Colloquia

UW Madison mathematics Colloquium is on Fridays at 4:00 pm.

February 3, 2023, Friday at 4pm Facundo Mémoli (Ohio State University)

(host: Lyu)

The Gromov-Hausdorff distance between spheres.

The Gromov-Hausdorff distance is a fundamental tool in Riemanian geometry (through the topology it generates) and is also utilized in applied geometry and topological data analysis as a metric for expressing the stability of methods which process geometric data (e.g. hierarchical clustering and persistent homology barcodes via the Vietoris-Rips filtration). In fact, distances such as the Gromov-Hausdorff distance or its Optimal Transport variants (i.e. the so-called Gromov-Wasserstein distances) are nowadays often invoked in applications related to data classification.

Whereas it is often easy to estimate the value of the Gromov-Hausdorff distance between two given metric spaces, its precise value is rarely easy to determine. Some of the best estimates follow from considerations related to both the stability of persistent homology and to Gromov's filling radius. However, these turn out to be non-sharp.

In this talk, I will describe these estimates and also results which permit calculating the precise value of the Gromov-Hausdorff between pairs of spheres (endowed with their usual geodesic distance). These results involve lower bounds which arise from a certain version of the Borsuk-Ulam theorem that is applicable to discontinuous maps, and also matching upper bounds which are induced from specialized constructions of (a posteriori optimal) ``correspondences" between spheres.

February 24, 2023, Cancelled/available

March 3, 2023, Friday at 4pm Stefan Steinerberger (University of Washington)

Title: How curved is a combinatorial graph?

Abstract: Curvature is one of the fundamental ingredients in differential geometry. People are increasingly interested in whether it is possible to think of combinatorial graphs as behaving like manifolds and a number of different notions of curvature have been proposed. I will introduce some of the existing ideas and then propose a new notion based on a simple and explicit linear system of equations that is easy to compute. This notion satisfies a surprisingly large number of desirable properties -- connections to game theory (especially the von Neumann Minimax Theorem) and potential theory will be sketched; simultaneously, there is a certain "magic" element to all of this that is poorly understood and many open problems remain. I will also sketch some curious related problems that remain mostly open. No prior knowledge of differential geometry (or graphs) is required.

(hosts: Shaoming Guo, Andreas Seeger)

March 8, 2023, Wednesday at 4pm Yair Minsky (Yale University)

Distinguished lectures

Title: Surfaces and foliations in hyperbolic 3-manifolds

Abstract: How does the geometric theory of hyperbolic 3-manifolds interact with the topological theory of foliations within them? Both points of view have seen profound developments over the past 40 years, and yet we have only an incomplete understanding of their overlap. I won't have much to add to this understanding! Instead, I will meander through aspects of both stories, saying a bit about what we know and pointing out some interesting questions.

(host: Kent)

March 10, 2023, Friday at 4pm Yair Minsky (Yale University)

Distinguished lectures

Title: End-periodic maps, via fibered 3-manifolds

Abstract: In the second lecture I will focus on some joint work with Michael Landry and Sam Taylor. Thurston showed how a certain ``spinning'' construction in a fibered 3-manifold produces a depth-1 foliation, which is described by an end-periodic map of an infinite genus surface. The dynamical properties of such maps were then studied by Handel-Miller, Cantwell-Conlon-Fenley and others. We show how to reverse this construction, obtaining every end-periodic map from spinning in a fibered manifold. This allows us to recover the dynamical features of the map, and more, directly from the more classical theory of fibered manifolds.

(host: Kent)

March 24, 2023 , Friday at 4pm Carolyn Abbott (Brandeis University)

(host: Dymarz, Uyanik, WIMAW)

Title: Boundaries, boundaries, and more boundaries Abstract: It is possible to learn a lot about a group by studying how it acts on various metric spaces.

One particularly interesting (and ubiquitous) class of groups are those that act nicely on negatively curved spaces, called hyperbolic groups. Since their introduction by Gromov in the 1980s, hyperbolic groups and their generalizations have played a central role in geometric group theory. One fruitful tool for studying such groups is their boundary at infinity. In this talk, I will discuss two generalizations of hyperbolic groups, relatively hyperbolic groups and hierarchically hyperbolic groups, and describe boundaries of each. I will describe various relationships between these boundaries and explain how the hierarchically hyperbolic boundary characterizes relative hyperbolicity among hierarchically hyperbolic groups. This is joint work with Jason Behrstock and Jacob Russell.