# Algebraic Geometry Seminar

#### Local-Global Principles Over Arithmetic Function Fields

**Speaker:**
David Harbater, University of Pennsylvania

**Location:**
Warren Weaver Hall 201

**Date:**
Tuesday, April 18, 2017, 3:30 p.m.

**Synopsis:**

Many local-global principles in algebra can be phrased as asserting the existence of a rational point on a variety V over a field F provided that there are points defined over certain overfields of F that are obtained by taking completions. When the variety V is a torsor, the local-global principle can be stated in terms of first Galois cohomology, via a version of the Tate-Shafarevich group. Classically, F is a global field; but generalizations have been considered for more general fields and for higher cohomology. After reviewing known results, the talk will present recent developments in the situation where F is the function field of a variety over a complete discretely valued field.