Graduate Course Schedule

Spring 2018

MAT 515 Topics in Number Theory and Related Analysis: Spectral Theory of Automorphic Forms The course covers the basics of the spectral theory of automorphic forms, including the analytic continuation of Eisenstein series, the trace formula (in some basic cases) and the general Ramanujan conjectures. Applications to number theory are highlighted . Instructor(s): Peter Clive Sarnak
Schedule
C01 W 01:30 PM - 04:20 PM
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): Peter Constantin
Schedule
C01 T Th 01:30 PM - 02:50 PM
MAT 522/APC 522 Introduction to PDE The course is an introduction to partial differential equations, problems associated to them and methods of their analysis. Topics may include: basic properties of elliptic equations, wave equation, heat equation, Schr\"{o}dinger equation, hyperbolic conservation laws, Fokker-Planck equation, basic function spaces and inequalities, regularity theory for linear PDE, De Giorgi method, basic harmonic analysis methods, existence results and long time behavior for classes of nonlinear PDE including the Navier-Stokes equations. Instructor(s): Sergiu Klainerman
Schedule
C01 M W 01:30 PM - 02:50 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: Arithmetic Algebraic Geometry We discuss open problems in equidistribution which arise in looking at quite concrete and explicit situations of everyday algebro-geometric life over finite fields. This is a continuation from fall 2017. Instructor(s): Nicholas Michael Katz
Schedule
C01 T Th 01:30 PM - 02:50 PM
MAT 549 Topics in Algebra: Moduli of Varieties of General Type This course covers current topics in Algebra. More specific topic details provided when the course is offered. Instructor(s): Gabriele Di Cerbo
Schedule
C01 T Th 01:30 PM - 02:50 PM
MAT 558 Topics in Conformal and Cauchy-Rieman (CR) Geometry: Conformal and CR invariants We go over the key steps in the solutions of the Yamabe equation and the Q-curvature equations in dimensions three and four. Instructor(s): Paul Chien-Ping Yang
Schedule
C01 T Th 11:00 AM - 12:20 PM
MAT 559 Topics in Geometry: Picard-Lefschetz Theory and Floer Homology The course covers the symplectic geometry of Lefschetz fibrations and pseudoholomorphic curve theory as applied to such fibrations, including Hamiltonian Floer cohomology and Fukaya category. Instructor(s): Paul A. Seidel
Schedule
C01 T Th 11:00 AM - 12:20 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): Henry Theodore Horton
Schedule
C01 T Th 11:00 AM - 12:20 PM
MAT 568 Topics in Knot Theory: Knot Floer homology 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. Instructor(s): Zoltán Szabó
Schedule
C01 T Th 09:30 AM - 10:50 AM
MAT 569 Topics in Topology: Contact and symplectic topology The course covers Fukaya categories, generation criterion, Hochschild homology, categorical localization, Stein manifolds, and Lagrangian spines. Instructor(s): John Vincent Pardon
Schedule
C01 M W 01:30 PM - 02:50 PM
MAT 577 Topics in Combinatorics: The Probabilistic Method This course covers current topics in Combinatorics. More specific topic details are provided when the course is offered. Instructor(s): Noga Mordechai Alon
Schedule
C01 T Th 09:30 AM - 10:50 AM
MAT 586/APC 511/MOL 511/QCB 513 Computational Methods in Cryo-Electron Microscopy 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. Instructor(s): Amit Singer
Schedule
C01 T Th 11:00 AM - 12:20 PM
MAT 589 Topics in Probability, Statistics and Dynamics: Stochastic processes on graphs Over the course of the semester, we cover the fundamentals of random graph models and probabilistic models of discrete variables on graphs and their phase transitions. We particularly focus on random constraint satisfaction problems and spin glasses. Instructor(s): Allan M. Sly
Schedule
C01 M W 01:30 PM - 02:50 PM
PHY 521/MAT 597 Introduction to Mathematical Physics An interdisciplinary introduction to statistical mechanics. An attempt is made to present the physics embodied in the subject, along with mathematical methods (from probability and analysis) for rigorous results about some of its key models. Topics covered include phase transitions, critical phenomena, emergent structures and scaling limits. Instructor(s): Michael Aizenman
Schedule
L01 T Th 11:00 AM - 12:30 PM