Fall 2018

Topics in Arithmetic Geometry: Heights of Algebraic Cycles For a smooth and projective variety X over a global field of dimension n with an adelic polarization, we propose canonical local and global height pairings for two cycles Y, Z of pure dimension p, q satisfying p + q = n*1. We give some explicit arichmedean local pairings by writing down explicit formula for the diagonal green current for some Shimura varieties. Instructor(s): Shou-Wu Zhang
Schedule
C01 M W 01:30 PM - 02:50 PM
Topics in Number Theory: Arithmetic Statistics This course covers current topics in number theory. Specific topic information will be provided when the course is offered.
Schedule
C01 01:00 AM - 01:00 AM
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): Alexandru D. Ionescu
Schedule
C01 T Th 11:00 AM - 12:20 PM
Topics in Nonlinear Analysis: Topics in General Relativity This course covers some of the main open mathematical problems in general relativity such as the rigidity, stability and formation of black holes as well as regularity issues. There is a quick introduction to black holes in general relativity and relevant topics on the long time behavior of nonlinear wave equations. Instructor(s): Sergiu Klainerman
Schedule
C01 M W 01:30 PM - 02:50 PM
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. Instructor(s): Nicholas Michael Katz
Schedule
C01 T 01:30 PM - 04:20 PM
Topics in Algebra: Moduli of Varieties of General Type We study the moduli of varieties of general type. Half the course is devoted to the general theory and half to recent developing topics. Instructor(s): János Kollár
Schedule
C01 T Th 01:30 PM - 02:50 PM
Topics in Conformal and Cauchy-Rieman (CR) Geometry: Recent Developments in Conformal Geometry We cover basic background and then some recent developments in conformal geometry. The following are discussed: (1) Background of fully non-linear elliptic PDE. (2) Garding inequality (3) Study of $\sigma_2$ equation on 4-manifold, recent developments. (4) Study of the singular Yamabe problem. (5) Study of fractional Yamabe problem. We also discuss open problems in this research area. Instructor(s): Sun-Yung Alice Chang
Schedule
C01 T Th 11:00 AM - 12:20 PM
Topics in Low Dimensional Topology: Smooth surfaces in 4-manifolds We discuss various classical constructions of spheres in the 4-sphere and slice discs in the 4-ball as well as some classical invariants that smoothly distinguish them. We look at a recent construction of slice knots due to Gompf, Scharlemann and Thompson as well as the theory immersions, crossing changes and moves on surfaces based on the work of Hirsch-Smale, Giller and Roseman. Instructor(s): David Gabai
Schedule
C01 Th 01:30 PM - 04:20 PM
Topics in Discrete Mathematics: Induced Subgraphs This is a general introduction to the topic of induced subgraphs. We survey some of the main theorems (perfect graph theorem, chi-boundedness results, clawfree graphs), and then focus on several open questions, including Erdos-Hajnal conjecture, Fox conjecture, Gyarfas-Sumner conjecture and Liebenau-Pilipczuk conjecture. More specific details are provided. Instructor(s): Paul Douglas Seymour
Schedule
C01 T Th 04:30 PM - 06:20 PM