Mathematics

Academic Year: 
2019-20
Address: 
Fine Hall
Phone: 
609-258-4443
Website: 
Department of Mathematics
Program offerings: 
Ph.D.
Director of Graduate Studies
Mihalis Dafermos
Javier Gomez-Serrano
Graduate Program Administrator
Jill LeClair

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

Application Deadline: 
December 15 - 11:59PM Eastern Standard Time
Program Length: 
4 years
Application Fee: 
$90
Application Requirements: 
Statement of Academic Purpose
Resume/Curriculum Vitae
Recommendation Letters
Transcripts
Fall Semester Grades
Required Tests
English Language Tests
GRE : 
General Test and Mathematics Subject test required

Ph.D.

Ph.D. Requirements: 
Courses:

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.

Language(s):

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.

Pre-Generals Requirement(s):

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.

General Exam:

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.

Qualifying for the M.A.:

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.

Teaching:

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.

Post-Generals Requirement(s):

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

Dissertation and FPO:

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.

COS 522 Computational Complexity (also
MAT 578
) Introduction to research in computational complexity theory. Computational models: nondeterministic, alternating, and probabilistic machines. Boolean circuits. Complexity classes associated with these models: NP, Polynomial hierarchy, BPP, P/poly, etc. Complete problems. Interactive proof systems and probabilistically checkable proofs: IP=PSPACE and NP=PCP (log n, 1). Definitions of randomness. Pseudorandomness and derandomizations. Lower bounds for concrete models such as algebraic decision trees, bounded-depth circuits, and monotone circuits.
MAT 500 Effective Mathematical Communication This course is for second-year graduate students to help them develop their writing and speaking skills for communicating mathematics in a wide variety of settings, including teaching, grant applications, teaching statement, research statement, talks aimed at a general mathematical audience, and seminars, etc. In addition, responsible conduct in research (RCR) training is an integral part of this course.
MAT 511 Class Field Theory This course will describes abelian extensions of number fields and function fields of curves over finite fields. One example is the celebrated Kronecker-Weber theorem stating that any abelain extension of Q is contained in a field generated by roots of unity. Another example is Kronecker's Jugendtraum stating that all abelian extensions of imaginary quadratic fields can be obtained analogously using torsion points of elliptic curves with complex multiplications. Prerequisites: Galois Theory (such as MAT 322) and MAT 419.
MAT 514 Topics in Algebra Covers areas of current interest in Algebra.
MAT 515 Topics in Number Theory and Related Analysis This course covers current topics in Number Theory and Related Analysis. Specific topic information provided when course is taught.
MAT 516 Topics in Algebraic Number Theory This course covers current topics in Algebraic Number Theory. More specific topic details provided when the course is taught.
MAT 517 Topics in Arithmetic Geometry This course covers current topics in Arithmetic Number Theory. Specific topic information provided when course is offered.
MAT 518 Topics in Automorphic Forms This course covers current topics in Automorphic Forms. Specific topic information provided when the course is taught.
MAT 519 Topics in Number Theory This course covers current topics in number theory. Specific topic information will be provided when the course is offered.
MAT 520 Functional Analysis Basic introductory course to modern methods of analysis. Specific applications of methods to problems in other fields, such as partial differential equations, probability, & number theory are presented. Topics include Lp spaces, tempered distribution, Fourier transform, Riesz interpolation theorem, Hardy-Littlewood maximal function, Calderon-Zygmund theory, the spaces H1 and BMO, oscillatory integrals, almost orthogonality, restriction theorems & applications to dispersive equations, law of large numbers & ergodic theory. Course also discusses applications of Fourier methods to discrete counting problems, using Poisson summation formula.
MAT 522 Introduction to PDE (also
APC 522
) The course is a basic introductory graduate course in partial differential equations. Topics include: Laplacian, properties of harmonic functions, boundary value problems, wave equation, heat equation, Schrodinger equation, hyperbolic conservation laws, Hamilton-Jacobi equations, Fokker-Planck equations, basic function spaces and inequalities, regularity theory for second order elliptic linear PDE, De Giorgi method, basic harmonic analysis methods, linear evolution equations, existence, uniqueness and regularity results for classes of nonlinear PDE with applications to equations of nonlinear and statistical physics.
MAT 523 Advanced Analysis The course covers the essentials of the first eleven chapters of the textbook, "Analysis" by Lieb and Loss. Topics include Lebesque integrals, Measure Theory, L^p Spaces, Fourier Transforms, Distributions, Potential Theory, and some illustrative examples of applications of these topics.
MAT 526 Topics in Geometric Analysis and General Relativity This course covers current topics in Geometric Analysis and General Relativity. More specific topic details provided when the course is offered.
MAT 527 Topics in Differential Equations This course covers current topics in Differential Geometry. (More details provided the course is offered/scheduled.)
MAT 528 Topics in Nonlinear Analysis This course covers current topics in Nonlinear Analysis. More specific details will be provided when the course is offered.
MAT 529 Topics in Analysis This course covers current topics in Analysis. Specific topic details provided when offered.
MAT 531 Introduction to Riemann Surfaces This course is an introduction to the theory of compact Riemann surfaces, including some basic properties of the topology of surfaces, differential forms and the basis existence theorems, the Riemann-Roch theorem and some of its consequences, and the general uniformization theorem if time permits.
MAT 547 Topics in Algebraic Geometry This course covers current topics in Algebraic Geometry. Specific topic details provided when course is offered.
MAT 549 Topics in Algebra This course covers current topics in Algebra. More specific topic details provided when the course is offered.
MAT 550 Differential Geometry This is an introductory graduate course covering questions and methods in differential geometry. As time permits, more specialized topics will be covered as well, including minimal submanifolds, curvature and the topology of manifolds, the equations of geometric analysis and its main applications, and other topics of current interest.
MAT 554 Algebraic Geometry 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 Topics in Differential Geometry This course covers current topics in differential geometry. Specific topic information will be provided when the course is offered.
MAT 556 Analytical Methods in Algebraic Geometry 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 Conformal and Cauchy-Rieman (CR) Geometry This course covers current topics in Conformal and Cauchy-Rieman (CR) Geometry. More specific topic details are provided when the course is offered.
MAT 559 Topics in Geometry This course covers current topics in Geometry. More specific topic details provided when course is offered.
MAT 560 Algebraic Topology The aim of the course is to study some of the basic algebraic techniques in Topology, such as homology groups, cohomology groups and homotopy groups of topological spaces.
MAT 566 Topics in Differential Topology This course covers current topics in Differential Topology. More specific topic details provided when the course is offered.
MAT 567 Topics in Low Dimensional Topology This course covers current topics in Low Dimensional Topology. Specific topic information provided when the course is taught.
MAT 568 Topics in Knot Theory Knot theory involves the study of smoothly embedded circles in three-dimensional manifolds. There are lots of different techniques to study knots: combinatorial invariants, algebraic topology, hyperbolic geometry, Khovanov homology and gauge theory. This course will cover some of the modern techniques and recent developments in the field.
MAT 569 Topics in Topology This course covers current topics in Topology. More specific topic details provided when the course is offered.
MAT 572 Topics in Combinatorial Optimization (also
APC 572
) This course covers current topics in combinatorial optimization. More specific topic details are provided when the course is offered.
MAT 576 Topics in Computational Complexity This course covers current topics in Computational Complexity. More specific topic details are provided when the course is offered.
MAT 577 Topics in Combinatorics This course covers current topics in Combinatorics. More specific topic details are provided when the course is offered.
MAT 579 Topics in Discrete Mathematics This course covers current topics in Discrete Mathematics. Specific topic information provided when the course is taught.
MAT 582 Dynamical Systems This course in Dynamical Systems will include differentiable actions of Z or R on manifolds, with emphasis on asymptotic properties of orbits, as well as stability (KAM theory), instability (chaos), and ergodic properties.
MAT 585 Mathematical Analysis of Massive Data Sets (also
APC 520
) This course focuses on spectral methods useful in the analysis of big data sets. Spectral methods involve the construction of matrices (or linear operators) directly from the data and the computation of a few leading eigenvectors and eigenvalues for information extraction. Examples include the singular value decomposition and the closely related principal component analysis; the PageRank algorithm of Google for ranking web sites; and spectral clustering methods that use eigenvectors of the graph Laplacian.
MAT 586 Computational Methods in Cryo-Electron Microscopy (also
APC 511
/
MOL 511
/
QCB 513
) This course focuses on computational methods in cryo-EM, including three-dimensional ab-initio modelling, structure refinement, resolving structural variability of heterogeneous populations, particle picking, model validation, and resolution determination. Special emphasis is given to methods that play a significant role in many other data science applications. These comprise of key elements of statistical inference, image processing, and linear and non-linear dimensionality reduction. The software packages RELION and ASPIRE are routinely used for class demonstration on both simulated and publicly available experimental datasets.
MAT 587 Topics in Ergodic Theory This course covers current topics in Ergodic Theory. More specific topic details provided when course is offered.
MAT 589 Topics in Probability, Statistics and Dynamics This course covers current topics in Probability, Statistics and Dynamics. More specific topic details provided when the course is offered.
MAT 590 Topics in Arithmetic Geometry The course covers the recent work about rational points and algebraic cycles on algebraic varieties defined over number fields. General questions addressed: the height or measure theoretical distribution of rational or algebraic points, and the connection of L-functions to the group of cycles. The tools studied include: Arakelov thoery, diophantine approximation, Shimura variety, and automorphic L-functions. Some familiarity with algebraic geometry, class field theory, and automorphic representations required.
MAT 595 Topics in Mathematical Physics (also
PHY 508
) The course covers current topics in Mathematical Physics. More specific topic details provided when the course is offered.
MAT 599 Extramural Summer Research Project 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.
PHY 521 Introduction to Mathematical Physics (also
MAT 597
) An introduction to mathematically rigorous methods in physics. Topics to be covered include classical and quantum statistical mechanic, quantum many-body problem, group theory, Schroedinger operators, and quantum information theory.