# 2018-2019 Bowen Lectures

**The 2018-2019 Bowen Lectures** were 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