Spring 2024

MAT 515 Topics in Number Theory and Related Analysis: Spectral Theory of Locally Uniform Geometries We introduce and discuss problems connected with prescribing the differential operator spectra of locally uniform geometries, primarily Euclidean hyperbolic and regular graphs. Various applications are given. Most of the tools from number theory, analysis, geometry, combinatorics and group theory are introduced as needed. Instructor(s): Peter Clive Sarnak
Schedule
C01 T 01:30 PM - 04:20 PM
MAT 517 Topics in Arithmetic Geometry: Diophantine Geometry and Modular Forms We cover three topics: The geometric Bombieri-Lang conjecture and explain a proof of the geometric Bombieri-Lang conjecture for ramified covers of abelian varieties by J. Xie and X. Yuan; Faltings heights of abelian varieties and explain a proof of the Northcott property of Faltings heights in an isogeny class of abelian variety by M. Kisin and L. Mocz; and Kroneck limit formula for SL(3) and explain a proof of this formula and an application to the Stark conjecture for cubic fields by N. Bergeron, P. Charollois, and L. Garcia. Instructor(s): Shou-Wu Zhang
Schedule
C01 M W 01:30 PM - 02:50 PM
MAT 519 Topics in Number Theory: Sieves and Algebraic Number Theory We discuss recent developments in arithmetic statistics. Topics include the study of squarefree values of polynomials, Galois groups of random polynomials, field extensions with given Galois group, sizes and structure of class groups, Selmer groups, Tate-Shafarevich groups, and related problems involving sieve theory and algebraic number theory. Instructor(s): Manjul Bhargava
Schedule
C01 Th 01:30 PM - 04:20 PM
MAT 527 Topics in Differential Equations: Parabolic PDE This is an introductory course to Parabolic PDE. We start with a brief review of 2nd order elliptic PDE, then discuss basic theory about the heat equation on the Euclidean space, including fundamental solution, Schauder and Lp estimates, maximal principle, construction and estimate of the heat kernel, energy methods. We then discuss heat equation on manifolds, Harnack principle of Li-Yau. As applications, we plan to cover a derivation of the log-Sobolev inequality on the Euclidean space via parabolic estimates, and the connectionof the inequality to Perelman's W-functional. Instructor(s): Sun-Yung Alice Chang
Schedule
C01 T Th 01:30 PM - 02:50 PM
MAT 528 Topics in Nonlinear Analysis: Topics in General Relativity This course will concern advanced topics in general relativity, including the mathematics of gravitational collapse, the dynamics of rapidly rotating and extremal black holes and the structure of spacetime singularities. Instructor(s): Mihalis Dafermos
Schedule
C01 F 03:00 PM - 05:20 PM
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 basic existence theorems, the general uniformization theorem, and the Riemann-Roch theorem and some of its consequences. Instructor(s): Paul Chien-Ping Yang
Schedule
C01 M W 11:00 AM - 12:20 PM
MAT 547 Topics in Algebraic Geometry: Boundedness of Varieties In the last a few decades, results on boundedness plays a central role in the progress of higher dimensional geometry. This course aims to give a comprehensive treatment of the topic. We cover results on the Fujita Conjecture, uniform bounds on birational maps for general type varieties, boundedness of log general type pairs/ACC Conjecture, and boundedness of complements/BAB Conjecture etc. Instructor(s): Chenyang Xu
Schedule
C01 T Th 01:30 PM - 02:50 PM
MAT 549 Topics in Algebra: Equidistribution via \ell-adic cohomology This course pursues the study of equidistribution questions using the ideas and techniques of \ell-adic cohomology. Instructor(s): Nicholas Michael Katz
Schedule
C01 T Th 03:00 PM - 04:20 PM
MAT 559 Topics in Geometry: Lipschitz Extension, Reverse Isoperimetry and Rounding Suppose that we are given a Lipschitz function f from a subset S of a metric space X to a metric space Y. Can we extend f to a Y-valued Lipschitz function that is defined on all of X? This depends on geometric properties of X,Y,S,f, and it is typically impossible, but for over a century a range of creative methods were devised to prove such extension theorems in many settings. The course describes results on extending Lipschitz functions, starting from the classical and arriving to current research. This includes links to questions on how one can reverse the isoperimetric inequality and ways to round continuous space to a discrete subset. Instructor(s): Assaf Naor
Schedule
C01 W 01:30 PM - 04:20 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): Ian Michael Zemke
Schedule
C01 T Th 11:00 AM - 12:20 PM
MAT 569 Topics in Topology: Topology of 4-manifolds The aim of this course is to give an introduction to smooth 4-manifolds. Topics include elliptic surfaces, symplectic 4-manifolds, the symplectic sum, the knot surgery construction of Fintushel and Stern, Seiberg-Witten invariants, and exotic structures on simply-connected 4-manifolds with small Euler characteristics. Instructor(s): Zoltán Szabó
Schedule
C01 T Th 01:30 PM - 02:50 PM
MAT 577 Topics in Combinatorics: The Probabilistic Method Probabilistic methods in Combinatorics and their applications in theoretical Computer Science. The topics include linearity of expectation, the second moment method, the local lemma, correlation inequalities, martingales, large deviation inequalities, geometry, the VC dimension of a range space and its applications, and possibly more as time permits. 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, optimization, and 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: Modern Discrete Probability Theory The aim of this course is to survey some of the fundamentals of modern discrete probability, particularly random processes on networks and discrete structures. The course introduces topics including concentration of measure, random graphs, percolation, convergence of Markov chains and other topics depending on student interest. Instructor(s): Allan M. Sly
Schedule
C01 M W 01:30 PM - 02:50 PM
MAT 595/PHY 508 Topics in Mathematical Physics: Mathematical Aspects of Condensed Matter Physics The course discusses rigorous results in quantum mechanics, relevant for condensed matter physics. Topics to be covered include: Effect of disorder on quantum dynamics, quantum transport and linear response theory, topological phases of matter, quantum Hall effect and topological classification of insulators (via K-theory or otherwise). Instructor(s): Jacob Shapiro
Schedule
C01 T Th 01:30 PM - 02:50 PM