Department of Mathematics

  • Chair

    David Gabai

  • Associate Chair

    János Kollár

  • Departmental Representative

    János Kollár 

    Jennifer M. Johnson 

  • Director of Graduate Studies

    Javier Gómez Serrano

    Zoltán Szabó

  • Professor

    Michael Aizenman, also Physics

    Noga M. Alon, also Applied and Computational Mathematics

    Manjul Bhargava

    Sun-Yung Alice Chang

    Maria Chudnovsky, also Applied and Computational Mathematics

    Fernando Codá Marques

    Peter Constantin, also Applied and Computational Mathematics

    Mihalis C. Dafermos

    Weinan E, also Applied and Computational Mathematics

    Charles L. Fefferman

    David Gabai

    Robert C. Gunning

    Alexandru D. Ionescu

    Nicholas M. Katz

    Sergiu Klainerman

    János Kollár

    Sophie Morel

    Assaf Naor

    Peter S. Ozsváth

    John V. Pardon

    Igor Y. Rodnianski

    Peter C. Sarnak

    Paul D. Seymour, also Applied and Computational Mathematics

    Yakov G. Sinai

    Amit Singer, also Applied and Computational Mathematics

    Christopher M. Skinner

    Allen M. Sly

    Zoltán Szábo

    Gang Tian

    Paul C. Yang

    Shou-Wu Zhang

  • Associate Professor

    Zeev Dvir, also Computer Science

  • Assistant Professor

    Tristan J. Buckmaster

    Gabriele Di Cerbo

    Javier Gómez Serrano

    Jonathan Hanselman

    Adam W. Marcus, also Applied and Computational Mathematics

    Ana Menezes

    Fabio G. Pusateri

    Tetiana Shcherbyna

    Nicholas J. Sheridan

    Vlad Vicol

  • Instructor

    Nicolas A.S. Boumal

    Francesc Castella

    Otis Chodosh

    Hansheng Diao

    Ziyang Gao

    Mihaela Ignatova

    Daniel J. Ketover

    Ilya Khayutin

    Francesco Lin

    Yueh-Ju Lin

    Chun-Hung Liu

    Rafael Montezuma

    Evita Nestoridi

    Oanh Nguyen

    Yakov Shlapentokh-Rothman

    Yunqing Tang

    Konstantin Tikhomirov

    Joseph A. Waldron

  • Senior Lecturer

    Jennifer M. Johnson

    Mark W. McConnell

    Christine J. Taylor

  • Associated Faculty

    Emmanuel A. Abbe, Electrical Engineering and Applied and Computational Mathematics

    John P. Burgess, Philosophy

    René A. Carmona, Operations Research and Financial Engineering

    Bernard Chazelle, Computer Science

    Hans P. Halvorson, Philosophy

    William A. Massey, Operations Research and Financial Engineering

    Frans Pretorius, Physics

    Robert E. Tarjan, Computer Science

    Ramon van Handel, Operations Research and Financial Engineering

    Robert J. Vanderbei, Operations Research and Financial Engineering

    Sergio Verdu, Electrical Engineering

Information and Departmental Plan of Study

Most freshmen and sophomores interested in science, engineering, or finance take courses from the standard calculus and linear algebra sequence 103-104-201-202, which emphasizes concrete computations over more theoretical considerations. Note that 201 and 202 can be taken in either order.

Students who are not prepared to begin with 103 may take 100, a rigorous precalculus/prestatistics refresher offered only in the fall semester and intended for students whose highest math SAT score is below 650.

Prospective economics majors can minimally fulfill their mathematics prerequisites with (100)-103-175. Note that 175 covers selected topics from 201, with biology and economics applications in mind. Prospective math-track economics/finance majors will need the standard sequence 103-104-201-202 instead of 175.

More mathematically inclined students, especially prospective physics majors, may opt to replace 201-202 with 203-204, for greater emphasis on theory and more challenging computational problems.

Prospective mathematics majors must take at least one course introducing formal mathematical argument and rigorous proofs. The recommended freshman sequence for prospective majors is 215-217. Prospective majors who already have substantial experience with university-level proof-based analysis courses may consider the accelerated sequence 216-218 instead. Other possible sequences for prospective majors include 214-204-203 and 203-204-215, although the latter two are relatively rare. Note that 203 and 204 can be taken in either order.

Placement. Students with little or no background in calculus are placed in 103, or in 100 if their SAT mathematics scores indicate insufficient background in precalculus topics. To qualify for placement in 104 or 175, a student should score 5 on the AB Advanced Placement Examination or a 4 on the BC Advanced Placement Examination. To qualify for placement into 201 or 202, a student should have a score of 5 on the BC Examination. Students who possess in addition a particularly strong interest in mathematics as well as a SAT mathematics score of at least 750 may opt for 203 or 214 or 215 or 216 instead. For more detailed placement information, consult the Department of Mathematics home page or placement officer.

Advanced Placement

One unit of advanced placement credit is granted when a student is placed in MAT 104 or 175. Two units of advanced placement credit are granted when a student is placed in MAT 201, 203, or 217.

Prerequisites

Generally, either 215-217 or 216-218 or 203-204-215 are strongly recommended for admission to the department. Prospective mathematics majors should consult the department early and plan a program that includes as much of the 215-217 or 216-218 sequence as possible. Most majors begin taking courses at the 300-level by the second semester of the sophomore year, in preparation for their junior independent work.

Further information for prospective majors is available on the department home page.

Program of Study

Students must complete four core requirements:

  • one course in real analysis (e.g. 320 or 325 or 425 or 385)
  • one course in complex analysis (e.g. 330 or 335)
  • one course in algebra (e.g. 340 or 345)
  • one course in geometry or topology (e.g. 350 or 355 or 365 or 560)

It is recommended that students complete some of these core requirements by the end of the sophomore year. Completing these core courses early gives more options for junior and senior independent work.

Note: One course in discrete mathematics (e.g. 375, 377 or 378) can replace the geometry/topology core requirement, if desired.

In addition to the four core requirements, students must complete an additional four courses at the 300 level or higher, up to three of which may be cognate courses outside the mathematics department, with permission from the junior or senior advisers or departmental representative.

The departmental grade (the average grade of the eight departmental courses) together with grades and reports on independent work is the basis on which honors and prizes are awarded on graduation.

Students should refer to Course Offerings to check which courses are offered in a given term. Programs of study in various fields of pure mathematics and applied mathematics are available. Appropriate plans of study may be arranged for students interested in numerical analysis, discrete mathematics, optimization, physics, the biological sciences, probability and statistics, finance, economics, or computer science. For students interested in these areas, a coherent program containing up to three courses in a cognate field may be approved.

Independent Work

All departmental students engage in independent work, supervised by a member of the department chosen in consultation with a departmental adviser. The independent work of the junior year generally consists of participating actively in a junior seminar in both the fall and the spring semesters. Alternatively, a student may opt to replace one junior seminar with supervised reading in a special subject and then writing a paper based on that reading. The independent work in the senior year centers on writing a senior thesis. A substantial percentage of our majors work with faculty in other departments on their senior project.

Senior Departmental Examination

Each senior takes an oral examination based on the senior thesis and the broader subfield to which it contributes. A departmental committee conducts the examination in May.

Courses