# 2016-2017 Bowen Lectures

**The 2016-2017 Bowen Lectures** were given by Michael Harris of Columbia University, on February 22, 23 and 24, 2017. Each lecture begins at 4:10pm and ends at 5:00pm.

Series Title: **Automorphic Galois representations and Langlands correspondences**

Series Abstract:

The Langlands program can be seen as a framework for using sophisticated methods of analysis, geometry, and group theory to derive insights into fundamental questions of number theory, specifically the structure of Galois groups of polynomial equations with rational coefficients. It can also be seen as a way to use Galois theory to find some order in the infinite-dimensional representation theory of important classes of topological groups, including semisimple Lie groups and their p-adic counterparts. The past few years has seen spectacular progress in both perspectives, thanks in large part to new ideas introduced by Peter Scholze and Vincent Lafforgue, based on p-adic geometry and geometric representation theory. The lecture series will introduce some of the goals of the Langlands program, and will review how much closer we are to achieving them than we were a few years ago, and how much remains to be accomplished.

Wednesday February 22, 2017

Lecture 1: *Galois representations and automorphic forms: an introduction to the Langlands program*

Room 50, Birge Hall

Automorphic forms are functions on a class of homogeneous spaces that arise naturally naturally in geometry and number theory and that enjoy particularly strong symmetry properties (the simplest non-trivial example is the circle). Galois representations are efficient ways of packaging meaningful information about solutions to polynomial equations; the cohomology of algebraic varieties provides a rich supply of such packages. The Langlands program is predicated on the insight that these two branches of mathematics are to a large extent about the same thing.

Thursday February 23, 2017

Lecture 2: *Attaching Galois representations to automorphic forms, and vice versa: recent progress*

Room 60, Evans Hall

Most of the Galois representations whose properties can be studied are obtained by analyzing the cohomology of *Shimura varieties*, and of the moduli spaces (stacks) of shtukas, which are families of vector bundles over curves over finite fields with extra structure. This lecture reports on recent work on these *automorphic* Galois representations, with an emphasis on the contributions of Scholze and V. Lafforgue.

Friday February 24, 2017

Lecture 3: *An idiosyncratic survey of open problems*

Room 740, Evans Hall

The final lecture will focus on a few questions in the Langlands program that may be on the verge of solution, and on other questions that at present appear completely out of reach.