Mathematics Academic Year 2022 – 2023 Jump To: General Information Address Fine Hall Phone 609-258-4443 Website Department of Mathematics Program Offerings: Ph.D. Director of Graduate Studies: Chenyang Xu Lue Pan 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. Our students are extremely motivated and come from a wide variety of backgrounds. While we urge independent work and research, a real sense of camaraderie exists among our graduate students. As a result, the atmosphere created is one of excitement and stimulation and 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 graduates are successfully launched into academic positions at premier mathematical institutions and industry. Apply Application deadline December 15, 11:59 p.m. Eastern Standard Time (This deadline is for applications for enrollment beginning in fall 2023) Program length 4 years Fee $75 GRE General Test not accepted; Subject Test in Mathematics not accepted Program Offerings Ph.D. 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. Students must pass the language test by the end of the first year and before standing for the general exam. Additional pre-generals requirements 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. Without fees or formalities, students may also attend 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 research topics 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. It is earned once a student successfully passes the language requirement and the general exam, and the faculty recommends it. 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 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 requirements 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 the 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. 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 Igor Rodnianski Associate Chair Zoltán Szabó Director of Graduate Studies Lue Pan Chenyang Xu Director of Undergraduate Studies Ana Menezes (acting) Zoltán Szabó Professor Michael Aizenman Noga M. Alon Manjul Bhargava Sun-Yung A. Chang Maria Chudnovsky Fernando Codá Marques Peter Constantin Mihalis Dafermos Zeev Dvir Charles L. Fefferman David Gabai June E. Huh Alexandru D. Ionescu Nicholas M. Katz Sergiu Klainerman János Kollár Emmy Murphy Assaf Naor Peter Steven Ozsváth John V. Pardon Igor Rodnianski Peter C. Sarnak Paul Seymour Amit Singer Christopher M. Skinner Allan M. Sly Zoltán Szabó Chenyang Xu Paul C. Yang Shou-Wu Zhang Assistant Professor Jonathan Hanselman Casey L. Kelleher Ana Menezes Evita Nestoridi Lue Pan Jacob Shapiro Jakub Witaszek Ian M. Zemke Ruobing Zhang Associated Faculty John P. Burgess, Philosophy René A. Carmona, Oper Res and Financial Eng Bernard Chazelle, Computer Science Hans P. Halvorson, Philosophy William A. Massey, Oper Res and Financial Eng Frans Pretorius, Physics Robert E. Tarjan, Computer Science Robert J. Vanderbei, Oper Res and Financial Eng Ramon van Handel, Oper Res and Financial Eng Instructor David Boozer Matija Bucic Alan Chang Jennifer Li Paul David Timothy William Minter Jean Pierre Mutanguha Laurel A. Ohm Sarah Peluse Semon Rezchikov Ravi Shankar Artane Siad Fan Wei Liyang Yang Andrew V Yarmola University Lecturer Jennifer M. Johnson Senior Lecturer Mark W. McConnell Lecturer Bjoern Bringmann Allen J. Fang Jonathan M. Fickenscher Tangli Ge Daniel Ginsberg Xiaoyu He Wei Ho Henry Theodore Horton Tatiana K. Howard Jef C. Laga Samuel Mundy Andrew O'Desky Eden Prywes Samuel Pérez-Ayala Hannah Schwartz John T. Sheridan Rita Teixeira da Costa David Villalobos Visiting Professor Bhargav B. Bhatt Visiting Lecturer with Rank of Professor Camillo De Lellis Jacob A. Lurie Akshay Venkatesh For a full list of faculty members and fellows please visit the department or program website. 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 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 525 - Topics in Harmonic Analysis This course covers current topics in Harmonic Analysis. More specific topic information is provided when the course is offered. 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 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 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 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 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.