# The Bowen Lectures

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

**The 2018-2019 Bowen Lectures** will be given by James McKernan of University of California, San Diego and Christopher Hacon of University of Utah on February 19, 20 and 21, 2019. Each lecture begins at 4:10pm and ends at 5:00pm.

Series Title: **Geometry of Algebraic Varieties**

**Tuesday February 19th**

Lecture 1: *Symmetries of polynomial equations.*

**Banatao Auditorium, Sutardja Dai Hall**

*Abstract*:

The symmetries of systems of polynomial equations can be be understood in terms of the geometry of the variety of zeroes (or solution set) of the polynomials. Roughly speaking, there are 3 kinds of geometries corresponding to positive, zero and negative curvature giving rise to 3 different kinds of symmetry groups. In this lecture, I will discuss recent advances in algebraic geometry that lead to very precise results on the structure of these symmetry groups.

**Wednesday February 20th**

Lecture 2: *On the birational classification of algebraic varieties.***Banatao Auditorium, Sutardja Dai Hall**

*Abstract*:

Algebraic varieties are geometric objects defined by polynomial equations. The minimal model program (MMP) is an ambitious program that aims to classify algebraic varieties. According to the MMP, there are 3 building blocks: Fano varieties, Calabi-Yau varieties and varieties of general type which are higher dimensional analogs of Riemann Surfaces of genus 0,1 or at least 2 respectively. In this talk I will recall the general features of the MMP and discuss recent advances in our understanding of Fano varieties and varieties of general type.

**Thursday February 21st**

Lecture 3: *Birational geometry in characteristic $p>0$.***60 Evans Hall**

*Abstract*:

After recent spectacular progress in the classification of varieties over an algebraic closed field of characteristic 0 (e.g. the solution set of a system of polynomial equations defined by $p_1,...,p_r$ in $C[x_1,...,x_n]$) it is natural to try and understand the geometry of varieties defined over an algebraically closed field of characteristic $p>0$. Many technical difficulties arise in this context. Nevertheless, there has been much progress recently. In particular, the MMP was established for 3-folds in characteristic $p>5$ by work of Birkar, Hacon, Xu and others. In this talk, we will explain some of the challenges and the recent progress in this active area of research.

**Biographies of Speakers:**

**Christopher Hacon** was born in Manchester, UK but grew up in Pisa, Italy where he attended the Scuola Normale Superiore di Pisa and the University of Pisa. He received his PhD from UCLA in 1998 under the direction of Robert Lazarsfeld. He is currently a distinguished professor at the University of Utah.
He is a member the American Academy of Arts and Sciences and the National Academy of Sciences. He is also a recipient of the Clay, Cole and Breakthrough prizes.

**James McKernan** was born in London, UK. He went to Trinity College in
Cambridge, where he obtained a BA in mathematics and then took Part
III. He first went to Brown and then transferred to Harvard where he
obtained his PhD under the direction of Joe Harris. He is currently a
professor at the University of San Diego. He is a Fellow of the
Royal Society and a recipient of the Clay, Cole and Breakthrough

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

2016-17 Michael Harris

2017-18 Avi Wigderson