# The Bowen Lectures

The Bowen Lectures are supported by anonymous donors, one of whom was an undergraduate student of Rufus Bowen.

**The 2016-2017 Bowen Lectures** will be 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.

Born in 1947 in Vallejo, California, Robert Edward (Rufus) Bowen was awarded the AB with prizes for scholarship by the University of California at Berkeley in 1967. His doctorate in Mathematics was completed in Berkeley in 1970 under the direction of Stephen Smale. In that year he was appointed to the faculty of the Department of Mathematics at Berkeley. He was promoted to the rank of Professor in 1977.

Bowen worked in mathematical dynamics systems theory. His pioneering studies of topological entropy, symbolic dynamics, Markov partitions, and invariant measures are of lasting importance; much of today's research is inspired by his ideas.

### Past Bowen Lecturers

1981-82 Dennis Sullivan

1982-83 Anatol Katok

1983-84 Michael Atiyah

1984-85 John Franks

1985-86 William Parry

1986-87 Nancy Kopell

1987-88 Blaine Lawson

1988-89 David Ruelle

1989-90 Yuri Manin

1990-91 John Milnor

1991-92 Philip Holmes

1992-93 Israel M. Gelfand

1993-94 Alain Connes

1994-95 Shing-Tung Yau

1995-96 Peter Sarnak

1996-97 Vladimir Arnold

1997-98 Simon Donaldson

1998-99 Barry Mazur

1999-00 Cliff Taubes

2000-01 Don Zagier

2001-02 Yakov G. Sinai

2002-03 Roger Penrose

2003-04 Richard Hamilton

2004-05 Curt McMullen

2005-06 John Conway

2006-07 Edward Witten

2007-08 Michael Freedman

2008-09 Hillel Furstenberg

2009-10 Christophe Soulé

2010-11 Michael Hopkins

2011-12 Cédric Villani

2012-13 Benedict Gross

2013-14 Jeff Cheeger

2014-15 Dusa McDuff

2015-16 Jacob Lurie