Saturday, November 12, 2022 at the University of Maryland

Tony Feng (Berkeley)

Title for part 1: A brief history of theta series
Abstract:  Theta series were introduced by Jacobi as generating functions for counting lattice vectors. They turn out to enjoy a symmetry properties called modularity, which has many interesting applications. Kudla introduced an incarnation of theta series in arithmetic geometry, called arithmetic theta series, which are also expected to enjoy modularity, although this is conjectural in general. I will survey this story, as well as recent function field version which is joint work with Zhiwei Yun and Wei Zhang, wherein we construct generalizations that we call “higher theta series”, and conjecture their modularity properties.

Title for part 2: Derived Fourier analysis and modularity
Abstract:  The modularity of classical theta series can be proved using the Poisson summation formula, a tool in Fourier analysis. I will explain an approach to the modularity conjecture for higher theta series, which is joint with Zhiwei Yun and Wei Zhang. It is based on a sheaf-cycle correspondence generalizing the classical sheaf-function correspondence, plus a theory of Fourier analysis on derived vector spaces.

Michael Larsen (Indiana)

Title for Part 1: Character bounds for finite simple groups

Abstract: The absolute value of any character value of a finite group is bounded above by the degree of the character. There has been a concerted effort to find stronger bounds, with an eye to applications in probability theory and algebraic group theory. I will talk about recent work on character bounds for finite simple groups and say something about typical applications.

Title for Part 2: Strongly dense subgroups of algebraic groups

Abstract: Breuillard, Green, Guralnick, and Tao introduced the notion of strongly dense free subgroups of algebraic groups and proved that for every simple algebraic group G over an algebraically closed field of sufficiently high transcendence degree, G has subgroups of this kind. I will describe this work and some recent improvements and generalizations.

Burt Totaro (UCLA)

Title: Algebraic varieties with extreme behavior
Abstract: The “volume” is the basic discrete invariant for an algebraic variety of general type, analogous to the genus of a curve. We construct varieties of general type with the smallest known volume. These can be viewed as varieties that are “barely” of general type, generalizing curves of genus 2. We also construct algebraic varieties of several other types (such as Fano and Calabi-Yau varieties) with extreme behavior. (Joint with Louis Esser and Chengxi Wang.)