Graduate Course Schedule

Spring 2019

COS 522/MAT 578 Computational Complexity Computational complexity theory is a mathematical discipline that studies efficient computation. The course covers some of the truly beautiful ideas of modern complexity theory such as: approaches to the famous P vs NP question and why they are stuck; complexity classes and their relationship; circuit lower bounds; proof systems such as zero knowledge proofs, interactive proofs and probabilistically checkable proofs; hardness of approximation; de-randomization and the hardness vs randomness paradigm; quantum computing. Instructor(s): Gillat Kol
Schedule
L01 T Th 03:00 PM - 04:20 PM
MAT 517 Topics in Arithmetic Geometry: Rigid Analytic Geometry The goal of this class is to learn the basic theory of Huber's adic spaces and to see how it simplifies in particular cases, such as rigid analytic varieties, formal schemes, perfectoid spaces. Knowledge of commutative algebra and algebraic geometry over a general basis (i.e. scheme theory) is assumed. Instructor(s): Sophie Marguerite Morel
Schedule
C01 M W 01:30 PM - 02:50 PM
MAT 518 Topics in Automorphic Forms: Automorphic forms and special values of L-functions This course will cover some recent applications of the arithmetic of automorphic forms to questions about special values of L-functions. Particular attention will be given to L-functions of elliptic curves and modular forms. Instructor(s): Christopher McLean Skinner
Schedule
C01 M W 03:00 PM - 04:20 PM
MAT 519 Topics in Number Theory: Arithmetic Statistics We discuss squarefree values of polynomials and related topics in sieve theory and arithmetic statistics. MAT 520 Functional Analysis Basic introductory course to modern methods of analysis. Topics include Lp spaces, Banach spaces, uniform boundedness principle, closed graph theorem, locally convex spaces, distributions, Fourier transform, Riesz interpolation theorem, Hardy-Littlewood maximal function, Calderon-Zygmund theory, oscillatory integrals, almost orthogonality, Sobolev spaces, restriction theorems, spectral theory of compact operators. Applications to partial differential equations and probability theory are presented. Instructor(s): Tristan J. Buckmaster
Schedule
C01 M W 01:30 PM - 02:50 PM
MAT 527 Topics in Differential Equations: Dynamics of Nonlinear PDE We study long-time behavior of solutions of nonlinear PDE with dynamic boundary interactions. Applications include Navier-Stokes equations and related systems. Instructor(s): Peter Constantin
Schedule
C01 T Th 11:00 AM - 12:20 PM
MAT 529 Topics in Analysis: Metric Dimension Reduction This course is devoted to geometric realizations of metric spaces, including how to construct them and invariants that serve as obstructions to their existence. We focus on embeddings into low-dimensions, aiming to tackle the important question of how information about pairwise distances among finitely many points influences the magnitude of the dimension of the ambient 'host space'. We develop useful mathematical tools that are needed to address them, but are powerful in other areas as well, leading to investigations in probability, harmonic analysis, geom group theory, the linear theory of Banach spaces, and the Ribe program. Instructor(s): Assaf Naor
Schedule
C01 T 01:30 PM - 04:20 PM
MAT 531 Introduction to Riemann Surfaces After a survey of some useful tools, including sheaves and line bundles, the course will cover some of the basic results about compact Riemann surfaces, including holomorphic and meromorphic differentials, the Riemann-Roch Theorem, and Abel's Theorem; then some special topic will be discussed, such as the role of theta functions, the Hurwitz moduli space, fine structure of the Brill-Noether diagram, pseudogroup structures, Prym differentials, uniformization, or other current topics. Instructor(s): Robert Clifford Gunning
Schedule
C01 T Th 01:30 PM - 02:50 PM
MAT 547 Topics in Algebraic Geometry: Introduction to \ell-adic etale cohomology This course is an introduction to \ell-adic etale cohomology and some of its applications. We will continue from Fall 2018. Instructor(s): Nicholas Michael Katz
Schedule
C01 T 01:30 PM - 04:20 PM
MAT 558 Topics in Conformal and Cauchy-Rieman (CR) Geometry: Recent Developments in Conformal Geometry We will cover some recent developments in conformal geometry, mainly on 4-manifolds. Topics included: 1) Study of integral conformal invariants, /sigma_2 functional on 4-manifolds, issues related to compactness, and study of blow-up limits. (Recent joint work of Chang-Yang-Ruobing Zhang), and 2) Study of conformal 'filling in' of manifolds as boundary of conformal compact Einstein manifolds, the questions of existence and non-existence; unique and non-uniqueness and compactness with given boundary data. We will cover recent developments in this field, including works of Qing-Li-Shi, Gursky-Han, Gursky-Han-Strolz, Chang-Ge and Chang-Ge-Qing. Instructor(s): Sun-Yung Alice Chang
Schedule
C01 M W 01:30 PM - 02:50 PM
MAT 559 Topics in Geometry: Discrete geometry: Incidence theorems and their applications Given a family of geometric objects, what can we say about their incidence structure? How many pairs of points/lines can intersect in the real plane? Over finite fields? These questions and others come up in various guises in many areas of mathematics and theoretical computer science and often have powerful and surprising applications. Topics include Szemeredi-Trotter over the reals, Guth-Katz solution to Erdos' distinct distance problem, Kakyea estimates over the reals and over finite fields, applications to pseudo-randomness, Bourgain-Katz-Tao sum-product theorem, Sylvester-Gallai theorem and variations, etc. Instructor(s): Zeev Dvir
Schedule
C01 T Th 01:30 PM - 02:50 PM
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. Instructor(s): Peter Steven Ozsváth
Schedule
C01 M W 11:00 AM - 12:20 PM
MAT 568 Topics in Knot Theory: Khovanov homology and knot Floer homology This course covers some of the modern techniques and recent developments in knot theory. The focus this semester is on Khovanov homology and knot Floer homology with the following weekly topics: the Jones polynomial, the geometry of alternating knots and Tait conjectures, Seifert surfaces and the Alexander polynomial, Khovanov homology, smoothly slice knots and Rasmussen's s-invariant, knot Floer homology, computing knot Floer homology for (1,1)-knots, Heegaard Floer homology, surgery problems for knots, L-space knots, applications of knot Floer homology, and relations between Knot Floer homology and Khovanov homology. Instructor(s): Zoltán Szabó
Schedule
C01 T Th 09:30 AM - 10:50 AM
MAT 569 Topics in Topology: Elliptic operators and topology The Atiyah-Patodi-Singer index theorem describes, for a Riemannian manifold with boundary, a deep relation between the geometry of the manifold, its topology, and the spectral theory of its boundary. With the understanding of this theorem as a motivating guideline, this course focuses on the circle of ideas surrounding it: spin geometry, heat kernels on Riemannian manifolds, spectral geometry, and related topics. Instructor(s): Francesco Lin
Schedule
C01 M W 09:30 AM - 10:50 AM
MAT 577 Topics in Combinatorics: Extremal Combinatorics The course covers topics in Extremal Combinatorics, including ones motivated by questions in computer science, information theory, number theory and geometry. Subjects covered include graph powers, the Shannon capacity and the Witsenhausen rate of graphs, Szemeredi's regularity lemma and its applications in graph property testing and in the study of sets with no. 3 term arithmetic progressions, the Combinatorial Nullstellensatz and its applications, the capset problem, containers and list coloring, and related topics as time permits. Instructor(s): Noga Mordechai Alon
Schedule
C01 T Th 09:30 AM - 10:50 AM
MAT 579 Topics in Discrete Mathematics: Structure and Algorithms We discuss graph algorithms that rely heavily on the structure of the input graph. The focus is on recent developments in coloring algorithms for graphs with certain Induced subgraphs excluded. Instructor(s): Maria Chudnovsky
Schedule
C01 T Th 11:00 AM - 12:20 PM
MAT 588/APC 588 Topics in Numerical Analysis: Optimization On Smooth Manifolds This course covers current topics in numerical analysis. Specific topic information is provided when the course is offered. For the first topic we study theory and algorithms for optimization on smooth manifolds. The course mixes mathematical analysis and coding, aiming for methods that are actually practical and that we truly understand. No prior background in Riemannian geometry or optimization is assumed. Students should be comfortable with linear algebra and multivariable calculus. Instructor(s): Nicolas Boumal
Schedule
C01 M W 11:00 AM - 12:20 PM