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- Dr. Casey Rodriguez
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- Mathematics
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- Mathematical Analysis
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Real analysis, course description.
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Understanding the process: Admissions statistics
We love data at MIT. Reliable data, properly contextualized , can help people understand complex systems and make informed decisions. So, a few years ago, we began publishing our own admissions statistics which went beyond the stats already contributed to the MIT’s standard Common Data Set .
Holistic admissions
It is important to understand that these numbers do not determine our admissions process, but are the result of our process. In our holistic admissions process, we consider quantifications like test scores , but we also care deeply about factors like your match with MIT . Qualitative characteristics like these are much harder to quantify and are therefore not included in the tables below despite their centrality to our process.
The most important thing to remember is that at MIT we admit people, not numbers . With that in mind, here are some numbers about the people we admit.
Admissions statistics for the Class of 2027
See also First-year class profile .
Early Action
Regular action, u.s. citizens/permanent residents, international students, middle 50% score range of admitted students (25th and 75th percentiles), other sources of data about mit.
- Common Data Set
- Registrar’s Enrollment Statistics
- International Students Office Statistics
- College Board
- College Navigator
- College Results Online
- College Scorecard
Doctoral Degrees
A doctoral degree requires the satisfactory completion of an approved program of advanced study and original research of high quality..
Please note that the Doctor of Philosophy (PhD) and Doctor of Science (ScD) degrees are awarded interchangeably by all departments in the School of Engineering and the School of Science, except in the fields of biology, cognitive science, neuroscience, medical engineering, and medical physics. This means that, excepting the departments outlined above, the coursework and expectations to earn a Doctor of Philosophy and for a Doctor of Science degree from these schools are generally the same. Doctoral students may choose which degree they wish to complete.
Applicants interested in graduate education should apply to the department or graduate program conducting research in the area of interest. Some departments require a doctoral candidate to take a “minor” program outside of the student’s principal field of study; if you wish to apply to one of these departments, please consider additional fields you may like to pursue.
Below is a list of programs and departments that offer doctoral-level degrees.
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Math Major Roadmaps
This document outlines “roadmaps” of course options for undergraduates interested in particular fields and applications of mathematics. Each roadmap consists of three stages:
- Stage 1: Introductory courses with few prerequisites, accessible to a typical sophomore.
- Stage 2: More advanced classes for students who have mastered several stage-1 classes.
- Stage 3: The most advanced classes, often beginning graduate-level subjects, for students who have mastered many stage-1 and stage-2 classes. (Many math majors will never take a stage-3 course, and that’s okay!)
The stages also reflect priorities —for example, students interested in pursuing analysis probably want to take 18.100 as early as possible, whereas a student pursuing mathematical finance might take it later to deepen their understanding.
In each specialized field or application of mathematics below, we list courses that could be relevant for a career in that field —not just courses narrowly in that specialization, but also useful courses in broadly related areas. On the other hand, these specific courses are not requirements to pursue that field.
Note that a math degree requires 18.06/18.C06/18.700/701 (or approved substitutions thereof), but these are not necessarily listed in every roadmap below, nor do we list GIRs like 18.02.
Pure Mathematics
General comments: Below is a list of research areas. As an undergraduate, however, you should not conceive of yourself as specializing in one or another of these areas . A much better idea is to gain experience in several of them. You may find yourself taking more courses in one or another area, but all of these fields of study reinforce each other.
Stage 2, but not specific to any particular area: 18.821 (Project Lab in Mathematics) is a great introduction to some essential skills (research, teamwork, and communication).
- Stage 1: 18.090 , 18.700 + 18.703 , 18.700 + 18.701 + 18.702 or 18.701 + 18.702
- Stage 2: 18.704 , 18.715 , 18.721 , 18.782
- Stage 3: 18.705 , 18.706 , 18.725 , 18.745 , 18.755 , 18.783
Analysis & Geometry
- Stage 1: 18.090 , 18.100 , 18.700 / 18.701
- Stage 2: 18.101 , 18.102 , 18.103 , 18.104 , 18.112 , 18.152 , 18.950 , 18.994
- Stage 3: 18.125 , 18.155 , 18.156 , 18.952 , 18.965 , 18.966
- Stage 1: 18.090 , 18.510 , 18.701 or 18.700 + 18.703
- Stage 2: 18.400 , 18.404 , 18.504
- Stage 3: 18.515 (not offered regularly)
The MIT Philosophy department also offers subjects in logic: 24.241–24.245 and 24.711 . Also consider logic classes at Harvard .
Number Theory
- Stage 1: 18.090 , 18.700 + 18.703 or 18.701 + 18.702 , 18.781
- Stage 2: 18.704 , 18.721 , 18.782 , 18.784
- Stage 3: 18.705 , 18.725 , 18.783 , 18.785
Probability & Statistics
- Stage 1: 18.090 , 18.065 , 18.100 , 18.600 , 18.700 / 18.701
- Stage 2: 18.102 , 18.103 , 18.112 , 18.200 , 18.211 , 18.615 , 18.642 , 18.650
- Stage 3: 18.125 , 18.675 , 18.676 , 18.677 , 18.338 , 18.416 , 18.424 , 18.655 , 18.657
Topology & Geometry
- Stage 1: 18.090 , 18.100 , 18.112 , 18.701 + 18.702 or 18.700 + 18.703 , 18.900 , 18.950
- Stage 2: 18.101 , 18.102 , 18.901 , 18.904 , 18.952 , 18.994
- Stage 3: 18.116 , 18.155 , 18.721 , 18.755 , 18.782 , 18.905 , 18.906 , 18.965 , 18.966
Applied Mathematics
Combinatorics.
- Stage 1: 18.090 , 18.100 , 18.200 , 18.400 , 18.600 , 18.701 + 18.702 or 18.700 + 18.703 , 18.900
- Stage 2: 18.204 , 18.211 , 18.212 , 18.112 , 18.404 , 18.410 , 18.453 , 18.721 , 18.781 , 18.901 , 18.950
- Stage 3: 18.217 , 18.218 , 18.225 , 18.226 , 18.338 , 18.455 , 18.615 , 18.705 , 18.715 , 18.725 , 18.745
Computer Science
- Stage 1: 18.06 / 18.C06 / 18.700 , 18.062 , 18.090 , 18.200 , 18.330 , 18.600 , 18.650 , 18.701
- Stage 2: 18.204 , 18.211 , 18.400 , 18.404 , 18.410 , 18.424 , 18.434 , 18.453
- Stage 3: 18.337 , 18.415 , 18.416 , 18.425 , 18.435 , 18.437 , 18.455 , 18.783
Students in this area should strongly consider supplementing their math courses with several courses in computer science; see the 18c major requirements for typical choices.
Economics and Finance
- Stage 1: 18.03 , 18.05 , 18.06 / 18.C06 / 18.700 , 18.065 , 18.090 , 18.300 , 18.330 , 18.600 , 18.650
- Stage 2: 18.100 , 18.152 , 18.303 , 18.453 , 18.642
- Stage 3: 18.103 , 18.125 , 18.675 , 18.676 , 18.677 , 18.335 , 18.337 , 18.338 , 18.352 , 18.353 , 18.355 , 18.615 , 18.655
Students interested in economics and finance should strongly consider supplementing their math courses with several classes in course 14 and 15, if not a minor or double major.
Computational Science and Engineering
- Stage 1: 18.03 , 18.06 / 18.C06 / 18.700 , 18.04 , 18.05 , 18.062 , 18.065 , 18.090 , 18.200 , 18.303 , 18.330 , 18.600 , 18.650 , 18.S190 , 18.S191
- Stage 2: 18.100 , 18.404 , 18.434 , 18.410 , 18.701
- Stage 3: 18.335 , 18.336 , 18.337 , 18.338 , 18.367 , 18.415 , 18.437
Students in this area should consider supplementing their math courses with several courses in computer science, both for software engineering (e.g. 6.100A , 6.1010 ) and numerical methods or optimization (e.g. 6.7201 , 6.7330 ). For students interested in computational modeling of physical systems, see also the Physical Applied Math classes below.
Physical Applied Math
- Stage 1: 18.03 , 18.04 , 18.06 / 18.C06 / 18.700 , 18.090 , 18.300 , 18.303 , 18.330 , 18.352 , 18.353 , 18.354 , 18.600 , 18.650
- Stage 2: 18.100 , 18.112 , 18.152 , 18.384 , 18.417
- Stage 3: 18.102 , 18.103 , 18.305 , 18.306 , 18.335 , 18.355 , 18.357 , 18.367 , 18.369 , 18.376 , 18.377
Students interested in physical applied mathematics should strongly consider supplementing their math courses with at least 2–3 non-GIR courses in physics and/or engineering, depending on their field of interest, if not a minor or double major.
Statistics and Data Science
- Stage 1: 18.05 , 18.06 / 18.C06 / 18.700 , 18.062 , 18.065 , 18.090 , 18.100 , 18.200 , 18.330 , 18.600 , 18.650 , 18.701
- Stage 2: 18.102 , 18.112 , 18.204 , 18.642
- Stage 3: 18.125 , 18.675 , 18.676 , 18.335 , 18.338 , 18.615 , 18.657
Students in this area should consider supplementing their math courses with courses in computer science on machine learning and courses in economics on econometrics.
When the PhD path leads to career struggles
A doctoral degree is a major commitment. Think carefully.
I appreciated reading Kara Miller’s The Big Idea column “PhD: Pretty heavily disappointed” (Business, May 22), about people with doctoral degrees struggling to build careers in academia. It made me think back to a conversation I had when I was about to graduate from high school.
I happened to run into a former track coach of mine, and as we were reminiscing he asked me what I planned as a major in college. “History,” I responded. He said, “Why don’t you take some computer classes also? It never hurts to be able to do something useful.”
I did not reflect on his motivation at the time, but my track coach was a young guy, and he was probably giving me advice straight from his own life, as a parent trying to raise his own young children. I did take computer classes in college and ultimately received a PhD in chemical engineering. I always remember that conversation as being a kind of turning point.
Earning a doctoral degree is a life commitment of great proportion. It can take, as Miller notes, between four and seven years. If we think of working life as roughly between the ages of 22 and 65, then a PhD requires more than 10 percent of a person’s working life. People need to think carefully about that investment.
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Two powerful arguments in favor of the path of science, technology, engineering, and math are that there tend to be more STEM jobs for PhDs, and many universities’ STEM departments are generous in covering their PhD students’ tuition and cost of studies, including a stipend toward food, rent, and other expenses.
Stuart Gallant
Not much has changed in 30 years
As I prepared to graduate in 1995 with a doctor of education degree from the Harvard Graduate School of Education, my mother memorably said to me, “Of my four children, you are the one with the most education and the smallest salary.” Apparently not much has changed in 30 years.
I must congratulate these students, however, on following their passion rather than following the money. I can’t help but think that their lives, though stressful, may contain greater happiness.
Peggy Clark
Lawyers & electricians & philosophers, oh my!
Kara Miller’s column on the career challenges for people with doctoral degrees generated more than 260 comments on Boston.Globe.com. The following is an edited sample of readers’ reactions:
Lots of law school grads are underemployed as well. (PL)
So true, PL. The market in Massachusetts is flooded with talented lawyers seeking work. (Roforma)
Supply and demand, the market at work. (guk)
Investing in education and research in all fields is the hallmark of a society with staying power. Disinvesting from these endeavors signals decline and decay. (Massachusetts citizen)
Electricians, plumbers, mechanics, and other skilled technical professions have no problems getting $100k jobs with great benefits. (ramsen)
Not enough turnover from tenured professors, leaving little space for new faculty. Although the tenured, well-established professors are needed, it’s the junior faculty who are hungry and with new ideas that help build new programs. The whole graduate program model is a bad model. I worked two jobs, had my tuition and some type of minimal student health insurance and could barely cover the rent with my stipend, and the second job paid for everything else. Though I was working on many faculty projects, it was the faculty who said this would be good for me. Never did they say it was also good for them. (TravelerofNJ2)
I just retired from a tenured faculty position in science. I’m in my early 70s. I have colleagues who are still doing what they do well into their 70s, a couple approaching 80. There is no active incentive from the university to move the older faculty on, to make way for a new generation. (Lola-lola)
The next step is for adjuncts to go on strike across the nation and hold colleges and universities accountable. The current system is completely absurd. (Wordsmith2358)
Universities should be required to release disclosure data about the fate of their PhD graduates. (davidman820)
I knew an attorney who managed a Cheesecake Factory. She had worked in food services through school. As an attorney, she really did not make that much money and was not doing the field of law of her choice. How many real estate closings can you do without dying of boredom? She went into management in the food industry and makes the same salary. (Antietem)
It was always a question and puzzling to me why people study philosophy. (Blazer27)
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MIT's math program is insanely competitive. I wouldn't feel bad about it. People with good GPAs, publications, and lots of research experience get rejected every year. Getting rejected from there doesn't mean you won't get accepted into a good program or have a great mathematical career. I'm sure you'll get into another program.
I went to a decent undergrad (top-15 in math) so I can give you a vague idea about the background of the 5 or so students my year who got into at least one top graduate program (best is not a good word to use - in the US for example Harvard, Princeton, MIT, UChicago, Berkeley, and Stanford are all considered top institutions, and if you focused on a subfield there would be other programs just ...
1. Utrecht (Netherlands) 1. Wisconsin. 1. Some comments: As you may have expected, MIT, Harvard, and Princeton (widely regarded as the three "best" math departments in the United States) make up the three most common schools where faculty members received their PhDs from, although I was a bit surprised that the school that produced the greatest ...
Revlong57. •. Honestly, if you're looking at doing a PhD in applied math, it will probably have value in a future career outside of academia. Pure math is basically only useful for academic jobs. Now, let's look at the pros and cons of getting a PhD in applied math. Source, I'm getting a PhD in an applied math area.
The classes don't care at all about competition math, all you have to do is learn the material and you will do fine. That being said, you should be cognizant of the wide gap of math skill in MIT. You will be in classes with people who are ranked internationally in math. It will be like comparing a international grandmaster with a person who won ...
On another thread I left a comment with some data I'd compiled about grad students in math at Harvard. I went through the list of grad students there and compiled the undergraduate universities for those that I could find. I'll copy the results over here, for ease (but see this comment for some updates to this from a Harvard PhD student): . I just went through the list of Harvard grad students.
For the US, particularly in engineering, most PhD programs allow direct admissions, without a Master's. The most important thing for graduate admissions is your research experience and match between that and a professor at your intended school. For grad school, you pretty much get in to do research with a specific person in a specific lab. Reply.
0. Roughly: good grades (3.8+ GPA) in difficult courses, good test scores (80+ percentile on math GRE subject test [not the regular GRE math, which you should get a ~perfect score on without studying]), strong research background and good letters corresponding to it. That will get you into schools in the top ~30.
Compared to Maxim's readership, I'd bet he's a math genius. Also, the only quantitative definition of genius is three standard deviations above the mean, which just means top .15℅. I am pretty sure far less than one person in a thousand has the mathematical ability to do publication-worthy work, so he probably really should be called a math ...
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 ...
Most graduate students are supported through teaching assistantships (TAs). They work by teaching recitations, grading papers, etc., in return for tuition and living expenses. A teaching assistantship is really a partial fellowship. Stipends are usually enough to live on, and they are generous in terms of the work expected.
Bad news: I got a 1430 on the SAT I. NOTE:I do not personally think that a 1430 SAT I (which I guess would be, say, a 2150 or so now) is a bad score. But there are a lot of people who think that it is a bad score, and there are certainly a lot of people who think you can't get into MIT with a 1430/2150, especially if your math score is sub-700.
You self-reported your grades in step 1, but we require an official transcript for all admitted students. If/when we request this, arrange for an official copy of your college transcript to be sent to: Academic Services, Room 2-110. Dept of Mathematics, MIT. 77 Massachusetts Ave. Cambridge MA 02139-4307.
Strictly speaking, stats is a subset of math. Math is a HUGE umbrella. If you study the pidgeon hole principle, hairy ball theory and reimann H. You will likely be useless for ML but those are all math. If you study calculus, linear algebra, numerical methods and optimization methods youll be well set.
John Urschel. urschel AT mit DOT edu. I am an assistant professor in the MIT Math department. My research is focused on matrix analysis and numerical analysis, with an emphasis on theoretical results and provable guarantees for practical problems. I am also a Junior Fellow at the Harvard Society of Fellows (currently on leave).
Hi guys, I have a question about the admission to a generic phd program in AI in a good university (not necessarily a top tier university like MIT, Stanford oxford ecc). In particular how is importa the GPA to be admitted compared to other research activities and/or pubblications during the master degree? Thank you very much
No reason to take any more courses in econ either unless you need to get references. PhD micro and PhD metrics are usually most valuable. I was in your position: math, CS, and then stats masters before my econ PhD. Definitely take the core micro / metrics PhD sequence. You don't need more math.
I'm currently studying maths and philosophy, just finishing my first year. I know it is very early and a phd is a long way away and I may change my mind. But I also know that it is incredibly competitive. I love maths and I really think I want to do a phd.
Most PhD programs will make you take the GRE, which consists of a math section (about the same difficulty as the SAT), a verbal (aka reading) section (harder than the SAT), and an essay-writing section; No one gives a rat's ass about what you score as long as you score "high enough" (which varies depending on the program.
Download Course. This course covers the fundamentals of mathematical analysis: convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, and the interchange of limit operations. It shows the utility of abstract concepts through a study of real numbers, and teaches an ...
Classes taken at MIT While we are happy to consider Transfer Credit for work done elsewhere, at least half of the subjects beyond 18.03 used to fulfill the requirements for the Mathematics major must be taken at MIT; i.e. at least four of the eight 12-unit subjects required for the course 18 options, and at least six of the twelve required for ...
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. ... SAT Math [780, 800] SAT ERW [740, 780] ACT Math [35, 36] ACT Reading [34, 36] ACT English [34, 36] ACT Science [34, 36] ACT ...
A doctoral degree requires the satisfactory completion of an approved program of advanced study and original research of high quality. Please note that the Doctor of Philosophy (PhD) and Doctor of Science (ScD) degrees are awarded interchangeably by all departments in the School of Engineering and the School of Science, except in the fields of biology, cognitive science, neuroscience, medical ...
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:[email protected]. ...
It can take, as Miller notes, between four and seven years. If we think of working life as roughly between the ages of 22 and 65, then a PhD requires more than 10 percent of a person's working life.