Colloquia/Spring 2025

date	speaker	title	host(s)
Feb 7 in VV 911	Peter Smillie (MPI) (Job Talk)	Harmonic maps and geometrization	Waldron
Feb 21	Alex Wright (Michigan)	Curve graphs and totally geodesic subvarieties of moduli spaces of Riemann surfaces	Apisa
Feb 24 (Monday)	Daniel Bragg (Univ. of Utah)	Murphy’s Law for the moduli stack of curves	Caldararu
Feb 28		TBA
Mar 7	Daniel Groves (UIC)	TBA	Uyanik
Mar 14	Yue M. Lu (Harvard)	TBA	Li
March 19 (Wed)	Xuan-Hien Nguyen (Iowa State)		Tran	Ph.D. Prospective Student Visit Day
Mar 21	Aaron Brown (Northwestern)	Schneider LAA Lecture	Zimmer
Mar 28	Spring Break
April 4	Caglar Uyanik	TBA
April 11	Special Colloquium (combined with Differential Geometry Workshop)	TBA	Zhang
April 18	Jack Xin (UC Irvine)		Tran
April 29	Mladen Bestvina (Utah)	Distinguished Lecture Series Part I (room B130)	Uyanik
April 30	Mladen Bestvina (Utah)	Distinguished Lecture Series Part II (room B239)	Uyanik
May 1	Mladen Bestvina (Utah)	Distinguished Lecture Series Part III (room B130)	Uyanik
May 2	Henri Berestycki (Maryland–College Park / EHESS)		Graham

Abstracts

February 7: Peter Smillie (MPI) (Job Talk)

Title: Harmonic maps and geometrization

Abstract: Many problems in differential geometry can be studied via the space of representations from the fundamental group $\Gamma$ of a manifold $M$ to a Lie group $G$. Conversely, much of what we know about the space of representations is through this sort of geometrization. For $M$ a closed surface, two fields have emerged in the last thirty years with distinct yet overlapping methods: Higher Teichm\"uller theory focusing more on dynamics and coarse geometry, and Non-Abelian Hodge Theory more algebro-geometric and analytic. A central point of overlap between these two fields is the study of equivariant harmonic maps.

I will give an introduction to both fields, and explain two foundational conjectures of Higher Teichm\"uller theory on the relationship between them. I will then present the resolution of one of these conjectures in the negative (joint work with Nathaniel Sagman) and ongoing work on the resolution of the other in the positive (joint with Max Riestenberg). Time permitting, I will also explain the solution (joint with Philip Engel) of a problem in carbon chemistry, and how it fits into this picture.

February 21: Alex Wright (Michigan)

Title: Curve graphs and totally geodesic subvarieties of moduli spaces of Riemann surfaces

Abstract: Given a surface, the associated curve graph has vertices corresponding to certain isotopy classes of curves on the surface, and edges for disjoint curves. Starting with work of Masur and Minsky in the late 1990s, curve graphs became a central tool for understanding objects in low dimensional topology and geometry. Since then, their influence has reached far beyond what might have been anticipated. Part of the talk will be an expository account of this remarkable story.

Much more recently, non-trivial examples of totally geodesic subvarieties of moduli spaces have been discovered, in work of McMullen-Mukamel-Wright and Eskin-McMullen-Mukamel-Wright. Part of the talk will be an expository account of this story and its connections to dynamics.

The talk will conclude with new joint work with Francisco Arana-Herrera showing that the geometry of totally geodesic subvarieties can be understood using curve graphs, and that this is closely intertwined with the remarkably rigid structure of these varieties witnessed by the boundary in the Deligne-Mumford compactification.

February 24: Daniel Bragg (Utah)

Title: Murphy’s Law for the moduli stack of curves

Abstract: Murphy's Law states "Anything that can go wrong will go wrong". In the context of algebraic geometry, "Murphy's Law" is used to refer to the philosophy that moduli spaces of algebro-geometric objects should be expected to have arbitrarily complicated structure, absent a good a-priori reason to think otherwise. In this talk I will explain my work verifying that a certain precise formulation of this philosophy holds for the moduli of curves, as well as a number of other natural moduli problems. This implies that the moduli space of curves fails to be a fine moduli space in every possible way, and that there exist curves which are obstructed from being defined over their fields of moduli by every possible mechanism. This is joint work with Max Lieblich.