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
Mar 5 (Wednesday)	Eli Weinstein (Columbia)	Probabilistic Experimental Design for Petascale DNA Synthesis	Cochran
Mar 7	Daniel Groves (UIC)	3-manifold groups?	Uyanik
Mar 14	Yue M. Lu (Harvard)	Nonlinear Random Matrices in Estimation and Learning: Equivalence Principles and Applications	Li
March 19 (Wed)	Xuan-Hien Nguyen (Iowa State)	The fundamental gap in Euclidean, spherical, and hyperbolic spaces	Tran	Ph.D. Prospective Student Visit Day
Mar 21	Aaron Brown (Northwestern)	Matrix groups: dynamics, geometry, and rigidity (Schneider LAA Lecture)	Zimmer
Mar 28	Spring Break
April 11	Special Colloquium (combined with Differential Geometry Workshop) Claude LeBrun (Stony Brook)	Einstein Constants and Differential Topology	Zhang
April 18	Jack Xin (UC Irvine)	Lagrangian and Game Theoretic Methods for Multi-scale and Multi-Dimensional Problems	Tran
April 25	Caglar Uyanik (Madison)	Madison Experimental Mathematics Lab: An Invitation
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 2	Mladen Bestvina (Utah)	Distinguished Lecture Series Part III (room B239)	Uyanik

Abstracts

February 7: Peter Smillie (MPI)

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.

March 5: Eli Weinstein (Columbia)

Title: Probabilistic Experimental Design for Petascale DNA Synthesis

Abstract: Generative modeling offers a powerful paradigm for designing novel functional DNA, RNA and protein sequences. In this talk, I introduce experimental design methods to efficiently manufacture samples from a distribution over biomolecules in the real world. The algorithms implement numerical techniques for approximate sampling using stochastic chemical reactions. I demonstrate synthesizing ~10^16 samples from a generative model of human antibodies, at a sample quality comparable to state-of-the-art protein language models, and a cost of ~\$10^3. The library yields candidate therapeutics for "undruggable" cancer targets. Using previous methods, manufacturing a DNA library of the same size and quality would cost roughly ~\$10^15.

March 7: Daniel Groves (UIC)

Title: 3-manifold groups?

Abstract: Due to a vast amount of work over the last decades, the fundamental groups of 3-manifolds are by now very well understood. I will focus on the following (wide open) question: When is a discrete group the fundamental group of a compact 3-manifold? I'll discuss the background to this question, what is known in various dimensions, and then focus on the case of greatest interest in 3-dimensional topology - the hyperbolic case. Finally, I'll report on some recent work around this question in joint work with Haissinsky, Manning, Osajda, Sisto, and Walsh.

March 14: Yue M. Lu (Harvard)

Title: Nonlinear Random Matrices in Estimation and Learning: Equivalence Principles and Applications

Abstract: In recent years, new classes of structured random matrices have emerged in statistical estimation and machine learning. Understanding their spectral properties has become increasingly important, as these matrices are closely linked to key quantities such as the training and generalization performance of large neural networks and the fundamental limits of high-dimensional signal recovery. Unlike classical random matrix ensembles, these new matrices often involve nonlinear transformations, introducing additional structural dependencies that pose challenges for traditional analysis techniques.

In this talk, I will present a set of equivalence principles that establish asymptotic connections between various nonlinear random matrix ensembles and simpler linear models that are more tractable for analysis. I will then demonstrate how these principles can be applied to characterize the performance of kernel methods and random feature models across different scaling regimes and to provide insights into the in-context learning capabilities of attention-based Transformer networks.

March 19 (Wed): Xuan-Hien Nguyen (Iowa State)

Title: The fundamental gap in Euclidean, spherical, and hyperbolic spaces

Abstract: The fundamental gap is the difference between the first two eigenvalues of the Dirichlet problem for the Laplace operator. We will give a brief history of the problem, state the main conjecture, and give a survey of recent results for the subject.

March 21: Aaron Brown (Northwestern)

Title: Matrix groups: dynamics, geometry, and rigidity

Abstract: I will discuss certain matrix groups, namely, lattices in SL(n, R) and discuss some striking differences between the rank-1 (n=2) and the higher-rank (n at least 3) settings.  I'll describe certain standard (projective and affine) actions of such groups, their dynamical properties, and discuss some recent results aiming to classify actions of higher-rank lattices.

April 11: Claude LeBrun (Stony Brook University)

Title: Einstein constants and differential topology

Abstract: A Riemannian metric is said to be  Einstein if it has constant Ricci curvature. In dimensions 2 or 3, this is actually equivalent  requiring the metric to have constant sectional curvature. However,  in dimensions 4 and higher, the Einstein condition becomes significantly weaker that constant sectional curvature, and this has rather dramatic consequences. In particular, it turns out that there are  high-dimensional smooth closed manifolds that admit pairs of Einstein metrics with Ricci curvatures of opposite signs. After explaining how one constructs such examples, I will then discuss some recent results exploring the coexistence of Einstein metrics with zero and positive Ricci curvatures.

April 18: Jack Xin (UC Irvine)

Title: Lagrangian and Game Theoretic Methods for Multi-scale and Multi-Dimensional Problems

Abstract: In this talk, we discuss some recent development of Lagrangian and game theoretic (i.e. stochastic and two-player control generalizations of the method of characteristics) approaches for multi-scale and multi-dimensional reaction-diffusion-advection equations. Through two case studies, we show how stochastic interacting particle methods (IPM) work out as a mesh-free and self-adaptive computational tool. The first case, dated back to Kolmogorov 1937, is concerned with entropy production of reverse-time diffusion processes, and the resulting principal eigenvalue problem of a non-self-adjoint advection-diffusion operator. At a linear complexity rate, the IPM, derived from the Feynman-Kac formula with a genetic interpretation, computes the eigenfuction as a concentrated invariant measure of particle population evolution up to dimension 16. In the second case study of a haptotaxis advection-diffusion system modeling cancer cell spreading, an IPM with a field coupling captures cell merging and expanding dynamics in 3 space dimensions. The third study aims to address a fundamental problem in turbulent combustion by analyzing a curvature dependent level set Hamilton-Jacobi equation (a.k.a. curvature G-equation), and proving the existence of effective front speeds in a cellular flow. To overcome non-coercivity and non-convexity of the Hamiltonian, we combine a one-sided reachability estimate based on the Kohn-Serfaty deterministic two player game characterization, the streamline structure of the flow and a minimum value principle.

April 25: Caglar Uyanik (Madison)

Title: Madison Experimental Mathematics Lab: An Invitation

Abstract: This is going to be a non-standard colloquium talk focusing on the work we do at the MXM, and how you, whether graduate student, postdoc, faculty, or staff, can get involved. We will talk about our experiences and accomplishments to date, and mention what we plan on doing going forward expanding the mission of MXM.

Mladen Bestvina

Talk 1: Asymptotic dimension, mapping class groups and Out(F_n)

Asymptotic dimension, introduced by Gromov, is a basic large scale invariant of metric spaces. The first part of the talk will be a leisurely introduction to this notion and some methods of computing it. Then I will present some ideas leading to the proof that mapping class groups have finite asymptotic dimension, and finally I will describe an attempt to show that the automorphism group of a free group has finite asymptotic dimension. The main goal of the second part of the talk is to introduce the basic spaces on which mapping class groups and Out(F_n) act.

Talk 2: Automatic continuity of big groups

A Polish group satisfies automatic continuity (AC) if every homomorphism to a separable group is continuous. In a recent preprint with George Domat and Kasra Rafi we classified those stable surfaces (of infinite type) whose mapping class groups satisfy AC. In the talk I will try to outline a proof of the simpler result (also in the paper) that the homeomorphism group of every stable Stone space (e.g. the endspace of a stable surface) is AC. Our work builds on the previous work of Rosendal, Rosendal-Solecki, K. Mann and others.

Talk 3: Non-unique ergodicity in strata of measured geodesic laminations

It follows from the work of Gabai and Lenzhen-Masur that on a surface of genus g>1 there are geodesic laminations with 3g-3 projectively distinct ergodic measures, and this number is maximal possible. All such laminations have ideal triangles as complementary components. In the work in progress with Jon Chaika and Sebastian Hensel we find the maximal number of ergodic measures when the types of complementary components are prescribed. The key is a combinatorial reformulation of "having k ergodic measures" in terms of a certain pattern of multi-curves. If there is time left I will also compare this to a theorem, joint with Elizabeth Field and Sanghoon Kwak, giving an estimate of the maximal number of ergodic length functions on an arational tree in the boundary of Outer space.