** Daniel Litt, Columbia University **

Title: Fundamental groups in arithmetic and geometry

Abstract: Let X be an algebraic variety -- that is, the solution set to a system of polynomial equations. Then the fundamental group of X has several incarnations, reflecting the geometry, topology, and arithmetic of X. This talk will discuss some of these incarnations and the subtle relationships between them, and will describe an ongoing program which aims to apply the study of the fundamental group to classical problems in algebraic geometry and number theory.