Department of Mathematics
Faculty
Faculty | |
Chair
David Gabai
Associate Chair
Christopher M. Skinner
Directors of Graduate Studies
Alexandru D. IonescuNicolas P. Templier Professor
Michael Aizenman, also PhysicsManjul Bhargava Sun-Yung Alice Chang Peter Constantin, also Applied and Computational Mathematics John H. Conway 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 Elliott H. Lieb, also Physics John N. Mather Sophie Morel Edward Nelson Peter S. Ozsváth Igor Y. Rodnianski Peter C. Sarnak Paul D. Seymour, also Applied and Computational Mathematics Yakov G. Sinai Christopher M. Skinner Zoltán Szábo Gang Tian Paul C. Yang Shou-Wu Zhang Associate Professor
Amit Singer, also Applied and Computational Mathematics
Assistant Professor
Szu-Yu ChenZeev Dvir, also Computer Science Rupert Frank Choonghong Oh
Sucharit Sarkar |
Assistant Professor (continued)
Alexander SodinClaus M. Sorensen Nicolas P. Templier Stefan H.M. van Zwam Vlad Vicol Anna K. Wienhard Instructor
Stefanos AretakisCostante Bellettini Jeffrey S. Case Michael Damron Jonathan Fickenscher Aurel Mihai Fulger Penka V. Georgieva David Geraghty Tasho Kaletha Victor Lie Yu-Han Liu Niels Møller Luc L. Nguyen Zsolt Patakfalvi Oana Pocovnicu Claudiu C. Raicu Nicholas Sheridan Kevin F. Tucker Bart Vandereycken Fang Wang Senior Lecturer
Jennifer M. Johnson
Associated Faculty
John P. Burgess, PhilosophyRené A. Carmona, Operations Research and Financial Engineering Bernard Chazelle, Computer Science Erhan Çınlar, Operations Research and Financial Engineering Hans Halvorson, Philosophy Philip J. Holmes, Mechanical and Aerospace Engineering Yannis G. Kevrekidis, Chemical and Biological Engineering William Massey, Operations Research and Financial Engineering Frans Pretorius, Physics Robert E. Tarjan, Computer Science Robert J. Vanderbei, Operations Research and Financial Engineering Sergio Verdú, Electrical Engineering |
Requirements
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.
Ph.D. Requirements
To earn the Doctor of Philosophy (Ph.D.), students must pass a language requirement and both portions of the general examinations, submit an acceptable dissertation and sustain a final public oral examination.
M.A. Requirements
To qualify for the Master of Arts (M.A.), the student must pass the language requirement and the first part of the general examination, and be recommended by the faculty.
General Examination
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; it also 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 for more information.
Teaching Requirements
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 teach at least once before they graduate.
Language Requirements
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.
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.
Financial Support
All first-year students receive full tuition, health plan coverage and a first-year fellowship in science and engineering. First-year students are not required to grade or teach. All second-, third- and fourth-year students are paid as assistants in instruction (AI)/assistants in research (AR). Stipends are paid at a “pre-generals” and “post-generals” rate; the latter being a higher amount. Students are encouraged to apply for funding from outside funding sources.
Courses
Mathematics
MAT 501 Mathematical Logic
Simon B. Kochen
A course selected from one or more of the following topics, which include model theory, decision problems, set theory, recursion theory, and other branches of logic are studied.
MAT 502 Mathematical Logic
Simon B. Kochen
A course selected from one or more of the following topics, which include model theory, decision problems, set theory, recursion theory, and other branches of logic are studied.
MAT 503 Selected Topics in Logic
Edward Nelson
The course covers areas of current interest in mathematical logic.
MAT 504 Selected Topics in Logic
Edward Nelson
The course covers areas of current interest in mathematical logic.
MAT 506 Gauge Theory & Low-Dimensional Topology
Zoltán Szabó
No Description Available
MAT 507 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 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 Algebraic Number Theory
Jordan S. Ellenberg
Ideal theory, adèles and idèles, and zeta functions; Galois cohomology and class field theory; and laws of reciprocity and other applications of class field theory are studied.
MAT 510 Algebraic Number Theory
Goro Shimura
Ideal theory, adèles and idèles, and zeta functions; Galois cohomology and class field theory; and laws of reciprocity and other applications of class field theory are studied.
MAT 511 Analysis and Number Theory
Peter C. Sarnak
The general analytic theory of L-functions. Analytic continuation, zeros, magnitude, and related issues are studied. The emphasis of the course is on applications to problems in number theory automorphic forms and mathematical physics.
MAT 512 Analysis and Number Theory
Peter C. Sarnak
The general analytic theory of L-functions. Analytic continuation, zeros, magnitude, and related issues are studied. The emphasis of the course is on applications to problems in number theory automorphic forms and mathematical physics.
MAT 514 Topics in Algebra
John H. Conway
Covers areas of current interest in Algebra.
MAT 515 Topics in Number Theory
Ramin Takloo-Bighash
Covers areas of current interest in number theory.
MAT 513 Quadratic Forms
John H. Conway
The course will focus on the invariant theory of integral 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 invariants" (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 will depend on the size and interests of the class.
MAT 521 Introduction to Harmonic Analysis
Wilhelm Schlag
An introductory course to basic approaches to problems in harmonic analysis.
MAT 523 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 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 Fourier Analysis on Euclidean Spaces
Elias M. Stein
Fourier transforms, singular integrals, pseudodifferential operators, harmonic functions, L^{p} estimates, and applications to partial differential equations are studied.
MAT 526 Fourier Analysis on Euclidean Spaces
Elias M. Stein
Fourier transforms, singular integrals, pseudodifferential operators, harmonic functions, L^{p} estimates, and applications to partial differential equations are studied.
MAT 527 Fourier Analysis on Groups
Charles L. Fefferman
Group representations, Fourier analysis and potential theory on symmetric spaces, analogues of singular integrals and pseudodifferential operators in the noncommutative case, and applications to several complex variables are studied.
MAT 528 Fourier Analysis on Groups
Charles L. Fefferman
Group representations, Fourier analysis and potential theory on symmetric spaces, analogues of singular integrals and pseudodifferential operators in the noncommutative case, and applications to several complex variables are studied.
MAT 529 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 C R-functions. The aim is to present the background for current research and then to present some of the problems that arise.
MAT 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 C R-functions. The aim is to present the background for current research and then to present some of the problems that arise.
MAT 531 Complex Analytic Varieties
Staff
Topics in complex manifolds such as Riemann surfaces, Kaehler manifolds, complex algebraic manifolds, and the singularities of analytic varieties.
MAT 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 Elliptic and Parabolic Partial Differential Equations
Sun-Yung A. Chang
Basic, classical results in elliptic and parabolic partial differential equations (PDE). Topics include Laplace equation, heat equation, Sobolov spaces, Holder regularity, maximal principles, Harnack inequality for second order elliptic and parabolic PDE of divergence and non divergence type.
MAT 534 Elliptic and Parabolic Partial Differential Equations
Sun-Yung A. Chang
Topics from fully non-linear elliptic PDE. This may include the study of Monge-Ampere equations, of non-linear PDE from geometric considerations, and topics related to recent research.
MAT 535 Nonlinear Wave Equations
Igor Rodnianski
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 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
Sergiu Klainerman
Areas of current interest in analysis.
MAT 539 Topics in Harmonic Analysis
Alex Ionescu
Course explains the main techniques in this area and their applications to relevant examples.
MAT 540/APC 520 Mathematical Analysis of Massive Data Sets
Amit Singer
Analysis of high dimensional data sets using spectral methods, how to perform tasks such as cluserting and classification, regression and out-of-sample extenion of empirical functions, semi-supervised learning, and independent component analysis (ICA). Applications to image and signal processing, structural biology, dynamical systems, text and document analysis, search and data mining, and finance, among others, will be covered.
MAT 542 Introduction to Riemannian Geometry & General Relativity
Hale F. Trotter
Metrics and connections; first variation of arc length, geodesics; completeness, the Hopf-Rinow theorem; the cut locus; second variation and curvature; synge theorem; the index theorem; Rauch and Bishop comparison theorems; Hadamard-Cartan and Bonnet-Myers theorems; the Toponogov comparison theorem; closed geodesics; the injectivity radius; and the isoperimetric inequality are studied.
MAT 549 Differential Geometry
Gang Tian
This course concerns problems in differential geometry. The course will start with an introduction on basic concepts and tools in geometry and then go on to discuss in details selected topics in geometry. The topics include the Hodge theory, Curvature and topology of manifolds, Vanishing theorems, Gauge theory, Geometric equations and geometric flows, Einstein metrics and special holonomy, Complex manifolds. There will be one or two selected topics each semester. The topics may vary in each semester.
MAT 550 Differential Geometry
Gang Tian
This course concerns problems in differential geometry. The course will start with an introduction on basic concepts and tools in geometry and then go on to discuss in details selected topics in geometry. The topics include the Hodge theory, Curvature and topology of manifolds, Vanishing theorems, Gauge theory, Geometric equations and geometric flows, Einstein metrics and special holonomy, Complex manifolds. There will be one or two selected topics each semester. The topics may vary in each semester.
MAT 551 Riemannian Geometry
Paul C. Yang
This is an introductory course about the analytic aspects of Riemannian geometry.
MAT 553 Algebraic Geometry
Staff
This year-long course covers the basic theory of varieties and schemes as well as a large number of more specialized topics, including varieties over number fields and arithmetic questions, cohomol-ogy theories (etale cohomology, de Rham cohomology, p-adic cohomology), the relation between complex projective varieties and complex analysis, and group schemes and p-divisible groups.
MAT 554 Algebraic Geometry
János Kollár
This year-long course covers the basic theory of varieties and schemes as well as a large number of more specialized topics, including varieties over number fields and arithmetic questions, cohomol-ogy theories (etale cohomology, de Rham cohomology, p-adic cohomology), the relation between complex projective varieties and complex analysis, and group schemes and p-divisible groups.
MAT 555 Analytical Methods in Algebraic Geometry
Nicholas M. Katz
Arithmetic algebraic geometry, number theory, and arithmetic aspects of differential equations are studied. The course usually treats topics of current student interest in arithmetic algebraic geometry and number theory.
MAT 556 Analytical Methods in Algebraic Geometry
Nicholas M. Katz
Arithmetic algebraic geometry, number theory, and arithmetic aspects of differential equations are studied. The course usually treats topics of current student interest in arithmetic algebraic geometry and number theory.
MAT 558 Topics in Geometry
Staff
Areas of current interest in geometry.
MAT 560 Topics in Representation Theory
Andrei Okounkov
This course covers current areas of interest in representation theory.
MAT 561 Topology of Manifolds
Staff
Homology groups, hotropy groups, cohomology, CW-complexes, Poincare duality, Lefschetz fixed-point theoren, classification of surfaces, Morse theory. Additional material may include fibrations, covering spaces, spectral sequences, and DeRham cohomology.
MAT 562 Topology of Manifolds
Zoltán Szabó
Characteristic classes, vector bundles, Stiefel-Whitney classes, Chern classes, Pontrjagin classes, Cobodism groups, Milnor's exotic seven-dimensional spheres. Hirzebruch signature theorem. Additional material may include the Atiyah-Singer index theorem.
MAT 563 Dynamical Systems
John N. Mather
Topics in differential dynamical systems, singularities of mappings, structures on manifolds, and related areas.
MAT 564 Dynamical Systems
John N. Mather
Topics in differential dynamical systems, singularities of mappings, structures on manifolds, and related areas.
MAT 565 Algebraic Methods in Topology
Staff
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 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 Topology of Algebraic Varieties
Robert D. MacPherson
Some of the most interesting topological spaces are algebraic varieties. Examples include toric varieties, flag varieties, Schubert varieties, and modular varieties. Algebraic varieties admit special topological methods of study, such as stratifications, intersection, homology, the weight filtration, and Lefschetz theorems. This course is an introduction to some of these examples and topological methods.
MAT 568 Topology of Algebraic Varieties
Robert D. MacPherson
Some of the most interesting topological spaces are algebraic varieties. Examples include toric varieties, flag varieties, Schubert varieties, and modular varieties. Algebraic varieties admit special topological methods of study, such as stratifications, intersection, homology, the weight filtration, and Lefschetz theorems. This course is an introduction to some of these examples and topological methods.
MAT 569 Gauge Theory & Low Dimensional Topology
Zoltán Szabó
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, Floer-homologies, and Topological Quantum Field Theories in dimension 4.
MAT 570 Gauge Theory and Low Dimensional Topology
Zoltán Szabó
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 572 Low Dimensional Topology
David Gabai
This course will focus on foliations and laminations on 3-manifolds. Starting at the very beginning we will rapidly develop the basic theory of the subject. Then we will 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
The course covers areas of current interest in topology.
MAT 579 Additive and Combinatorial Number Theory
W. T. Gowers
Course focuses mainly on results that lie in the intersection of number theory, harmonic analysis, and combinatorics. Three of the main results covered will be Vinogradov's three-primes theorem, Freiman's theoren, and a special case of Szemeredi's theorem. These results give a good introduction to some important and useful techniques and lead to very interesting open problems.
MAT 581 Stochastic Processes
Staff
Martingales, Markov processes, and ergodic theory; and relations to potential theory, boundary value problems, and functional analysis are studied.
MAT 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 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/PHY 521 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 with some subject in mathematical physics, such as constructive quantum field theory, the theory of Schrödinger operators, or the statistical mechanics of lattice systems.
MAT 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 with some 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
The course covers areas of current interest in Probability.
MAT 591 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 Schrodinger 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. The course will consist of two main parts. The first part deals with sharp transition layers such as the boundary layer or domain-wall type of behavior. The second part deals with oscil
MAT 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 Schrodinger 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/APC 583 Wavelets
Ingrid C. Daubechies
Wavelet analysis, especially wavelet bases, is a functional analytic tool having applications in many fields. The course is aimed at building a bridge between mathematics and engineering, and should be accessible to students from either discipline, provided they have sufficient background in mathematical analysis. This is a graduate course open to undergraduate students who have taken Math 314, 315. If in doubt about having the necessary background, the student should consult the instructor before signing up.
MAT 594/APC 584 Wavelets
Radu V. Balan
Course covers topics of wavelet and time-frequency analysis, with special emphasis on wavelet basis construction and filterbanks. It aims at building a bridge between the mathematics of harmonic analysis and its applications in engineering sciences. Two-thirds of the time will be spent on theory, with remaining one-third to be devoted to applications.
MAT 595/APC 595 Topics in Discrete Mathematics
Paul D. Seymour
Various branches of discrete mathematics and combinatorial theory. Material treated is relevant to dsicrete optimization and algorithmic complexity but focuses most strongly on combinatorial theory. Student 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 advisor and an industrial, private or government sponsor that will provide practical experience relevant to the student's research area. Start no earlier than June 1. A final written report is required.