UW Madison mathematics Colloquium is on Fridays at 4:00 pm.
February 3, 2023, Friday at 4pm Facundo Mémoli (Ohio State University)
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)
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.
March 10, 2023, Friday at 4pm Yair Minsky (Yale University)
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.
March 24, 2023 , Friday at 4pm Carolyn Abbott (Brandeis University)
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.
March 31, 2023 , Friday at 4pm Bálint Virág (University of Toronto)
Title: Random plane geometry -- a gentle introduction
Abstract: Consider Z^2, and assign a random length of 1 or 2 to each edge based on independent fair coin tosses. The resulting random geometry, first passage percolation, is conjectured to have a scaling limit. Most random plane geometric models (including hidden geometries) should have the same scaling limit. I will explain the basics of the limiting geometry, the "directed landscape", and its relation to traffic jams, tetris, coffee stains and random matrices.
April 7, 2023, Friday at 4pm Rupert Klein (FU Berlin)
Title: Mathematics: A key to climate research
Abstract: Mathematics in climate research is often thought to be mainly a provider of techniques for solving, e.g., the atmosphere and ocean flow equations. Three examples elucidate that its role is much broader and deeper:
1) Climate modelers often employ reduced forms of “the flow equations” for efficiency. Mathematical analysis helps assessing the regimes of validity of such models and defining conditions under which they can be solved robustly.
2) Climate is defined as “weather statistics”, and climate research investigates its change in time in our “single realization of Earth” with all its complexity. The required reliable notions of time dependent statistics for sparse data in high dimensions, however, remain to be established. Recent mathematical research offers advanced data analysis techniques that could be “game changing” in this respect.
3) Climate research, economy, and the social sciences are to generate a scientific basis for informed political decision making. Subtle misunderstandings often hamper systematic progress in this area. Mathematical formalization can help structuring discussions and bridging language barriers in interdisciplinary research.
(hosts: Smith, Stechmann)
April 21, 2023, Friday at 4pm Peter Sternberg (Indiana University)
(hosts: Feldman, Tran)
Title: A family of toy problems modeling liquid crystals exhibiting large disparity in the elastic coefficients.
Abstract: Certain classes of liquid crystals have been found to strongly favor particular types of deformations over others; for example, the cost of splay may greatly exceed the cost of bend or twist. In a series of studies with Dmitry Golovaty (Akron), Michael Novack (UT Austin) and Raghav Venkatraman (Courant), we explore the implications of assuming various asymptotic regimes for the elastic constants. Through a mixture of formal and rigorous analysis, along with computations, we identify the limiting behavior of minimizers to the associated energies. We find that a variety of singular structures emerge corresponding to jumps in the profile of these limiting minimizers that effectively save on the cost of splay, bend or twist—whichever is assumed to be most expensive.
April 28, 2023, Friday at 4pm Nam Q. Le (Indiana University)
Title: Hessian eigenvalues and hyperbolic polynomials
Abstract: Hessian eigenvalues are natural nonlinear analogues of the classical Dirichlet eigenvalues. The Hessian eigenvalues and their corresponding eigenfunctions are expected to share many analytic and geometric properties (such as uniqueness, stability, max-min principle, global smoothness, Brunn-Minkowski inequality, convergence of numerical schemes, etc) as their Dirichlet counterparts. In this talk, I will discuss these issues and some recent progresses in various geometric settings. I will also explain the unexpected role of hyperbolic polynomials in our analysis. I will not assume any familiarity with these concepts.
May 5, 2023, Friday at 4pm Janko Gravner (UC Davis)
Title: Long-range nucleation
Abstract: Nucleation is a mechanism by which one equilibrium displaces another through formation of small unstoppable nuclei. Typically, nucleation is local, as the size of the nuclei is much smaller than the time scale of convergence to the new state. We will discuss a few simple models where nuclei are not small in diameter but instead are a result of lower-dimensional structures that grow and interact significantly before most of the space is affected. Analysis of such models includes a variety of combinatorial and probabilistic methods.