Department of Mathematics

Chair

Andrew J. Wiles

Associate Chair

Simon B. Kochen

Director of Graduate Studies

Natasa Pavlovic

Andrei Okounkov

Professor

Michael Aizenman, also Physics

Manjul Bhargava

William Browder

A. Robert Calderbank, also Electrical Engineering

Sun-Yung Alice Chang

John H. Conway

Ingrid C. Daubechies

Weinan E

Charles L. Fefferman

David Gabai

Robert C. Gunning

Nicholas M. Katz

Sergiu Klainerman

Simon B. Kochen

Joseph J. Kohn

János Kollár

Elliott H. Lieb, also Physics

Elon Lindenstrauss

John N. Mather

Edward Nelson

Andrei Okounkov

Rahul Pandharipande

Igor Y. Rodnianski

Peter C. Sarnak

Paul D. Seymour

Yakov G. Sinai

Christopher Skinner

Elias M. Stein

Zoltán Szábo

Gang Tian

Andrew J. Wiles

Paul C. Yang

Assistant Professor

Thomas Chen

Maria Chudnovsky

Samuel Grushevsky

Fengbo Hang

Bo’az Klartag

Leonid Koralov

Maryam Mirzakhani

Natasa Pavlovic

Pierre Raphael

Jacob A. Rasmussen

Benjamin Sudakov

Ramin Takloo-Bighash

Simone Warzel

Instructor

Spyridon Alexakis

Shiri Artstein

Dmitri Beliaev

Daniel Katz

Max Lieblich

Hossein Namazi

André Neves

Alexei A. Oblomkov

Alireza Salehi Golsefidy

Claus Sorenson

Jeremie Szeftel

Michael Usher

Senior Lecturer

Anna M. Cannas da Silva

Jennifer Johnson

Associated Faculty

John P. Burgess, Philosophy

René A. Carmona, Operations Research and Financial Engineering

Bernard Chazelle, Computer Science

Erhan Çinlar, Operations Research and Financial Engineering

Philip J. Holmes, Mechanical and Aerospace Engineering

Yannis G. Kevrekidis, Chemical Engineering

Robert E. Tarjan, Computer Science

Robert J. Vanderbei, Operations Research and Financial Engineering

Sergio Verdú, Electrical Engineering

 

The Department of Mathematics offers graduate courses on various levels, all of which are oriented toward research. There are numerous seminars that encourage research even more directly. The content of courses varies considerably from year to year, and the course descriptions below should be read only as a rough guide. Students usually acquire the standard beginning graduate material primarily through independent study and consultations with the faculty.

To earn the Doctor of Philosophy (Ph.D.) degree, students must pass a language requirement and both portions of the general examinations, submit an acceptable dissertation, and sustain a final public oral examination. To qualify for the Master of Arts (M.A.) degree, the student must pass the language requirement and the first part of the general examination, and be recommended by the faculty.

In the first two years, students acquire a background in mathematics. Depending upon individual preparation, a student may take the general examination in the first or the second year of study. The general examination covers real variables, complex variables, and algebra; and it includes two additional topics of a more specialized nature. These two additional 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 in the second year, and sometimes even in the first, students begin investigations of their own that lead to the doctoral dissertation.

The department collaborates with the Department of Physics in offering work in mathematical physics and leading to an advanced degree. For a mathematics student interested in mathematical physics, the general examination is adjusted to include mathematical physics as one of the two special topics.

A plan of study also may be coordinated with the Program in Applied and Computational Mathematics. See their program description in this book for more information.

The student satisfies the language requirement by demonstrating to a member of the mathematics faculty a reasonable ability to read ordinary mathematical texts in at least two of the following three languages: French, German, and Russian. One language test must be passed before the end of the first year, and the second before completing the general examination.

Equipment and Facilities

The Department of Mathematics is located in Fine Hall. The low wing of this building contains the Common Room, the center of informal mathematical activities for faculty and students; classrooms for graduate and some advanced undergraduate mathematics classes; study rooms for beginning graduate students and undergraduate mathematics majors; computer facilities for graduate students and undergraduate mathematics majors; and smaller rooms for advanced graduate students. The Fine Hall Library of mathematics and physics occupies one of the lower floors of the hall and extends beneath the adjoining plaza to Jadwin Hall. The tower of Fine Hall contains faculty and graduate student offices and seminar rooms. Fine Hall also houses the main offices of the Program in Applied and Computational Mathematics.

Since 1911 Princeton University has published the Annals of Mathematics under the editorial direction of the Department of Mathematics. In 1933 an agreement was made whereby Princeton University and the Institute for Advanced Study jointly publish the Annals. Currently, it is edited by Professors Gabai, Katz, and Sarnak of the University, and Professors Bourgain and Griffiths of the Institute, with the cooperation of the department, the School of Mathematics of the Institute, and seven associate editors from various other institutions. Under the same joint auspices, the Annals of Mathematics Studies, a collection of research monographs, and the Princeton Mathematical Series are edited by Professors Mather and Stein of the University and Professor Caffarelli of the Institute.

Seminars

In addition to the course offerings listed below, 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 always 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 of the Institute for Advanced Study.

Courses

MAT 501, 502 Mathematical Logic

Simon B. Kochen

A course selected from one or more of several possible topics, which include model theory, decision problems, set theory, recursion theory, and other branches of logic.

MAT 503, 504 Selected Topics in Logic

Edward Nelson

Covers current areas of interest in mathematical logic.

MAT 505, 506 Algebra

Staff

Focuses on topics in algebra and number theory, which vary from year to year, including valuations and local fields, algebraic function fields of one variable (algebraic theory), formal groups, Galois cohomology, cyclotomic fields, and more.

MAT 507, 508 Number Theory

Andrew J. Wiles

Topics in number theory, such as class field theory, cyclotomic fields, p-adic L-functions, elliptic curves, and arithmetic questions relating to Abelian varieties.

MAT 509, 510 Algebraic Number Theory

Jordan S. Ellenberg

Ideal theory, adeles and ideles, and zeta functions; Galois cohomology and class field theory; and laws of reciprocity and other applications of class field theory.

MAT 511, 512 Analysis and Number Theory

Peter C. Sarnak

The general analytic theory of L-functions. Analytic continuation, zeros, magnitude, and related issues. The emphasis of the course is on applications to problems in number theory, automorphic forms, and mathematical physics.

MAT 513 Quadratic Forms

John H. Conway

Focuses on the invariant theory of intergral quadratic forms and the results that follow readily from that theory, e.g., re-enumerating the invariants and completing the proof that they are invariants by showing that almost all of them are “audible invarients” (properties of the representation numbers of the form). Kitaoka’s theorem on the audibility of the general of forms in, at most, four variables will be an immediate consequence. Discussion of further applications depends upon the size and the interests of the class.

MAT 514 Topics in Algebra

Staff

Covers current areas of interest in algebra.

MAT 515 Topics in Number Theory

Staff

Covers current areas of interest in number theory.

MAT 516 Introduction to Algebra

Staff

An introduction to algebra at the graduate level. Recommended for all first-year students; open to undergraduate math concentrators.

MAT 518 Introduction to Analysis

Staff

An introduction to analysis at the graduate level. Recommended for all first-year students; open to undergraduate math concentrators.

MAT 520 Introduction to Geometry

Staff

An introduction to geometry at the graduate level. Recommended for all first-year students; open to undergraduate math concentrators.

MAT 521, 522 Introduction to Harmonic Analysis

Staff

An introductory course to basic approaches to problems in harmonic analysis. More specific details will be provided when the course is again offered.

MAT 523, 524 Functional Analysis

Edward Nelson

The theory of linear operators and other aspects of analysis on infinite dimensional spaces or applications of functional analysis to probability theory and problems arising in mathematical physics.

MAT 525, 526 Fourier Analysis on Euclidean Spaces

Elias M. Stein

Fourier transforms, singular integrals, pseudodifferential operators, harmonic functions, Lp estimates, and applications to partial differential equations are studied.

MAT 527, 528 Fourier Analysis on Groups

Charlies L. Fefferman

Group representations, Fourier analysis and potential theory of symmetric spaces, analogues of singular integrals and pseudodifferential operators in the noncommunicative case, and the applications to several complex variables are studied.

MAT 529, 530 Several Complex Variables and Partial Differential Equations

Joseph J. Kohn

A consideration of the existence and regularity theorems that arise from the study of several complex variables. The main tool is the variational method associated with the ∂-Neumann and related problems. It usually deals with some of the following topics: pseudoconvexity, subellipticity, Sobolev spaces, pseudodifferential operators, and CR-functions. The aim is to present the background for current research and then to present some of the problems that arise.

MAT 531, 532 Complex Analytic Varieties

Robert C. Gunning

Topics in complex manifolds such as Riemann surfaces, Kaehler manifolds, complex algebraic manifolds, and the singularities of analytic varieties.

MAT 533, 534 Elliptic and Parabolic Differential Equations

Sun-Yung Alice Chang

Basic, classical results in elliptic and parabolic partial differential equations (PDE). Topics include the Laplace equation, the heat equation, Sobolov spaces, Holder regularity, maximal principles, and Harnack inequality for second order elliptic and parabolic PDE of divergence and non-divergence type. Later, topics from fully non-linear elliptic PDE. This may include the study of Monge-Ampere equations, other non-linear PDE from geometric considerations, and topics related to recent research work.

MAT 535, 536 Nonlinear Wave Equations

Sergiu Klainerman

The principal emphasis of the course is on developing analytic tools for treating the basic issues of regularity and breakdown of solutions of interesting equations. It discusses the main features of the nonlinear equations that arise in classical field theory. Though the course can change considerably from year to year, major examples include wave maps, gauge theories, and general relativity.

MAT 537 Topics in Analysis

Staff

Covers current areas of interest in analysis.

MAT 549, 550 Differential Geometry

Gang Tian

Concerns problems in differential geometry, beginning with basic concepts and tools and progressing to a detailed discussion of topics, including the Hodge theory, curvature and topology of manifolds, vanishing theorems, gauge theory, geometric equations and geometric flows, Einstein metrics and special holonomy, and complex manifolds. One or two topics are selected each semester.

MAT 551, 552 Riemannian Geometry

Paul C. Yang

An introductory course on the analytic aspects of Riemannian Geometry.

MAT 553, 554 Algebraic Geometry

János Kollár

The geometry of higher dimensional algebraic varieties. Includes various topics from classifications of surfaces and threefolds, minimal models, rational and rationally connected varieties, and birationality questions.

MAT 555, 556 Analytic Methods in Algebraic Geometry

Nicholas M. Katz

Arithmetic algebraic geometry, number theory, and arithmetic aspects of differential equations are studied. Usually treats topics of current student interest in arithmetic algebraic geometry and number theory.

MAT 557 Topics in Algebraic Geometry

Staff

Covers current areas of interest in algebraic geometry.

MAT 558 Topics in Geometry

Staff

Covers current areas of interest in geometry.

MAT 559 Topics in Differential Geometry

Staff

Covers current areas of interest in differential geometry.

MAT 560 Topics in Representation Theory

Andrei Okounkov

Covers current areas of interest in representation theory.

MAT 561, 562 Topology of Manifolds

Staff

Discussion includes homology groups, homotopy groups, cohomology, CW-complexes, Poincare duality, Lefschetz fixed-point theorem, classification of surfaces, Morse theory, fibrations, covering spaces, spectral sequences, and DeRham cohomology. Later discussion includes characteristic classes, vector bundles, Stiefel-Whitney classes, Chern classes, Pontrjagin classes, Cobodism groups, Milnor’s exotic 7-dimensional spheres, Hirzebruch signature theorem, and the Atiyah-Singer index theorem.

MAT 563, 564 Dynamical Systems

John N. Mather

Covers various topics concerning stability and randomness of motion in Hamiltonian systems.

MAT 565, 566 Algebraic Methods in Topology

William Browder

The development of algebraic topological approaches to problems in topology, utilizing ordinary and extraordinary cohomology theories (such as K-theory) and applied to geometric problems (such as classification problems for manifolds and surgery theory) is the focus of this course. The course content varies from year to year.

MAT 567, 568 Topology of Algebraic Varieties

Robert MacPherson

Some of the most interesting, topological spaces are algebraic varieties. Examples include toric, flag, Schubert, and modular varieties. Algebraic varieties admit special topological methods of study, such as stratifications, intersection, homology, the weight filtration, and Lefschetz theorems. Course is an introduction to some of these examples and topological methods.

MAT 569, 570 Gauge Theory and Low-Dimensional Topology

Zoltán Szábo

The geometry of 3- and 4-dimensional manifolds, by using gauge theory and symplectic geometry. Topics may include Donaldson Theory, Seiberg-Witten Theory, Gromov Invariants, and Topology Quantum Field Theories in dimension 4.

MAT 571, 572 Low-Dimensional Topology

David Gabai

Focuses on foliations and laminations on 3-manifolds. Starting at the very beginning, we will rapidly develop the basic theory of the subject. We then explore how the existence of a foliation gives rise to topological understanding of the underlying manifold. We plan to study various properties and characterizations of taut foliations—in particular their relation with the Thurston norm on the 2-dimensional homology and various existence and non-existence results.

MAT 573 Topics in Topology

Staff

Covers current areas of interest in topology.

MAT 581, 582 Stochastic Processes

Edward Nelson

Martingales, Markov processes, and ergodic theory, and relations to potential theory, boundary value problems, and functional analysis are studied.

MAT 583, 584 Statistical Mechanics

Elliott H. Lieb

Various topics in modern statistical mechanics and related areas of mathematical physics are discussed, including condensed-matter theory, field theory, and quantum mechanics of atoms and molecules. Guest speakers are invited to report on their current research.

MAT 585, 586 Mathematical Physics

Michael Aizenman

Consideration of either a single methodology applicable in mathematical physics, such as C*-algebras, probabilistic methods, or functional analytic methods, or a subject in mathematical physics, such as constructive quantum-field theory, the theory of Schrödinger operators, or the statistical mechanics of lattice systems.

MAT 587 Topics in Probability

Staff

Covers current areas of interest in probability.

MAT 591, 592 Applied Partial Differential Equations

Weinan E

Many differential equations in science and engineering come with small or large parameters, such as Planck’s constant in the Schrödinger equation or the Reynolds number in Navier-Stokes equations. Insight can be gained on the behavior of the solutions of these equations by studying the asymptotic limit when these parameters go to zero or infinity. Systematic asymptotic methods have been developed to study such limits. This course is an introduction to such methods.

MAT 593 Wavelets: Applications of Wavelets in Mathematics and Other Fields (see APC 583)

MAT 595 Topics in Discrete Mathematics

Paul D. Seymour, Staff

Various branches of discrete mathematics and combinatorial theory. Material treated is relevant to discrete optimization and algorithmic complexity, but focuses most strongly on combinatorial theory. Students should have already taken a course in elementary graph theory.

MAT 599 Extramural Summer Research Project

Staff

Summer research project designed in conjunction with the student’s adviser and an industrial, private, or government sponsor that will provide practical experience relevant to the student’s research area. Starts no earlier than June 1. A final written report is required. Students considering applying for this course should review the recommended guidelines before consulting their adviser and director of graduate studies.

 

Detailed descriptions of scheduled courses are offered on the department Web site at www.math.princeton.edu/graduate, and also on the bulletin board outside the Common Room on the 3rd floor of Fine Hall before the start of each semester.

Pertinent Courses in Allied Departments

Many graduate courses given by the Department of Physics, the Program in Applied and Computational Mathematics, and the appropriate section of the Department of Civil Engineering may be of interest to students of mathematics. Students should consult with these departments directly.