# Colloquium Abstracts Fall 2022

Abstracts will be posted here for the colloquium talks when they are available.

## Joseph Auslander

University of Maryland

October 27, 2022

*The Galois Theory of Minimal Flows*

A minimal flow is an action of a group on a compact Hausdorff space for which every orbit is dense. We will discuss equicontinuous and distal minimal flows as well as the Furstenberg structure theorem for the latter. A partial classification of minimal flows is achieved by the group of automorphisms of the universal minimal flow (the Galois theory).

## Patricio Gallardo Candela

University of California, Riverside

November 11, 2022*Unimodal Singularities and Boundary Divisors in the KSBA Moduli of a Class of Horikawa Surfaces*

Smooth surfaces of general type with K^{2}=1, pg=2, and q=0 constitute a fundamental example in the geography of algebraic surfaces. Its 28-dimensional moduli space admits a modular compactification via the minimal model program. We will describe eight new irreducible boundary divisors in such compactification parametrizing reducible stable surfaces. These divisors arise from degenerations to surfaces with an unimodal hypersurface singularity. We also study the GIT compactification, and the Hodge theory of the degenerate surfaces that the eight divisors parametrize. This is joint work with Patricio Gallardo, Gregory Pearlstein, and Zheng Zhang.

## Xiumin Du

Northwestern University

December 1, 2022*Falconer's Distance Set Problem*

A classical question in geometric measure theory, introduced by Falconer in the 80s is, how large does the Hausdorff dimension of a compact subset in Euclidean space need to be to ensure that the Lebesgue measure of its set of pairwise Euclidean distances is positive. In this talk, I'll report some recent progress on this problem, which combines several ingredients including Orponen's radial projection theorem, and the refined decoupling theory.