Title: Solving polynomials with (higher) positive curvature

Date: 09/14/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

A smooth solution set of a system of complex polynomials is a manifold that can be studied geometrically. About 15 years ago, two results proved the existence of solutions of the system over a "function field of a complex curve" (Graber-Harris-Starr) and over a finite field (Esnault) provided the associated complex manifolds have positive curvature in a weak sense (rational connectedness). More recently, when
the manifold satisfies a higher version of positive curvature (rational simple connectedness), a similar result was proved over a function field of a complex surface (de Jong-He-Starr). I will explain these results, some applications to algebra (Serre's "Conjecture II", "Period-Index"), and recent extensions, joint with Chenyang Xu, to "function fields over finite fields" and Ax's "PAC fields".