# 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.