assembly, and search; detecting genome rearrangements; constructing evolutionary trees; analyzing mass spectrometry data; connecting gene expression to cellular function; and machine learning for drug discovery. Prior knowledge of biology is not required. Particular emphasis on problem solving, collaborative learning, theoretical analysis, and practical implementation of algorithms. Students taking graduate version complete additional and more complex assignments.

18.415[J] Advanced Algorithms

Same subject as 6.5210[J] Prereq: 6.1220[J] and ( 6.1200[J] , 6.3700 , or 18.600 ) Acad Year 2023-2024: Not offered Acad Year 2024-2025: G (Fall) 5-0-7 units

See description under subject 6.5210[J] .

A. Moitra, D. R. Karger

18.416[J] Randomized Algorithms

Same subject as 6.5220[J] Prereq: ( 6.1200[J] or 6.3700 ) and ( 6.1220[J] or 6.5210[J] ) Acad Year 2023-2024: G (Spring) Acad Year 2024-2025: Not offered 5-0-7 units

See description under subject 6.5220[J] .

D. R. Karger

18.417 Introduction to Computational Molecular Biology

Subject meets with 18.413 Prereq: 6.1210 or permission of instructor Acad Year 2023-2024: Not offered Acad Year 2024-2025: G (Spring) 3-0-9 units

18.418[J] Topics in Computational Molecular Biology

Same subject as HST.504[J] Prereq: 6.8701 , 18.417 , or permission of instructor G (Fall) 3-0-9 units Can be repeated for credit.

Covers current research topics in computational molecular biology. Recent research papers presented from leading conferences such as the International Conference on Computational Molecular Biology (RECOMB) and the Conference on Intelligent Systems for Molecular Biology (ISMB). Topics include original research (both theoretical and experimental) in comparative genomics, sequence and structure analysis, molecular evolution, proteomics, gene expression, transcriptional regulation, biological networks, drug discovery, and privacy. Recent research by course participants also covered. Participants will be expected to present individual projects to the class.

18.424 Seminar in Information Theory

Prereq: ( 6.3700 , 18.05 , or 18.600 ) and ( 18.06 , 18.700 , or 18.701 ) U (Fall) 3-0-9 units

Considers various topics in information theory, including data compression, Shannon's Theorems, and error-correcting codes. Students present and discuss the subject matter. Instruction and practice in written and oral communication provided. Enrollment limited.

18.425[J] Cryptography and Cryptanalysis

Same subject as 6.5620[J] Prereq: 6.1220[J] G (Fall) 3-0-9 units

See description under subject 6.5620[J] .

S. Goldwasser, S. Micali, V. Vaikuntanathan

18.434 Seminar in Theoretical Computer Science

Prereq: 6.1220[J] U (Fall, Spring) 3-0-9 units

Topics vary from year to year. Students present and discuss the subject matter. Instruction and practice in written and oral communication provided. Enrollment limited.

Fall: E. Mossel. Spring: D. Minzer

18.435[J] Quantum Computation

Same subject as 2.111[J] , 6.6410[J] , 8.370[J] Prereq: 8.05 , 18.06 , 18.700 , 18.701 , or 18.C06[J] G (Fall) 3-0-9 units

Provides an introduction to the theory and practice of quantum computation. Topics covered: physics of information processing; quantum algorithms including the factoring algorithm and Grover's search algorithm; quantum error correction; quantum communication and cryptography. Knowledge of quantum mechanics helpful but not required.

I. Chuang, A. Harrow, P. Shor

18.436[J] Quantum Information Science

Same subject as 6.6420[J] , 8.371[J] Prereq: 18.435[J] G (Spring) 3-0-9 units

See description under subject 8.371[J] .

I. Chuang, A. Harrow

18.437[J] Distributed Algorithms

Same subject as 6.5250[J] Prereq: 6.1220[J] Acad Year 2023-2024: Not offered Acad Year 2024-2025: G (Fall) 3-0-9 units

See description under subject 6.5250[J] .

M. Ghaffari, N. A. Lynch

18.453 Combinatorial Optimization

Subject meets with 18.4531 Prereq: 18.06 , 18.700 , or 18.701 Acad Year 2023-2024: Not offered Acad Year 2024-2025: U (Spring) 3-0-9 units

Thorough treatment of linear programming and combinatorial optimization. Topics include matching theory, network flow, matroid optimization, and how to deal with NP-hard optimization problems. Prior exposure to discrete mathematics (such as 18.200 ) helpful.

Information: M. X. Goemans

18.4531 Combinatorial Optimization

Subject meets with 18.453 Prereq: 18.06 , 18.700 , or 18.701 Acad Year 2023-2024: Not offered Acad Year 2024-2025: G (Spring) 3-0-9 units

18.455 Advanced Combinatorial Optimization

Prereq: 18.453 or permission of instructor Acad Year 2023-2024: Not offered Acad Year 2024-2025: G (Spring) 3-0-9 units

Advanced treatment of combinatorial optimization with an emphasis on combinatorial aspects. Non-bipartite matchings, submodular functions, matroid intersection/union, matroid matching, submodular flows, multicommodity flows, packing and connectivity problems, and other recent developments.

M. X. Goemans

18.456[J] Algebraic Techniques and Semidefinite Optimization

Same subject as 6.7230[J] Prereq: 6.7210[J] or 15.093[J] Acad Year 2023-2024: Not offered Acad Year 2024-2025: G (Spring) 3-0-9 units

See description under subject 6.7230[J] .

18.504 Seminar in Logic

Prereq: ( 18.06 , 18.510 , 18.700 , or 18.701 ) and ( 18.100A , 18.100B , 18.100P , or 18.100Q ) Acad Year 2023-2024: Not offered Acad Year 2024-2025: U (Fall) 3-0-9 units

Students present and discuss the subject matter taken from current journals or books. Topics vary from year to year. Instruction and practice in written and oral communication provided. Enrollment limited.

18.510 Introduction to Mathematical Logic and Set Theory

Prereq: None Acad Year 2023-2024: U (Fall) Acad Year 2024-2025: Not offered 3-0-9 units

Propositional and predicate logic. Zermelo-Fraenkel set theory. Ordinals and cardinals. Axiom of choice and transfinite induction. Elementary model theory: completeness, compactness, and Lowenheim-Skolem theorems. Godel's incompleteness theorem.

18.515 Mathematical Logic

Prereq: Permission of instructor G (Spring) Not offered regularly; consult department 3-0-9 units

More rigorous treatment of basic mathematical logic, Godel's theorems, and Zermelo-Fraenkel set theory. First-order logic. Models and satisfaction. Deduction and proof. Soundness and completeness. Compactness and its consequences. Quantifier elimination. Recursive sets and functions. Incompleteness and undecidability. Ordinals and cardinals. Set-theoretic formalization of mathematics.

Information: B. Poonen

Probability and Statistics

18.600 probability and random variables.

Prereq: Calculus II (GIR) U (Fall, Spring) 4-0-8 units. REST Credit cannot also be received for 6.3700 , 6.3702

Probability spaces, random variables, distribution functions. Binomial, geometric, hypergeometric, Poisson distributions. Uniform, exponential, normal, gamma and beta distributions. Conditional probability, Bayes theorem, joint distributions. Chebyshev inequality, law of large numbers, and central limit theorem. Credit cannot also be received for 6.041A or 6.041B.

Fall: S. Sheffield. Spring: J. Kelner

18.615 Introduction to Stochastic Processes

Prereq: 6.3700 or 18.600 G (Fall) 3-0-9 units

Basics of stochastic processes. Markov chains, Poisson processes, random walks, birth and death processes, Brownian motion.

18.619[J] Discrete Probability and Stochastic Processes (New)

Same subject as 6.7720[J] , 15.070[J] Prereq: 6.3702 , 6.7700[J] , 18.100A , 18.100B , or 18.100Q G (Spring) 3-0-9 units

See description under subject 15.070[J] .

G. Bresler, D. Gamarnik, E. Mossel, Y. Polyanskiy

18.642 Topics in Mathematics with Applications in Finance

Prereq: 18.03 , 18.06 , and ( 18.05 or 18.600 ) U (Fall) 3-0-9 units

Introduction to mathematical concepts and techniques used in finance. Lectures focusing on linear algebra, probability, statistics, stochastic processes, and numerical methods are interspersed with lectures by financial sector professionals illustrating the corresponding application in the industry. Prior knowledge of economics or finance helpful but not required.

P. Kempthorne, V. Strela, J. Xia

18.650[J] Fundamentals of Statistics

Same subject as IDS.014[J] Subject meets with 18.6501 Prereq: 6.3700 or 18.600 U (Fall, Spring) 4-0-8 units

A rapid introduction to the theoretical foundations of statistical methods that are useful in many applications. Covers a broad range of topics in a short amount of time with the goal of providing a rigorous and cohesive understanding of the modern statistical landscape. Mathematical language is used for intuition and basic derivations but not proofs. Main topics include: parametric estimation, confidence intervals, hypothesis testing, Bayesian inference, and linear and logistic regression. Additional topics may include: causal inference, nonparametric estimation, and classification.

Fall: P. Rigollet. Spring: A. Katsevich

18.6501 Fundamentals of Statistics

Subject meets with 18.650[J] , IDS.014[J] Prereq: 6.3700 or 18.600 G (Fall, Spring) 4-0-8 units

18.655 Mathematical Statistics

Prereq: ( 18.650[J] and ( 18.100A , 18.100A , 18.100P , or 18.100Q )) or permission of instructor G (Spring) 3-0-9 units

Decision theory, estimation, confidence intervals, hypothesis testing. Introduces large sample theory. Asymptotic efficiency of estimates. Exponential families. Sequential analysis. Prior exposure to both probability and statistics at the university level is assumed.

P. Kempthorne

18.656[J] Mathematical Statistics: a Non-Asymptotic Approach

Same subject as 9.521[J] , IDS.160[J] Prereq: ( 6.7700[J] , 18.06 , and 18.6501 ) or permission of instructor G (Spring) 3-0-9 units

See description under subject 9.521[J] .

S. Rakhlin, P. Rigollet

18.657 Topics in Statistics

Topics vary from term to term.

P. Rigollet

18.675 Theory of Probability

Prereq: 18.100A , 18.100B , 18.100P , or 18.100Q G (Fall) 3-0-9 units

Sums of independent random variables, central limit phenomena, infinitely divisible laws, Levy processes, Brownian motion, conditioning, and martingales. Prior exposure to probability (e.g., 18.600 ) recommended.

Y. Shenfeld

18.676 Stochastic Calculus

Prereq: 18.675 G (Spring) 3-0-9 units

Introduction to stochastic processes, building on the fundamental example of Brownian motion. Topics include Brownian motion, continuous parameter martingales, Ito's theory of stochastic differential equations, Markov processes and partial differential equations, and may also include local time and excursion theory. Students should have familiarity with Lebesgue integration and its application to probability.

18.677 Topics in Stochastic Processes

Prereq: 18.675 G (Spring) 3-0-9 units Can be repeated for credit.

Algebra and Number Theory

18.700 linear algebra.

Prereq: Calculus II (GIR) U (Fall) 3-0-9 units. REST Credit cannot also be received for 6.C06[J] , 18.06 , 18.C06[J]

Vector spaces, systems of linear equations, bases, linear independence, matrices, determinants, eigenvalues, inner products, quadratic forms, and canonical forms of matrices. More emphasis on theory and proofs than in 18.06 .

18.701 Algebra I

Prereq: 18.100A , 18.100B , 18.100P , 18.100Q , 18.090 , or permission of instructor U (Fall) 3-0-9 units

18.701 - 18.702 is more extensive and theoretical than the 18.700 - 18.703 sequence. Experience with proofs necessary. 18.701 focuses on group theory, geometry, and linear algebra.

18.702 Algebra II

Prereq: 18.701 U (Spring) 3-0-9 units

Continuation of 18.701 . Focuses on group representations, rings, ideals, fields, polynomial rings, modules, factorization, integers in quadratic number fields, field extensions, and Galois theory.

18.703 Modern Algebra

Prereq: Calculus II (GIR) U (Spring) 3-0-9 units

Focuses on traditional algebra topics that have found greatest application in science and engineering as well as in mathematics: group theory, emphasizing finite groups; ring theory, including ideals and unique factorization in polynomial and Euclidean rings; field theory, including properties and applications of finite fields. 18.700 and 18.703 together form a standard algebra sequence.

18.704 Seminar in Algebra

Prereq: 18.701 , ( 18.06 and 18.703 ), or ( 18.700 and 18.703 ) U (Fall, Spring) 3-0-9 units

Topics vary from year to year. Students present and discuss the subject matter. Instruction and practice in written and oral communication provided. Some experience with proofs required. Enrollment limited.

18.705 Commutative Algebra

Prereq: 18.702 G (Fall) 3-0-9 units

Exactness, direct limits, tensor products, Cayley-Hamilton theorem, integral dependence, localization, Cohen-Seidenberg theory, Noether normalization, Nullstellensatz, chain conditions, primary decomposition, length, Hilbert functions, dimension theory, completion, Dedekind domains.

18.706 Noncommutative Algebra

Prereq: 18.702 Acad Year 2023-2024: Not offered Acad Year 2024-2025: G (Spring) 3-0-9 units

Topics may include Wedderburn theory and structure of Artinian rings, Morita equivalence and elements of category theory, localization and Goldie's theorem, central simple algebras and the Brauer group, representations, polynomial identity rings, invariant theory growth of algebras, Gelfand-Kirillov dimension.

R. Bezrukavnikov

18.708 Topics in Algebra

Prereq: 18.705 Acad Year 2023-2024: Not offered Acad Year 2024-2025: G (Fall) 3-0-9 units Can be repeated for credit.

18.715 Introduction to Representation Theory

Prereq: 18.702 or 18.703 Acad Year 2023-2024: Not offered Acad Year 2024-2025: G (Spring) 3-0-9 units

Algebras, representations, Schur's lemma. Representations of SL(2). Representations of finite groups, Maschke's theorem, characters, applications. Induced representations, Burnside's theorem, Mackey formula, Frobenius reciprocity. Representations of quivers.

18.721 Introduction to Algebraic Geometry

Prereq: 18.702 and 18.901 U (Spring) 3-0-9 units

Presents basic examples of complex algebraic varieties, affine and projective algebraic geometry, sheaves, cohomology.

18.725 Algebraic Geometry I

Prereq: None. Coreq: 18.705 G (Fall) 3-0-9 units

Introduces the basic notions and techniques of modern algebraic geometry. Covers fundamental notions and results about algebraic varieties over an algebraically closed field; relations between complex algebraic varieties and complex analytic varieties; and examples with emphasis on algebraic curves and surfaces. Introduction to the language of schemes and properties of morphisms. Knowledge of elementary algebraic topology, elementary differential geometry recommended, but not required.

18.726 Algebraic Geometry II

Prereq: 18.725 G (Spring) 3-0-9 units

Continuation of the introduction to algebraic geometry given in 18.725 . More advanced properties of the varieties and morphisms of schemes, as well as sheaf cohomology.

18.727 Topics in Algebraic Geometry

Prereq: 18.725 Acad Year 2023-2024: Not offered Acad Year 2024-2025: G (Spring) 3-0-9 units Can be repeated for credit.

18.737 Algebraic Groups

Prereq: 18.705 Acad Year 2023-2024: Not offered Acad Year 2024-2025: G (Spring) 3-0-9 units

Structure of linear algebraic groups over an algebraically closed field, with emphasis on reductive groups. Representations of groups over a finite field using methods from etale cohomology. Some results from algebraic geometry are stated without proof.

18.745 Lie Groups and Lie Algebras I

Prereq: ( 18.701 or 18.703 ) and ( 18.100A , 18.100B , 18.100P , or 18.100Q ) G (Fall) 3-0-9 units

Covers fundamentals of the theory of Lie algebras and related groups. Topics may include theorems of Engel and Lie; enveloping algebra, Poincare-Birkhoff-Witt theorem; classification and construction of semisimple Lie algebras; the center of their enveloping algebras; elements of representation theory; compact Lie groups and/or finite Chevalley groups.

18.747 Infinite-dimensional Lie Algebras

Prereq: 18.745 Acad Year 2023-2024: G (Fall) Acad Year 2024-2025: Not offered 3-0-9 units

18.748 Topics in Lie Theory

Prereq: Permission of instructor Acad Year 2023-2024: Not offered Acad Year 2024-2025: G (Spring) 3-0-9 units Can be repeated for credit.

18.755 Lie Groups and Lie Algebras II

Prereq: 18.745 or permission of instructor G (Spring) 3-0-9 units

A more in-depth treatment of Lie groups and Lie algebras. Topics may include homogeneous spaces and groups of automorphisms; representations of compact groups and their geometric realizations, Peter-Weyl theorem; invariant differential forms and cohomology of Lie groups and homogeneous spaces; complex reductive Lie groups, classification of real reductive groups.

18.757 Representations of Lie Groups

Prereq: 18.745 or 18.755 Acad Year 2023-2024: G (Fall) Acad Year 2024-2025: Not offered 3-0-9 units

Covers representations of locally compact groups, with emphasis on compact groups and abelian groups. Includes Peter-Weyl theorem and Cartan-Weyl highest weight theory for compact Lie groups.

18.781 Theory of Numbers

Prereq: None U (Spring) 3-0-9 units

An elementary introduction to number theory with no algebraic prerequisites. Primes, congruences, quadratic reciprocity, diophantine equations, irrational numbers, continued fractions, partitions.

M.-T. Trinh

18.782 Introduction to Arithmetic Geometry

Prereq: 18.702 Acad Year 2023-2024: Not offered Acad Year 2024-2025: U (Spring) 3-0-9 units

Exposes students to arithmetic geometry, motivated by the problem of finding rational points on curves. Includes an introduction to p-adic numbers and some fundamental results from number theory and algebraic geometry, such as the Hasse-Minkowski theorem and the Riemann-Roch theorem for curves. Additional topics may include Mordell's theorem, the Weil conjectures, and Jacobian varieties.

S. Chidambaram

18.783 Elliptic Curves

Subject meets with 18.7831 Prereq: 18.702 , 18.703 , or permission of instructor Acad Year 2023-2024: U (Fall) Acad Year 2024-2025: Not offered 3-0-9 units

Computationally focused introduction to elliptic curves, with applications to number theory and cryptography. Topics include point-counting, isogenies, pairings, and the theory of complex multiplication, with applications to integer factorization, primality proving, and elliptic curve cryptography. Includes a brief introduction to modular curves and the proof of Fermat's Last Theorem.

A. Sutherland

18.7831 Elliptic Curves

Subject meets with 18.783 Prereq: 18.702 , 18.703 , or permission of instructor Acad Year 2023-2024: G (Fall) Acad Year 2024-2025: Not offered 3-0-9 units

18.784 Seminar in Number Theory

Prereq: 18.701 or ( 18.703 and ( 18.06 or 18.700 )) U (Spring) 3-0-9 units

A. Landesman

18.785 Number Theory I

Dedekind domains, unique factorization of ideals, splitting of primes. Lattice methods, finiteness of the class group, Dirichlet's unit theorem. Local fields, ramification, discriminants. Zeta and L-functions, analytic class number formula. Adeles and ideles. Statements of class field theory and the Chebotarev density theorem.

18.786 Number Theory II

Prereq: 18.785 G (Spring) 3-0-9 units

Continuation of 18.785 . More advanced topics in number theory, such as Galois cohomology, proofs of class field theory, modular forms and automorphic forms, Galois representations, or quadratic forms.

18.787 Topics in Number Theory

Mathematics laboratory, 18.821 project laboratory in mathematics.

Prereq: Two mathematics subjects numbered 18.100 or above U (Fall, Spring) 3-6-3 units. Institute LAB

Guided research in mathematics, employing the scientific method. Students confront puzzling and complex mathematical situations, through the acquisition of data by computer, pencil and paper, or physical experimentation, and attempt to explain them mathematically. Students choose three projects from a large collection of options. Each project results in a laboratory report subject to revision; oral presentation on one or two projects. Projects drawn from many areas, including dynamical systems, number theory, algebra, fluid mechanics, asymptotic analysis, knot theory, and probability. Enrollment limited.

Fall: A. Negut. Spring: L. Piccirillo

18.896[J] Leadership and Professional Strategies & Skills Training (LEAPS), Part I: Advancing Your Professional Strategies and Skills

Same subject as 5.961[J] , 8.396[J] , 9.980[J] , 12.396[J] Prereq: None G (Spring; second half of term) 2-0-1 units

See description under subject 8.396[J] . Limited to 80.

18.897[J] Leadership and Professional Strategies & Skills Training (LEAPS), Part II: Developing Your Leadership Competencies

Same subject as 5.962[J] , 8.397[J] , 9.981[J] , 12.397[J] Prereq: None G (Spring; first half of term) 2-0-1 units

See description under subject 8.397[J] . Limited to 80.

Topology and Geometry

18.900 geometry and topology in the plane.

Prereq: 18.03 or 18.06 U (Spring) 3-0-9 units

Introduction to selected aspects of geometry and topology, using concepts that can be visualized easily. Mixes geometric topics (such as hyperbolic geometry or billiards) and more topological ones (such as loops in the plane). Suitable for students with no prior exposure to differential geometry or topology.

18.901 Introduction to Topology

Subject meets with 18.9011 Prereq: 18.100A , 18.100B , 18.100P , 18.100Q , or permission of instructor U (Fall, Spring) 3-0-9 units

Introduces topology, covering topics fundamental to modern analysis and geometry. Topological spaces and continuous functions, connectedness, compactness, separation axioms, covering spaces, and the fundamental group.

Fall: A. Pieloch. Spring: R. Jiang

18.9011 Introduction to Topology

Subject meets with 18.901 Prereq: 18.100A , 18.100B , 18.100P , 18.100Q , or permission of instructor G (Fall, Spring) 3-0-9 units

18.904 Seminar in Topology

Prereq: 18.901 U (Fall) 3-0-9 units

18.905 Algebraic Topology I

Prereq: 18.901 and ( 18.701 or 18.703 ) G (Fall) 3-0-9 units

Singular homology, CW complexes, universal coefficient and Künneth theorems, cohomology, cup products, Poincaré duality.

D. Alvarez-Gavela

18.906 Algebraic Topology II

Prereq: 18.905 and ( 18.101 or 18.965 ) G (Spring) 3-0-9 units

Continues the introduction to Algebraic Topology from 18.905 . Topics include basic homotopy theory, spectral sequences, characteristic classes, and cohomology operations.

T. S. Mrowka

18.917 Topics in Algebraic Topology

Prereq: 18.906 Acad Year 2023-2024: G (Spring) Acad Year 2024-2025: Not offered 3-0-9 units Can be repeated for credit.

Content varies from year to year. Introduces new and significant developments in algebraic topology with the focus on homotopy theory and related areas.

Information: T. Schlank

18.919 Graduate Topology Seminar

Prereq: 18.906 G (Spring) 3-0-9 units

Study and discussion of important original papers in the various parts of topology. Open to all students who have taken 18.906 or the equivalent, not only prospective topologists.

18.937 Topics in Geometric Topology

Content varies from year to year. Introduces new and significant developments in geometric topology.

18.950 Differential Geometry

Subject meets with 18.9501 Prereq: ( 18.06 , 18.700 , or 18.701 ) and ( 18.100A , 18.100B , 18.100P , or 18.100Q ) U (Fall) 3-0-9 units

Introduction to differential geometry, centered on notions of curvature. Starts with curves in the plane, and proceeds to higher dimensional submanifolds. Computations in coordinate charts: first and second fundamental form, Christoffel symbols. Discusses the distinction between extrinsic and intrinsic aspects, in particular Gauss' theorema egregium. The Gauss-Bonnet theorem. Geodesics. Examples such as hyperbolic space.

18.9501 Differential Geometry

Subject meets with 18.950 Prereq: ( 18.06 , 18.700 , or 18.701 ) and ( 18.100A , 18.100B , 18.100P , or 18.100Q ) G (Fall) 3-0-9 units

18.952 Theory of Differential Forms

Prereq: 18.101 and ( 18.700 or 18.701 ) U (Spring) Not offered regularly; consult department 3-0-9 units

Multilinear algebra: tensors and exterior forms. Differential forms on R n : exterior differentiation, the pull-back operation and the Poincaré lemma. Applications to physics: Maxwell's equations from the differential form perspective. Integration of forms on open sets of R n . The change of variables formula revisited. The degree of a differentiable mapping. Differential forms on manifolds and De Rham theory. Integration of forms on manifolds and Stokes' theorem. The push-forward operation for forms. Thom forms and intersection theory. Applications to differential topology.

18.965 Geometry of Manifolds I

Prereq: 18.101 , 18.950 , or 18.952 G (Fall) 3-0-9 units

Differential forms, introduction to Lie groups, the DeRham theorem, Riemannian manifolds, curvature, the Hodge theory. 18.966 is a continuation of 18.965 and focuses more deeply on various aspects of the geometry of manifolds. Contents vary from year to year, and can range from Riemannian geometry (curvature, holonomy) to symplectic geometry, complex geometry and Hodge-Kahler theory, or smooth manifold topology. Prior exposure to calculus on manifolds, as in 18.952 , recommended.

18.966 Geometry of Manifolds II

Prereq: 18.965 G (Spring) 3-0-9 units

Continuation of 18.965 , focusing more deeply on various aspects of the geometry of manifolds. Contents vary from year to year, and can range from Riemannian geometry (curvature, holonomy) to symplectic geometry, complex geometry and Hodge-Kahler theory, or smooth manifold topology.

18.968 Topics in Geometry

Prereq: 18.965 Acad Year 2023-2024: Not offered Acad Year 2024-2025: G (Spring) 3-0-9 units Can be repeated for credit.

18.979 Graduate Geometry Seminar

Prereq: Permission of instructor G (Spring) Not offered regularly; consult department 3-0-9 units Can be repeated for credit.

Content varies from year to year. Study of classical papers in geometry and in applications of analysis to geometry and topology.

18.994 Seminar in Geometry

Prereq: ( 18.06 , 18.700 , or 18.701 ) and ( 18.100A , 18.100B , 18.100P , or 18.100Q ) Acad Year 2023-2024: Not offered Acad Year 2024-2025: U (Spring) 3-0-9 units

Students present and discuss subject matter taken from current journals or books. Topics vary from year to year. Instruction and practice in written and oral communication provided. Enrollment limited.

18.999 Research in Mathematics

Prereq: Permission of instructor G (Fall, IAP, Spring, Summer) Units arranged Can be repeated for credit.

Opportunity for study of graduate-level topics in mathematics under the supervision of a member of the department. For graduate students desiring advanced work not provided in regular subjects.

18.C20[J] Introduction to Computational Science and Engineering

Same subject as 9.C20[J] , 16.C20[J] , CSE.C20[J] Prereq: 6.100A ; Coreq: 8.01 and 18.01 U (Fall, Spring; second half of term) 3-0-3 units Credit cannot also be received for 6.100B

See description under subject 16.C20[J] .

D. L. Darmofal, N. Seethapathi

18.C25[J] Real World Computation with Julia (New)

Same subject as 1.C25[J] , 6.C25[J] , 12.C25[J] , 16.C25[J] , 22.C25[J] Prereq: 6.100A , 18.03 , and 18.06 U (Fall) 3-0-9 units

Focuses on algorithms and techniques for writing and using modern technical software in a job, lab, or research group environment that may consist of interdisciplinary teams, where performance may be critical, and where the software needs to be flexible and adaptable. Topics include automatic differentiation, matrix calculus, scientific machine learning, parallel and GPU computing, and performance optimization with introductory applications to climate science, economics, agent-based modeling, and other areas. Labs and projects focus on performant, readable, composable algorithms, and software. Programming will be in Julia. Expects students to have some familiarity with Python, Matlab, or R. No Julia experience necessary.

A. Edelman, R. Ferrari, B. Forget, C. Leiseron,Y. Marzouk, J. Williams

18.UR Undergraduate Research

Undergraduate research opportunities in mathematics. Permission required in advance to register for this subject. For further information, consult the departmental coordinator.

18.THG Graduate Thesis

Program of research leading to the writing of a Ph.D. thesis; to be arranged by the student and an appropriate MIT faculty member.

18.S096 Special Subject in Mathematics

Prereq: Permission of instructor U (IAP, Spring) Units arranged Can be repeated for credit.

Opportunity for group study of subjects in mathematics not otherwise included in the curriculum. Offerings are initiated by members of the Mathematics faculty on an ad hoc basis, subject to departmental approval.

18.S097 Special Subject in Mathematics

Prereq: Permission of instructor U (IAP) Units arranged [P/D/F] Can be repeated for credit.

18.S190 Special Subject in Mathematics

Prereq: Permission of instructor U (IAP) Units arranged Can be repeated for credit.

18.S191 Special Subject in Mathematics

Prereq: Permission of instructor U (Fall) Not offered regularly; consult department Units arranged Can be repeated for credit.

18.S995 Special Subject in Mathematics

Prereq: Permission of instructor Acad Year 2023-2024: Not offered Acad Year 2024-2025: G (Spring) Units arranged Can be repeated for credit.

Opportunity for group study of advanced subjects in mathematics not otherwise included in the curriculum. Offerings are initiated by members of the mathematics faculty on an ad hoc basis, subject to departmental approval.

18.S996 Special Subject in Mathematics

Prereq: Permission of instructor G (Spring) Units arranged Can be repeated for credit.

Opportunity for group study of advanced subjects in mathematics not otherwise included in the curriculum. Offerings are initiated by members of the Mathematics faculty on an ad hoc basis, subject to Departmental approval.

18.S997 Special Subject in Mathematics

18.s998 special subject in mathematics.

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Elchanan Mossel Elected to AAAS

Elchanan Mossel has been elected to the American Academy of Arts & Sciences in 2024.

He and six other MIT faculty members are among 250 leaders from academia, the arts, industry, public policy, and research elected this year by one of the nation’s most prestigious honorary societies.

Congratulations, Elchanan!

Andre Lee Dixon Selected for School of Science Infinite Mile Award

The School of Science has selected Mathematics Program Coordinator André Lee Dixon as one of the recipients of the 2024 Infinite Mile Award!

“I have been consistently struck by the level of initiative and passion André brings to work,” says his nominator, John Urschel PhD ’21.

Infinite Mile Award winners are nominated by colleagues for going above and beyond in their roles at the Institute.

Congratulations, André!

John Urschel Receives Early Career Prize from SIAM Activity Group on Linear Algebra

John Urschel PhD ’21 will receive the 2024 SIAM Activity Group on Linear Algebra Early Career Prize. He will be awarded this May at the SIAM Conference on Applied Linear Algebra in Paris.

Established in 2017, this prize is awarded every three years to one post-PhD early-career researcher in the field of applicable linear algebra, for outstanding contributions within six years of receiving their PhD.

Congratulations, John!

PRIMES and RSI Students Awarded at Regeneron

PRIMES student Michelle Wei won 3rd Place ( $ 150,000 scholarship) for her project “Solving Second-Order Cone Programs Deterministically in Matrix Multiplication Time,” mentored by EECS’ Guanghao Ye.

PRIMES and RSI student Alan Bu earned 10th Place ( $ 40,000) for his RSI project, “On the Maximum Number of Spanning Trees in a Planar Graph with a Fixed Number of Edges: A Linear-Algebraic Connection,” mentored by Yuchong Pan . Each of them took prizes for math projects that finished within the top ten spots. Another PRIMES student became a finalist ( $ 25,000), and seven other PRIMES and RSI students won national scholar awards.

Congratulations to the winners, and a big thank you to their mentors, PRIMES Chief Research Advisor Pavel Etingof , RSI Faculty Advisor David Jerison , and PRIMES/RSI head mentor Tanya Khovanova !

MIT Students Take First Place in the 84th Putnam Math Competition

For the fourth time in the history of the annual William Lowell Putnam Mathematical Competition, and for the fourth year in a row, all five of the top spots in the contest, known as Putnam Fellows, came from a single school — MIT.

Putnam Fellows include three repeats, sophomores Papon Lapate and Luke Robitaille, and junior Brian Liu, plus junior Ankit Bisain and first-year Jiangqi Dai. Each receives an award of $2,500.

MIT’s 2023 Putnam team, made up of Bisain, Lapate, and Robitaille, also finished in first place — MIT’s eighth first-place win in the past 10 competitions. Teams are based on the three top scorers from each institution. The institution with the first-place team receives a $ 25,000 award, and each team member receives $ 1,000.

The top scoring female, first-year Isabella Zhu, received the Elizabeth Lowell Putnam Prize, which includes a $ 1,000 award. She is the seventh MIT student to receive this honor since the award began in 1992.

In total, 68 out of the top 100 test-takers who took the exam on December 2, 2023, were MIT students. Beyond the top 5 scorers, MIT students took 8 of the next 11 spots (each awarded $ 1,000), 7 of the next 10 after that (each awarded $ 250), and 48 out of a total of 75 honorable mentions.

“I am incredibly proud of our students’ amazing effort and performance at the Putnam Competition,” says Associate Professor of Mathematics Yufei Zhao ’10, PhD ’15. “MIT is truly a unique place to be a math major.”

Congratulations to everyone who participated in this year's exam!

A full list of the winners can be found on the Putnam website .

Grad Students Ishan Levy and Mehtaab Sawhney Receive Clay Research Fellowships

Ishan Levy and Mehtaab Sawhney have been awarded 2024 Clay Research Fellowships , for a term of five years.

Levy is known for his contributions to homotopy theory, and Sawhney is recognized for his breakthroughs on fundamental problems across extremal combinatorics, probability theory, and theoretical computer science.

Other current Fellows with MIT Math connections include researcher Yang Li , who received it in 2020; Assistant Professor Lisa Piccirillo , and former postdocs Maggie Miller and Alexander Smith , in 2021; and CLE Moore Instructor Ziquan Zhuang , in 2022.

Congratulations, Ishan and Mehtaab!

MIT Calendar
Conferences

Upcoming Conference

The many combinatorial legacies of richard p. stanley.

Immense Birthday Glory of the Epic Catalonian Rascal

June 3-7, 2024

Harvard Geological Lecture Hall

Applications

Mit graduate admissions is a decentralized network of departments that extends across all five academic schools and includes 46 departmental programs..

Each of these programs has an online application with a specific set of requirements and deadlines.

Answers to most common questions can be found online in the Frequently Asked Questions section. Specific questions related to department requirements, the review process, or test score requirements should be sent to the program to which you are applying.

Please send any general questions not addressed on our website to [email protected] .

Beginning the application process

To begin, select one of the 46 departmental programs , and click the link to “Apply Here.” Applicants must follow the application instructions provided by the department to which they are applying and send transcripts to the correct location, if required.

The MIT application process is dominated by two application platforms. The MIT Sloan School of Management, along with several other graduate programs, use a unified application portal called Slate to support their master’s and doctoral programs. Other programs at MIT utilize an online application system referred to as “GradApply.” The appropriate application portal is linked on each program’s webpage in the Degree Programs directory .

Requesting a waiver for the English language proficiency exam requirement

English proficiency exam requirements vary by department. Each department will list English language requirements and waiver opportunities on their admissions website or directly in the online application. Waiver requests should not be sent to the admissions email address.

No two departments at MIT are exactly alike; as such, there is no single application deadline or review period that fits all departments. However, there are general timeframes that most departments follow, with the exception of fields related to business and management.

September – Online applications become available
December/January – Deadlines to submit application approach
January/February/March – Departments review applications, conduct interviews as needed, and make application decisions
April 15 – Reply deadline utilized by all departments that offer financial support in coordination with the Council of Graduate Schools (CGS)

For more information about the April 15 deadline to reply to an admissions offer, please review the April 15 Resolution .

Visiting MIT

If you are interested in visiting MIT, the Institute Events office has created useful information to help you plan your visit. Please note, there is no general graduate admissions tour or sessions. Graduate applicants interested in arranging a visit should contact the department or program of interest directly to see if arrangements can be made.

Useful links

MIT Nondiscrimination Policy
MIT Annual Security and Fire Safety Report
Council of Graduate Schools April 15 Resolution

Page downloads

Download the April 15 Resolution [last updated in 2022]

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If you require further information, please visit the Privacy Policy page.

Current MIT Graduate Students

Doctoral Programs in Computational Science and Engineering

Application & admission information.

The Center for Computational Science and Engineering (CCSE) offers two doctoral programs in computational science and engineering (CSE) – one leading to a standalone PhD degree in CSE offered entirely by CCSE (CSE PhD) and the other leading to an interdisciplinary PhD degree offered jointly with participating departments in the School of Engineering and the School of Science (Dept-CSE PhD).

While both programs enable students to specialize at the doctoral level in a computation-related field via focused coursework and a thesis, they differ in essential ways. The standalone CSE PhD program is intended for students who plan to pursue research in cross-cutting methodological aspects of computational science. The resulting doctoral degree in Computational Science and Engineering is awarded by CCSE via the the Schwarzman College of Computing. In contrast, the interdisciplinary Dept-CSE PhD program is intended for students who are interested in computation in the context of a specific engineering or science discipline. For this reason, this degree is offered jointly with participating departments across the Institute; the interdisciplinary degree is awarded in a specially crafted thesis field that recognizes the student’s specialization in computation within the chosen engineering or science discipline.

Applicants to the standalone CSE PhD program are expected to have an undergraduate degree in CSE, applied mathematics, or another field that prepares them for an advanced degree in CSE. Applicants to the Dept-CSE PhD program should have an undergraduate degree in a related core disciplinary area as well as a strong foundation in applied mathematics, physics, or related fields. When completing the MIT CSE graduate application , students are expected to declare which of the two programs they are interested in. Admissions decisions will take into account these declared interests, along with each applicant’s academic background, preparation, and fit to the program they have selected. All applicants are asked to specify MIT CCSE-affiliated faculty that best match their research interests; applicants to the Dept-CSE PhD program also select the home department(s) that best match. At the discretion of the admissions committee, Dept-CSE PhD applications might also be shared with a home department beyond those designated in the application. CSE PhD admissions decisions are at the sole discretion of CCSE; Dept-CSE PhD admission decisions are conducted jointly between CCSE and the home departments.

Please note: These are both doctoral programs in Computational Science and Engineering; applicants interested in Computer Science must apply to the Department of Electrical Engineering and Computer Science .

Important Dates

September 15: Application Opens December 1: Deadline to apply for admission* December – March: Application review period January – March: Decisions released on rolling basis

*All supplemental materials (e.g., transcripts, test scores, letters of recommendation) must also be received by December 1. Application review begins on that date, and incomplete applications may not be reviewed. Please be sure that your recommenders are aware of this hard deadline, as we do not make exceptions. We also do not allow students to upload/submit material beyond what is required, such as degree certificates, extra recommendations, publications, etc.

A complete electronic CSE application includes the following:

Three letters of recommendation ;
Students admitted to the program will be required to supply official transcripts. Discrepancies between unofficial and official transcripts may result in the revocation of the admission offer.
Statement of objectives (limited to approximately one page) and responses to department-specific prompts for Dept-CSE PhD applicants;
Official GRE General Test score report , sent to MIT by ETS via institute code 3514 GRE REQUIREMENT WAIVED FOR FALL 2024 ;
Official IELTS score report sent to MIT by IELTS† (international applicants from non-English speaking countries only; see below for more information)
Resume or CV , uploaded in PDF format;
MIT graduate application fee of $75‡.

‡Application Fee

The MIT graduate application fee of $75.00 is a mandatory requirement set by the Institute payable by credit card. Please visit the MIT Graduate Admission Application Fee Waiver page for information about fee waiver eligibility and instructions.

Please note: CCSE cannot issue fee waivers; email requests for fee waivers sent to [email protected] will not be considered.

Admissions Contact Information

Email: [email protected]

► Current MIT CSE SM Students: Please see the page for Current MIT Graduate Students .

GRE Requirement

GRE REQUIREMENT WAIVED FOR FALL 2024 All applicants are required to take the Graduate Record Examination (GRE) General Aptitude Test. The MIT code for submitting GRE score reports is 3514 (you do not need to list a department code). GRE scores must current; ETS considers scores valid for five years after the testing year in which you tested.

†English Language Proficiency Requirement

The CSE PhD program requires international applicants from non-English speaking countries to take the academic version of the International English Language Testing System (IELTS). The IELTS exam measures one’s ability to communicate in English in four major skill areas: listening, reading, writing, and speaking. A minimum IELTS score of 7 is required for admission. For more information about the IELTS, and to find out where and how to take the exam, please visit the IELTS web site .

While we will also accept the TOEFL iBT (Test of English as a Foreign Language), we strongly prefer the IELTS. The minimum TOEFL iBT score is 100.

This requirement is waived for those who can demonstrate that one or more of the following are true:

English is/was the language of instruction in your four-year undergraduate program,
English is the language of your employer/workplace for at least the last four years,
English was your language of instruction in both primary and secondary schools.

Degree Requirements for Admission

To be admitted as a regular graduate student, an applicant must have earned a bachelor’s degree or its equivalent from a college, university, or technical school of acceptable standing. Students in their final year of undergraduate study may be admitted on the condition that their bachelor’s degree is awarded before they enroll at MIT.

Applicants without an SM degree may apply to the CSE PhD program, however, the Departments of Aeronautics and Astronautics and Mechanical Engineering nominally require the completion of an SM degree before a student is considered a doctoral candidate. As a result, applicants to those departments holding only a bachelor’s degree are asked in the application to indicate whether they prefer to complete the CSE SM program or an SM through the home department.

Nondiscrimination Policy

The Massachusetts Institute of Technology is committed to the principle of equal opportunity in education and employment. To read MIT’s most up-to-date nondiscrimination policy, please visit the Reference Publication Office’s nondiscrimination statement page .

Additional Information

For more details, as well as answers to most commonly asked questions regarding the admissions process to individual participating Dept-CSE PhD departments including details on financial support, applicants are referred to the website of the participating department of interest.

Understanding the process: What we look for

The match between you and mit.

Ask any admissions officer at MIT, and they will tell you that while grades and scores are important, it’s really the match between applicant and the Institute that drives our selection process.

Here are the key components:

Alignment with mit’s mission.

Remember that there are many ways to make the world better—we’re not looking for applicants to have cured all infectious disease in the world by the time they’re 15. Tutoring a single kid in math changes the world. Lobbying a senator to amend bad policy changes the world. There are thousands of examples.

Collaborative and cooperative spirit

The core of the MIT spirit is collaboration and cooperation; you can see it all over the Institute. Many of the problem sets (our affectionate term for homework) at MIT are designed to be worked on in groups, and cross-department labs are very common. MIT is known for its interdisciplinary research—passionate people working across their differences to tackle big questions and challenges together. If you enjoy working alone all the time, that’s completely valid, but you might not be particularly happy at MIT.

Opportunities are abundant at MIT, but they must be seized. Research projects, seed money, and interesting lectures aren’t simply handed to students on silver platters here. For those students who take initiative—who take advantage of what’s around them—MIT’s resources are unparalleled.

Risk-taking

MIT wants to admit people who are not only planning to succeed but who are also not afraid to fail. When people take risks in life, they learn resilience—because risk leads to failure as often as it leads to success. The most creative and successful people—and MIT is loaded with them—know that failure is part of life and that if you stay focused and don’t give up, goals are ultimately realized.

Hands-on creativity

MIT is an active, hands-on place. Innovation is risky and messy! Getting your hands dirty and trying something new is often the best way to achieve success. We apply theoretical knowledge to real-world problems here; our Latin motto means “Mind and Hand.” In other words, you shouldn’t just enjoy thinking, you should also enjoy doing.

Intensity, curiosity, and excitement

In a nutshell, you should be invested in the things that really mean something to you (we’re not particularly picky as to what). Explore! Choose quality over quantity—you don’t have to do a million things to get into college. Put your heart into a few things that you truly care about and that will be enough.

The character of the MIT community

Our community is comprised of thoughtful people from a wide variety of backgrounds and worldviews who take care of each other and lift each other up; they inspire each other to work and dream beyond their potential. Students regularly work alongside faculty and staff to shape MIT policies and further our mission to make the world a better place. We’re looking to admit people who feel responsible to their communities and will help sustain the heart of MIT’s.

The ability to prioritize balance

Despite what you may have heard, this place is NOT all about work. To be successful here, you must prioritize some measure of downtime. Therefore, we like to see that you’ve prioritized some downtime in high school as well. Our application’s essay question (Tell us about something you do simply for the pleasure of it.) is not a trick question. Answer it honestly.

Related blogs

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Smart. Open. Grounded. Inventive. Read our Ideas Made to Matter.

Which program is right for you?

Through intellectual rigor and experiential learning, this full-time, two-year MBA program develops leaders who make a difference in the world.

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Earn your MBA and SM in engineering with this transformative two-year program.

Combine an international MBA with a deep dive into management science. A special opportunity for partner and affiliate schools only.

A doctoral program that produces outstanding scholars who are leading in their fields of research.

Bring a business perspective to your technical and quantitative expertise with a bachelor’s degree in management, business analytics, or finance.

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This 20-month MBA program equips experienced executives to enhance their impact on their organizations and the world.

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A non-degree, customizable program for mid-career professionals.

Admissions Requirements

The following are general requirements you should meet to apply to the MIT Sloan PhD Program. Complete instructions concerning application requirements are available in the online application.

General Requirements

Bachelor's degree or equivalent
A strong quantitative background (the Accounting group requires calculus)
Exposure to microeconomics and macroeconomics (the Accounting group requires microeconomics)

A Guide to Business PhD Applications by Abhishek Nagaraj (PhD 2016) may be of interest.

Application Components

Statement of purpose.

Your written statement is your chance to convince the admissions committee that you will do excellent doctoral work and that you have the promise to have a successful career as an academic researcher.

GMAT/GRE Scores

We require either a valid GMAT or valid GRE score. At-home testing is allowed. Your unofficial score report from the testing institution is sufficient for application. If you are admitted to the program, you will be required to submit your official test score for verification.

We do not have a minimum score requirement. We do not offer test waivers. Registration information for the GMAT (code X5X-QS-21) and GRE (code 3510) may be obtained at www.mba.com and www.ets.org respectively.

TOEFL/IELTS Scores

We require either a valid TOEFL (minimum score 577 PBT/90 IBT ) or valid IELTS (minimum score 7) for all non-native English speakers. Your unofficial score report from the testing institution is sufficient for application. If you are admitted to the program, you will be required to submit your official test score for verification. Registration information for TOEFL (code 3510) and IELTS may be obtained at www.toefl.org and www.ielts.org respectively.

The TOEFL/IELTS test requirement is waived only if you meet one of the following criteria:

You are a native English speaker.
You attended all years of an undergraduate program conducted solely in English, and are a graduate of that program.

Please do not contact the PhD Program regarding waivers, as none will be discussed. If, upon review, the faculty are interested in your application with a missing required TOEFL or IELTS score, we may contact you at that time to request a score.

Transcripts

We require unofficial copies of transcripts for each college or university you have attended, even if no degree was awarded. If these transcripts are in a language other than English, we also require a copy of a certified translation. In addition, you will be asked to list the five most relevant courses you have taken.

Letters of Recommendation

We require three letters of recommendation. Academic letters are preferred, especially those providing evidence of research potential. We allow for an optional fourth recommendation, but no more than four recommendations are allowed.

Your resume should be no more than two pages. You may chose to include teaching, professional experience, research experience, publications, and other accomplishments in outside activities.

Writing Sample(s)

Applicants are encouraged to submit a writing sample. For applicants to the Finance group, a writing sample is required. There are no specific guidelines for your writing sample. Possible options include (but are not limited to) essays, masters’ theses, capstone projects, or research papers.

Video Essay

A video essay is required for the Accounting research group and optional for the Marketing and System Dynamics research groups. The essay is a short and informal video answering why you selected this research group and a time where you creatively solved a problem. The video can be recorded with your phone or computer, and should range from 2 to 5 minutes in length. There is no attention — zero emphasis! — on the production value of your video.

Nondiscrimination Policy: The Massachusetts Institute of Technology is committed to the principle of equal opportunity in education and employment. For complete text of MIT’s Nondiscrimination Statement, please click here .

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Admissions Information for Prospective Graduate Students

Thank you for considering the PhD program in Physics at MIT. Information regarding our graduate program and our application process can be found below and through the following webpages and other links on this page. If your questions are not answered after reviewing this information, please contact us at [email protected] .

Here are some links to pages relevant to prospective students:

Material Required for a Complete Application , and information about When/How to Apply can be found below on this page.
We have an FAQ which should help to answer many questions, and we provide Application Assistance from staff and students if you don’t find what you need in the FAQ.
Additional Guidance about the application itself, along with examples, can be found on a separate page. The graduate application is available at https://apply.mit.edu/apply/ .
General information about the graduate program and research areas in the physics department may also be of use.
MSRP (MIT Summer Research Program) is designed to give underrepresented and underserved students access to an MIT research experience, pairing each student with a faculty member who will oversee the student conducting a research project at MIT.

Statement regarding admissions process during COVID Pandemic (Updated Summer 2023)

MIT has adopted the following principle: MIT’s admissions committees and offices for graduate and professional schools will take the significant disruptions of the COVID-19 outbreak in 2020 into account when reviewing students’ transcripts and other admissions materials as part of their regular practice of performing individualized, holistic reviews of each applicant.

In particular, as we review applications now and in the future, we will respect decisions regarding the adoption of Pass/No Record (or Credit/No Credit or Pass/Fail) and other grading options during the unprecedented period of COVID-19 disruptions, whether those decisions were made by institutions or by individual students. We also expect that the individual experiences of applicants will richly inform applications and, as such, they will be considered with the entirety of a student’s record.

Ultimately, even in these challenging times, our goal remains to form graduate student cohorts that are collectively excellent and composed of outstanding individuals who will challenge and support one another.

Questions or concerns about this statement should be directed to the Physics Department ( [email protected] ).

Also, to stay up-to-date on the latest information on MIT and the COVID-19 pandemic at now.mit.edu .

Applying to the MIT Department of Physics

We know that the application process can be time-consuming, stressful, and costly. We are committed to reducing these barriers and to helping all applicants receive a full and fair assessment by our faculty reviewers. Help is available from the Physics Graduate Admissions Office at [email protected] and additional assistance from current students is offered during the admissions season. Further details are described at the end of this page in our Assistance for Prospective Applicants section.

The list below describes the important elements of a complete application. Please reach out to us at [email protected] if you have a concern or logistical difficulty that could prevent you from providing your strongest application.

Required for a Complete Application

1. online application and application fee.

MIT Graduate Admissions Online Graduate Application
Application Fee: $75 NOTE: Applicants who feel that this fee may prevent them from applying should send a short email to [email protected] to describe their general reasons for requesting a waiver. We will follow up with information about how to apply for a formal ‘application fee waiver’. Additional documents may be required, so additional time will be necessary to process requests. Either the fee or a formal fee waiver is required with a submitted application.

2. University Transcript(s)

Unofficial transcripts are sufficient for our initial review, with final transcripts required as a condition of matriculation for successful applicants. Applicants should include a scan of their transcript(s) and, if a degree is in progress, should include a list of the class subjects being taken in the current semester. The GradApply portal will allow applicants to log back into the application after the deadline to add their Fall term grades when they are available.

Note: We will respect decisions regarding the adoption of Pass/No Record (or Credit/No Credit or Pass/Fail) and other grading options during the unprecedented period of COVID-19 disruptions, whether those decisions were made by institutions or by individual students.

3. Standardized Test Results

GRE Tests are not required for graduate applications submitted in 2023. The Physics subject GRE (PGRE) will be optional in 2023 and our department does not require results from the General GRE test.
TOEFL or IELTS Test or a waiver is required for non-native English speakers. MIT’s TOEFL school code is 3514; the code for the Department of Physics is 76. IELTS does not require a code. Eligibility for TOEFL/IELTS waivers is in our FAQ section .
Self-reported scores are sufficient for our initial application screening, with official scores required for admitted students as a condition of their offer. Applicants should attach a scanned copy of their test score report.

4. Letters of Recommendation

Letters should include any individual work applicants have done and/or areas where they have special strengths. It is possible to submit up to 6 total letters, but 3 are sufficient for a complete application and committee members may evaluate applications based on the first three letters that they read.

5. Statement of Objectives

Research is central to graduate study in physics. The Statement of Objectives/Purpose should include descriptions of research projects, aptitude and achievements as completely as possible. This important part of the application provides an opportunity to describe any interests, skills, and background relative to the research areas selected on the application form. Applicants should share anything that prepares them for graduate studies and describe their proudest achievements.

Additional Application Materials

Research, Teaching, and Community Engagement – Any special background or achievement that prepares the applicant for Physics graduate studies at MIT. This may include research at their undergraduate school as part of their Bachelor or Master degree, or summer research at another program or school. We also value our student’s contributions to their community on a variety of scales (from institutional to societal) and we encourage applicants to tell us about their teaching and community engagement activities. The “experience” questions are intended to provide a CV-like listing of achievements, some of which may be elaborated on in the “Statement of Objectives” and/or the optional “Personal Statement”.
Publications, Talks, and Merit Based Recognition – Recognition of success in research, academics, and outreach can take many forms, including publications, talks, honors, prizes, awards, fellowships, etc. This may include current nominations for scholarships or papers submitted for publication.
Optional Personal Statement – Members of our community come from a wide variety of backgrounds and experiences. We welcome any personal information that will help us to evaluate applications holistically and will provide context for the applicant’s academic achievements. This statement may include extenuating circumstances, significant challenges that were overcome, a non-traditional educational background, description of any advocacy or values work, or other information that may be relevant.
Detailed instructions for each application section, and many examples , can be found on the “ Additional Guidance ” page. The detailed instructions are lengthy, and are intended to be read only “as needed” while you work on your application (i.e., you don’t need to go read the whole thing before you start).

When/How to Apply

When : Applications can be submitted between September 15 and December 15 by 11:59pm EST for the following year.

How : The application is online at https://apply.mit.edu/apply/

Application Assistance

Faculty, students, and staff have collaborated to provide extensive guidance to prospective applicants to our graduate degree program. Resources include several department webpages to inform prospective applicants about our PhD degree requirements and to help applicants as they assemble and submit their materials. In addition to staff responses to emails, current graduate students will answer specific individual questions, give one admissions-related webinar, and provide a mentorship program for selected prospective applicants.

During the application season, prospective students may request additional information from current students about the admissions process, graduate student life, or department culture, either as a response to a specific individual email question or for more in-depth assistance. Applicants will benefit most from contacting us early in the process, when current students and staff will be available to respond to questions and mentor selected applicants. After mid-November, department staff will continue to field questions through the admission process.

Here are some resources for prospective applicants:

Our website provides answers to many frequently-asked admissions questions .
Admissions staff are available for questions at [email protected] .
Current students collaborate with staff to answer specific questions emailed to [email protected] .
PhysGAAP Webinars are designed to provide student perspectives on the application and admissions processes in an interactive format. This year’s webinar will take place on Wednesday, Nov 1st, 2023 from 10am to 12pm EDT. Sign up here: https://mit.co1.qualtrics.com/jfe/form/SV_ah13eCcEh0cKW7I
PhysGAAP Mentoring provides in-depth guidance through the application process.

Student-led Q&A Service

A team of our current graduate students is available to share their experience and perspectives in response to individual questions which may fall under any of the following categories:

Coursework/research (e.g., How do I choose between two research areas and how do I find a potential research advisor?)
Culture (e.g., What is it like to be a student of a particular identity at MIT?)
Student life (e.g., What clubs or extracurriculars do graduate students at MIT take part in?)

To request a response from the current students, please send an email to [email protected] and indicate clearly in the subject line or first sentence that you would like your email forwarded to the PhysGAAP student team. Depending on the scope of your question, department staff will send your email to current students.

We encourage you to reach out as early as you can to maximize the benefit that this help can provide to you. While the admissions office staff will continue to field your questions throughout the admissions season, current students may not be available to respond to questions sent after November 15.

This student email resource is designed for individual basic questions. More in-depth guidance, especially about the application itself, will be available through the PhysGAAP Webinars and/or PhysGAAP Mentorship Program described below.

Student-led Webinar

A panel of our graduate students hosted a 2-hour long Zoom webinar in late October of 2022 to present information about the application and admissions processes, and to respond to questions on these topics. The webinar addressed general questions about preparing, completing, and submitting the application; what the Admissions Committee is looking for; and the general timeline for the admissions process.

Below is video from our latest webinar that took place on Wednesday, Nov 1st, 2023. Check back here in Fall 2024 for information on our next webinar.

Note: We have compiled a document containing supplementary material for previous PhysGAAP webinars.

Webinar Recordings

Past PhysGAAP Webinars

Please note that the two webinars below are from prior years and may contain outdated information about some topics, such as GRE requirements.

October 2022
December 2021
September 2021

Mentorship for Prospective Applicants

In addition to the materials available through this website, answers to emails sent to the department, or from our graduate student webinars, we also offer one-on-one mentoring for students who desire more in-depth individual assistance. Prospective applicants may apply to the PhysGAAP Mentoring program,, which pairs prospective graduate school applicants with current graduate students who can assist them through the application process, provide feedback on their application materials and insight into graduate school and the MIT Physics Department.

We welcome interest in the PhysGAAP Mentorship program and mentorship applications are open to any prospective applicant. However, our capacity is limited, so we will give preferential consideration to PhysGAAP Mentorship applicants who would most benefit from the program and can demonstrate that they are a good fit.

PhysGAAP Mentoring may a good fit for you if you

feel like you lack other resources to help you navigate the graduate school application process,
find the other forms of assistance (online webinars, email at [email protected] ) insufficient to address your needs, and
think you could benefit from one-on-one application mentorship.

PhysGAAP Mentoring may not be a good fit for you if you

only have one or two questions that could be answered elsewhere (online webinars, email at [email protected] , or online FAQs), or
feel like you already have sufficient resources to complete your application (e.g., the PhysGAAP webinars, access to other mentoring services or workshops)

Please note that:

PhysGAAP Mentoring is only open to students who are planning to apply to graduate schools in Fall 2024 .
Participation in PhysGAAP is not considered during admissions review. It helps applicants put forward their strongest materials, but does not guarantee admission into our graduate program.
Any information you submit in the PhysGAAP Mentoring application will only be seen by the PhysGAAP team and your matched mentor.

Admissions/Application FAQs

Our Frequently Asked Questions provide further information about degree requirements, funding, educational background, application deadlines, English language proficiency, program duration, start dates and deferrals, and fee waiver requests.

The MOST Frequently Asked Question…

What is included in a strong graduate application for physics at mit.

Applications are assessed holistically and many variables are considered in the application review process. The following four main factors are required for a complete application.

the applicant’s statement of objectives or purpose,
transcripts of past grades,
score reports of any required standardized tests,
three letters of reference.

In addition, any past research experience, publications, awards, and honors are extremely helpful, particularly if they are in the area(s) of the applicant’s interest(s). Applicants may also include a personal statement in their application to provide context as the materials are assessed.

Applications are routed to admission committee members and other faculty readers using the “areas of interest” and any faculty names selected from the menu as well as based on the research interests included in the statement of objectives. Please select the areas of interest that best reflect your goals.

Instructions are available in the application itself , with further guidance on our Additional Guidance page. The Physics Admissions Office will respond to questions sent to [email protected] .

General Questions Regarding the PhD Program in Physics

Must i have a degree in physics in order to apply to this graduate program.

Our successful applicants generally hold a Bachelor of Science degree in Physics, or have taken many Physics classes if they have majored in another discipline. The most common other majors are astronomy, engineering, mathematics, and chemistry. Bachelor of Science degrees may be 3-year or 4-year degrees, depending on the education structure of the country in which they are earned.

What are the requirements to complete a PhD?

The requirements for a PhD in Physics at MIT are the doctoral examination, a few required subject classes, and a research-based thesis. The doctoral examination consists of a written and an oral examination. The written component may be satisfied either by passing the 4 subject exams or by passing designated classes related to each topic with a qualifying grade; the oral exam will be given in a student’s chosen research area. The Physics Department also requires that each student take two classes in the field of specialization and two physics-related courses in fields outside the specialty. Research for the thesis is conducted throughout the student’s time in the program, culminating in a thesis defense and submission of the final thesis.

Can I take courses at other schools nearby?

Yes. Cross-registration is available at Harvard University and Wellesley College.

How many years does it take to complete the PhD requirements?

From 3 to 7 years, averaging 5.6 years.

How will I pay for my studies?

Our students are fully supported financially throughout the duration of their program, provided that they make satisfactory progress. Funding is provided from Fellowships (internal and external) and/or Assistantships (research and teaching) and covers tuition, health insurance, and a living stipend. Read more about funding .

Note: For more detailed information regarding the cost of attendance, including specific costs for tuition and fees, books and supplies, housing and food as well as transportation, please visit the Student Financial Services (SFS) website .

How many applications are submitted each year? How many students are accepted?

Although the number varies each year, the Department of Physics usually welcomes approximately 45 incoming graduate students each year. Last year we received more than 1,700 applications and extended fewer than 90 offers of admission.

What are the minimum grades and exam scores for admitted applicants?

There are no minimum standards for overall grade point averages/GPAs. Grades from physics and other related classes will be carefully assessed. Under a special COVID-19 policy, MIT will accept transcripts with a variety of grading conventions, including any special grading given during the COVID-19 pandemic. GRE Tests are not required for graduate applications submitted in 2023. The Physics subject GRE (PGRE) will be optional in 2023 and our department does not require results from the General GRE test.

Our program is conducted in English and all applicants must demonstrate their English language proficiency. Non-native English speakers should review our policy carefully before waiving the TOEFL/IELTS requirements. We do not set a minimum requirement on TOEFL/IELTS scores; however, students who are admitted to our program typically score above the following values:

IELTS – 7
TOEFL (computer based) – 200
TOEFL (iBT) – 100
TOEFL (standard) – 600

The Application Process

When is the deadline for applying to the phd program in physics.

Applications for enrollment in the fall are due each year by 11:59pm EST on December 15 of the preceding year. There is no admission cycle for spring-term enrollment.

The COVID-19 pandemic has made it difficult for me to take tests in person. Can I still apply?

GRE Tests are not required for graduate applications submitted in 2023. The Physics subject GRE (PGRE) will be optional in 2023 and our department does not require results from the General GRE test.Non-native English speakers who are not eligible for a test waiver should include their results from either an in-person or online version of the TOEFL or IELTS test.

Does the Department of Physics provide waivers for the English language exam (TOEFL/IELTS)?

An English language exam (IELTS, TOEFL, TOEFL iBT, or the C2 Cambridge English Proficiency exam) is required of all applicants who are from a country in which English is not the primary language. Exceptions to this policy will be considered for candidates who, at the start of their graduate studies in 2022, will have been in the US or in a country whose official language is English for three years or longer and who will have received a degree from a college or university in a country where the language of education instruction is English. An interview via telephone, Zoom, or Skype may be arranged at the discretion of the Admissions Committee. More information on a possible English Language Waiver Decision (PDF).

Does the Department of Physics provide application fee waivers?

Although we do not want the MIT application fee to be a barrier to admission, we cannot provide application fee waivers to all who request one. Under-resourced applicants, and applicants who have participated in the MIT Summer Research Program (MSRP), Converge, or another MIT program or an official MIT recruiting visit are eligible for a fee waiver from the MIT Office of Graduate Education (OGE). Please check MIT Graduate Diversity Programs for further details. Departmentally, we have allotted a small number of waivers for applicants who have completed an application (including transcript uploads, and requests for letters of recommendation), but do not qualify for a waiver from the OGE. Fee waiver requests will be considered on a first-come-first-served basis, and not after December 1. Furthermore, applications lacking the paid fee or a fee waiver by 11:59pm EST on December 15 will not be reviewed or considered for admission. Please complete the MIT Physics Departmental Fee Waiver Application Form when you are ready to apply for a departmental waiver. Waivers are not awarded until the application is complete.

Can I arrange a visit to the Physics Department or a specific research area?

Update as of September 23, 2021: In an effort to keep our community safe and healthy, we are not currently hosting or meeting with outside visitors in person, nor are we facilitating visits to our classrooms. Current graduate students and prospective applicants should direct any questions by email to [email protected] .

Applicants are invited to send specific questions to the Physics Admissions Office and some questions may be forwarded to current students for further information.

Can I receive an update on the status of my application?

Candidates will receive email acknowledgments from the Physics Academic Programs Office informing them whether their application is complete, is missing materials, or if further information is needed. Due to the high volume of applications that are received, no additional emails or telephone inquiries can be answered. It is the applicant’s responsibility to ensure that all items are sent.

When will I be notified of a final decision?

Applicants will be notified via email of decisions by the end of February. If you have not heard from us by March 1, please send email to [email protected] .

We do not provide results by phone.

Can admitted students start in a term other than the next Fall semester?

Applications submitted between September 15 and December 15 by 11:59pm EST are assessed for the following Fall semester. We do not provide a separate admission review cycle for the Spring semester. Individual research supervisors may invite incoming students to start their research during the summer term a few months earlier than their studies would normally begin. All other incoming students start their studies in late August for the Fall term.

Once admitted, applicants may request a one-year deferral to attend a specific academic program or for another approved reason, with single semester deferrals for the following Spring term granted only rarely.

Core Members
Affiliate Members
Interdisciplinary Doctoral Program in Statistics
Minor in Statistics and Data Science
MicroMasters program in Statistics and Data Science
Data Science and Machine Learning: Making Data-Driven Decisions
Norbert Wiener Fellowship
Stochastics and Statistics Seminar
IDSS Distinguished Seminars
IDSS Special Seminar
SDSC Special Events
Online events
IDS.190 Topics in Bayesian Modeling and Computation
Past Events
LIDS & Stats Tea
Interdisciplinary PhD in Mathematics and Statistics

Requirements: Students must complete their primary program’s degree requirements along with the IDPS requirements. Statistics requirements must not unreasonably impact performance or progress in a student’s primary degree program.

PhD Earned on Completion: Mathematics and Statistics

IDPS/Mathematics Chair : Philippe Rigollet

MIT Statistics + Data Science Center Massachusetts Institute of Technology 77 Massachusetts Avenue Cambridge, MA 02139-4307 617-253-1764

Accessibility
Interdisciplinary PhD in Aero/Astro and Statistics
Interdisciplinary PhD in Brain and Cognitive Sciences and Statistics
Interdisciplinary PhD in Economics and Statistics
Interdisciplinary PhD in Mechanical Engineering and Statistics
Interdisciplinary PhD in Physics and Statistics
Interdisciplinary PhD in Political Science and Statistics
Interdisciplinary PhD in Social & Engineering Systems and Statistics
LIDS & Stats Tea
Spring 2023
Spring 2022
Spring 2021
Fall – Spring 2020
Fall 2019 – IDS.190 – Topics in Bayesian Modeling and Computation
Fall 2019 – Spring 2019
Fall 2018 and earlier

IMAGES

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$mit math phd application$
Applied Math at MIT
$mit math phd application$
MIT Math PhD Acceptance Rate
MIT Application Procedure for Undergraduate International Students
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Applying to PhD programs for Mathematics

VIDEO

Harvard-MIT Math Tournament Vlog 2023 (TJHSST VMT)
Boost Your PhD Journey: 10 Essential Apps Every PhD Student Should Have! 📱🎓
Simon Locke NYU Mathematics PhD Video
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Part-1
How to Apply for PhD in Italy (All You Need to Know)

COMMENTS

Admission
Welcome to the MIT Mathematics Graduate Admissions page. This page explains the application process in general. For complete details, go to the on-line application which is available mid-September to December. These instructions are repeated there. MIT admits students starting in the Fall term of each year only.
Graduate
Graduate Students 2018-2019. The department offers programs covering a broad range of topics leading to the Doctor of Philosophy and the Doctor of Science degrees (the student chooses which to receive; they are functionally equivalent). Candidates are admitted to either the Pure or Applied Mathematics programs but are free to pursue interests ...
Applying to Grad School
The Basics. It's best to start preparing for the application process in the spring of your junior year. Getting ready involves writing essays, taking standardized test, asking professors for letters of recomendation, and filling out a lot of forms. Keep in mind, if you love math and want to go to grad school, you should apply.
Mathematics
77 Massachusetts Avenue Building 2-110 Cambridge MA, 02139. 617-253-2416 [email protected]. Website: Mathematics. Apply here. Application Opens: September 14
Procedures
Applications. Procedures. At MIT, a regular graduate student is one who is registered for a program of advanced study and research leading to a post-baccalaureate degree. A regular graduate student may concurrently hold an appointment as a research assistant, a teaching assistant, or an instructor. All graduate-level applicants must apply ...
Admissions < MIT
The application and additional information may be found on the Advanced Study Program website. Admission is valid only for one term; a student must seek readmission each term to continue at the Institute. Those applying for special graduate student status for the first time must pay an application fee. To be allowed to continue as a special ...
Department of Mathematics < MIT
The Mathematics with Computer Science degree is offered for students who want to pursue interests in mathematics and theoretical computer science within a single undergraduate program. At the graduate level, the Mathematics Department offers the PhD in Mathematics, which culminates in the exposition of original research in a dissertation.
MIT Mathematics
In total, 68 out of the top 100 test-takers who took the exam on December 2, 2023, were MIT students. Beyond the top 5 scorers, MIT students took 8 of the next 11 spots (each awarded $ 1,000), 7 of the next 10 after that (each awarded $ 250), and 48 out of a total of 75 honorable mentions. "I am incredibly proud of our students' amazing ...
Admissions statistics
At MIT Admissions, we recruit and enroll a talented and diverse class of undergraduates who will learn to use science, technology, and other areas of scholarship to serve the nation and the world in the 21st century. ... ACT Math [35, 36] ACT Reading [34, 36] ACT English [34, 36] ACT Science [34, 36] ACT Composite [34, 36] Other sources of data ...
Applications
MIT Graduate Admissions is a decentralized network of departments that extends across all five academic schools and includes 46 departmental programs. Each of these programs has an online application with a specific set of requirements and deadlines. Answers to most common questions can be found online in the Frequently Asked Questions section.
CSE PhD
The standalone CSE PhD program is intended for students who plan to pursue research in cross-cutting methodological aspects of computational science. The resulting doctoral degree in Computational Science and Engineering is awarded by CCSE via the the Schwarzman College of Computing. In contrast, the interdisciplinary Dept-CSE PhD program is ...
What we look for
Risk-taking. MIT wants to admit people who are not only planning to succeed but who are also not afraid to fail. When people take risks in life, they learn resilience—because risk leads to failure as often as it leads to success. The most creative and successful people—and MIT is loaded with them—know that failure is part of life and that ...
Graduate 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]. Graduate Admissions:[email protected]. Facilities: building2@mit ...
Interdisciplinary Doctoral Program in Statistics
Interdisciplinary Doctoral Program in Statistics. The Interdisciplinary PhD in Statistics (IDPS) is designed for students currently enrolled in a participating MIT doctoral program who wish to develop their understanding of 21st century statistics, using concepts of computation and data analysis as well as elements of classical statistics and probability within their chosen field of study.
Admissions Requirements
Admissions Requirements. The following are general requirements you should meet to apply to the MIT Sloan PhD Program. Complete instructions concerning application requirements are available in the online application. General Requirements. Bachelor's degree or equivalent. A strong quantitative background (the Accounting group requires calculus)
MIT Mathematics
The School of Science has selected Mathematics Program Coordinator André Lee Dixon as one of the recipients of the 2024 Infinite Mile Award! "I have been consistently struck by the level of initiative and passion André brings to work," says his nominator, John Urschel PhD '21. Infinite Mile Award winners are nominated by colleagues for going above and beyond in their roles at the ...
Graduate Admissions » MIT Physics
1. Online Application and Application Fee. MIT Graduate Admissions Online Graduate Application; Application Fee: $75 NOTE: Applicants who feel that this fee may prevent them from applying should send a short email to [email protected] to describe their general reasons for requesting a waiver. We will follow up with information about how to apply for a formal 'application fee waiver'.
Interdisciplinary PhD in Mathematics and Statistics
Interdisciplinary PhD in Mathematics and Statistics. Requirements: Students must complete their primary program's degree requirements along with the IDPS requirements. Statistics requirements must not unreasonably impact performance or progress in a student's primary degree program. PhD Earned on Completion: Mathematics and Statistics.