How will we do mathematics in 2030 ?

Michael R. Douglas
Simons Center for Geometry and Physics, Stony Brook
December 17, 2019

We make the case that over the coming decade, computer assisted reasoning will become far more widely used in the mathematical sciences. This includes interactive and automatic theorem verification, symbolic algebra, and emerging technologies such as formal knowledge repositories, semantic search and intelligent textbooks.

Thresholds Versus Fractional Expectation-Thresholds

Keith Frankston
Rutgers University
December 16, 2019

Given an increasing family F in {0,1}^n, its measure according to mu_p increases and often exhibits a threshold behavior, growing quickly as p increases from near 0 to near 1 around a specific value p_c. Thresholds of families have been of great historical interest and a central focus of the study of random discrete structures (e.g. random graphs and hypergraphs), with estimation of thresholds for specific properties the subject of some of the most challenging work in the area.


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.

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.