Colloquium Abstracts Fall 2024

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

Dougal Davis

University of Melbourne
October 24, 2024

Hodge Theory and Unitary Representations of Reductive Lie Groups

Classifying the irreducible unitary representations of a non-compact Lie group is one of the oldest open problems in representation theory. The problem seeks to generalise the theory of Fourier series to function spaces with complicated symmetry and has close ties to both number theory and mathematical physics. In this talk, I will explain joint work with Kari Vilonen, in which we prove a conjecture of Schmid and Vilonen linking unitary representations to another beautiful story in mathematics: Hodge theory and the topology of algebraic varieties. I’ll explain how this turns the ideas of some former residents of Lawrence, Kansas (Solomon Lefschetz and Alexander Grothendieck) into sharp tools for understanding unitary representations, and indicate how this has led to some rapid progress (joint with Lucas Mason-Brown) on previously intractable questions about them.

Botong Wang

University of Wisconsin
December 5, 2024

Positivity of Signed Euler Characteristics of Algebraic Varieties

The Chern-Hopf-Thurston conjecture says that for a closed 2d-dimensional manifold X, if the universal cover of X is contractible, then the Euler characteristic of X is always nonnegative, up to a factor (-1)^d. In this talk, we will explore various algebraic varieties with nonnegative signed Euler characteristics, and we will discuss how such positivity is related to vanishing theorems and nonabelian Hodge theory.