School of Mathematics

The h-principle in symplectic geometry

Emmy Murphy
Northwestern University; von Neumann Fellow, School of Mathematics
December 9, 2019

Symplectic geometry, and its close relative contact geometry, are geometries closely tied to complex geometry, smooth topology, and mathematical physics. The h-principle is a general method used for construction of smooth geometric objects satisfying various underdetermined properties. In the symplectic context, h-principles typically give constructions of surprising exotica, and methods for detecting the basic flexible objects. We survey a number of results from the previous decade.

The nonlinear stability of the Schwarzschild metric without symmetry

Mihalis Dafermos
Princeton University
December 6, 2019

I will discuss an upcoming result proving the full finite-codimension non-linear asymptotic stability of the Schwarzschild family as solutions to the Einstein vacuum equations in the exterior of the black hole region. 

 

No symmetry is assumed. The work is based on our previous understanding of linear stability of Schwarzschild in double null gauge. Joint work with G. Holzegel, I. Rodnianski and M. Taylor.

Topology of resolvent problems

Benson Farb
University of Chicago
December 6, 2019

In this talk I will describe a topological approach to some problems about algebraic functions due to Klein and Hilbert. As a sample application of these methods, I will explain the solution to the following problem of Felix Klein: Let $\Phi_{g,n}$ be the algebraic function that assigns to a (principally polarized) abelian variety its $n$-torsion points. What is the minimal $d$ such that, after a rational change of variables, $\Phi_{g,n}$ can be written as an algebraic function of $d$ variables? This is joint work with Mark Kisin and Jesse Wolfson.

Regularity lemma and its applications Part I

Fan Wei
Member, School of Mathematics
December 3, 2019

Szemeredi's regularity lemma is an important tool in modern graph theory. It and its variants have numerous applications in graph theory, which in turn has applications in fields such as theoretical computer science and number theory. The first part of the talk covers some basic knowledge about the regularity lemma and some of its applications, such as the graph removal lemma. I will also discuss some recent works related to the removal lemma.