# Mathematics

###### Director of Graduate Studies

###### Graduate Program Administrator

## Overview

The Department of Mathematics graduate program has minimal requirements and maximal research and educational opportunities. It differentiates itself from other top mathematics institutions in the U.S. in that the curriculum emphasizes, from the start, independent research. Each year, we have extremely motivated and talented students among our new Ph.D. candidates who, we are proud to say, will become the next generation of leading researchers in their fields. While we urge independent work and research, there exists a real sense of camaraderie among our graduate students. As a result, the atmosphere created is one of excitement and stimulation as well as of mentoring and support. There also exists a strong scholarly relationship between the department and the Institute for Advanced Study (IAS), located a short distance from campus. Students can contact IAS members as well as attend the IAS seminar series.

Students are expected to write a dissertation in four years but may be provided an additional year to complete their work if deemed necessary. Each year our Ph.D.s are successfully launched into academic positions at premier mathematical institutions as well as into industry.

## Applying

## Ph.D.

The department offers a broad variety of research-related courses as well as introductory (or “bridge”) courses in several areas, which help first-year students strengthen their mathematical background. Students also acquire standard beginning graduate material primarily through independent study, and consultations with the faculty and fellow students.

Students must satisfy the language requirement by demonstrating to a member of the mathematics faculty a reasonable ability to read ordinary mathematical texts in one of the following three languages: French, German, or Russian. The language test must be passed before the end of the first year, and before standing for the general exam.

*Seminars*

The department offers numerous seminars on diverse topics in mathematics. Some seminars consist of systematic lectures in a specialized topic; others present reports by students or faculty on recent developments within broader areas. There are regular seminars on topics in algebra, algebraic geometry, analysis, combinatorial group theory, dynamical systems, fluid mechanics, logic, mathematical physics, number theory, topology and other applied and computational mathematics. Students may also attend, without fees or formalities, seminars in the School of Mathematics at the IAS.

The department also facilitates several informal seminars specifically geared toward graduate students: (1) Colloquium Lunch Talk, where experts who have been invited to present at the department colloquium will give introductory talks, which allows graduate students to understand the afternoon colloquium more easily; (2) Graduate Student Seminar (GSS), which is organized and presented by graduate students and helps in creating a vibrant mathematical interaction among the graduate students; and, (3) What’s Happening in Fine Hall (WHIFH) seminar, where faculty members present talks in their own research areas specifically geared towards graduate students. Reading seminars are also organized and run by graduate students.

Beyond needing a strong knowledge of three more general subjects (algebra, and real and complex analysis), first-year students are set on the fast track of research by choosing two advanced topics of research as part of their general exam. The two advanced topics are expected to come from distinct major areas of mathematics, and the student’s choice is subject to the approval of the department. Usually by the second year, students will begin investigations of their own that lead to the doctoral dissertation.

*General Exam in Mathematical Physics*

For a mathematics student interested in mathematical physics, the general exam is adjusted to include mathematical physics as one of the two special topics.

The Master of Arts (M.A.) degree is considered an incidental degree on the way to full Ph.D. candidacy and is earned once a student successfully passes the language requirement and the general exam, and is recommended by the faculty. It may also be awarded to students who, for various reasons, may leave the Ph.D. program, provided that the following requirements are met: passing the language requirement as well as the three general subjects (algebra, and real and complex analysis) of the general exam, and receiving department approval.

During the second, third and fourth years, graduate students are expected to either grade or teach two sections of an undergraduate course, or the equivalent, each semester. Although students are not required to teach in order to fulfill department Ph.D. requirements, they are strongly encouraged to do so at least once before graduating. Teaching letters of recommendation are necessary for most postdoctoral applications.

*Selection of a Research Adviser*

Upon completion of the general exam, the student is expected to choose a thesis adviser.

Two to three years is usually necessary for completion of a suitable dissertation. Upon completion and acceptance of the dissertation by the department and Graduate School, the candidate is admitted to the final public oral examination, in which the dissertation is presented and defended by the candidate.

The Ph.D. is awarded after the candidate’s doctoral dissertation has been accepted and the final public oral examination sustained.

## Faculty

#### Chair

David Gabai

#### Associate Chair

János Kollár (Fall Semester)

Christopher M. Skinner (Spring Semester)

#### Director of Graduate Studies

Mihalis Dafermos

Javier Gómez-Serrano

#### Professor

Michael Aizenman

Noga M. Alon

Manjul Bhargava

Sun-Yung Alice Chang

Maria Chudnovsky

Fernando Codá Marques

Peter Constantin

Mihalis C. Dafermos

Weinan E

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

Yakov G. Sinai

Amit Singer

Christopher M. Skinner

Allen M. Sly

Zoltán Szábo

Paul C. Yang

Shou-Wu Zhang

#### Associate Professor

Zeev Dvir

#### Assistant Professor

Nicolas Boumal

Tristan J. Buckmaster

Gabriele Di Cerbo

Javier Gómez-Serrano

Jonathan Hanselman

Aleksandr Logunov

Adam W. Marcus

Ana Menezes

Evita Nestoridi

Tetiana Shcherbyna

Yakov M. Shlapentokh-Rothman

#### Instructor

Daniel Alvarez-Gavela

Kenneth Ascher

Clark Butler

Chiara Damiolini

Duncan Dauvergne

Theodore D. Drivas

Jiequn Han

Yusuf B. Kartal

Casey L. Kelleher

Chao Li

Joaquin Moraga

Oanh Nguyen

Sophie T. Spirkl

Yunqing Tang

Maxime C. R. Van De Moortel

Remy van Dobben de Bruyn

Joseph A. Waldron

Jingwei Xiao

Andrew V. Yarmola

Ian M. Zemke

Boyu Zhang

#### Visiting Lecturer with Rank of Professor

Camillo De Lellis

Helmut H. Hofer

Akshay Venkatesh

#### Lecturer

Tatiana Howard

## Courses

Permanent Courses

Courses listed below are graduate-level courses that have been approved by the program’s faculty as well as the Curriculum Subcommittee of the Faculty Committee on the Graduate School as permanent course offerings. Permanent courses may be offered by the department or program on an ongoing basis, depending on curricular needs, scheduling requirements, and student interest. Not listed below are undergraduate courses and one-time-only graduate courses, which may be found for a specific term through the Registrar’s website. Also not listed are graduate-level independent reading and research courses, which may be approved by the Graduate School for individual students.