# Colloquium; Academic Year 24-25

## “The Black Hole Photon Ring”

Alex Lupsasca- Vanderbilt University

What does a black hole look like? The first images of the supermassive black hole M87* display a bright ring encircling the event horizon, which appears as a dark patch in its surrounding emission. But Einstein’s theory of general relativity predicts that within this image there also lies a thin “photon ring” consisting of multiple mirror images of the main emission. These images arise from photons that orbited around the black hole multiple times, probing the warped space-time geometry just outside its horizon. The photon ring carries an imprint of the strong gravity in this region and encodes fundamental properties of the black hole. A measurement of this predicted (but not yet observed) ring could provide a precise test of general relativity and will be one of the main targets of a NASA mission proposed to fly within the next decade: the Black Hole Explorer (BHEX).

## Local Topological Quantum Codes – Location SC1308

David Penneys- The Ohio State

Quantum information is encoded in a state vector of a tensor product of Hilbert spaces. Quantum error correction codes are useful for correcting errors when transporting quantum information through a noisy channel. In this talk, we will discuss a family of 2D ‘‘local topological’’ quantum error correction codes which use the robustness of topology to deformation to protect quantum information. We will then explain how operator algebra and subfactor techniques can be used to analyze quasi-particle excitations called anyons.

## Amenability, optimal transport and abstract ergodic theorems- Location- SC 5211

Christian Rosendal, University of Maryland

The concept of amenability is ubiquitous in functional analysis, group theory and logic. In general, amenability of, for example, a group allows one to integrate bounded real valued functions on the group in a translation invariant manner, which is of great utility in many contexts. However, unbounded functions are a completely different matter. Nevertheless, by using tools from the theory of optimal transport, more specifically, optimal transportation cost spaces, we shall present a couple of results that show how one may integrate potentially unbounded Lipschitz functions defined on amenable groups as long as the latter admit no non-trivial homomorphism to $\mathbb R$. This is related to previous results of Schneider–Thom and Cuth–Doucha in the bounded setting. The talk will be aimed at a general mathematical audience.

## Talk by Mark Ellingham and Rares Rasdeaconu

Mark Ellingham and Rares Rasdeaconu, Vanderbilt University

**“Maximum genus directed embeddings of digraphs”**

**Mark Ellingham, Vanderbilt University**

In topological graph theory we often want to find embeddings of a given connected graph with minimum genus, so that the underlying compact orientable surface of the embedding is as simple as possible. If we restrict ourselves to cellular embeddings, where all faces are homeomorphic to disks, then it is also of interest to find embeddings with maximum genus. For undirected graphs this is a very well-solved problem. For digraphs we can consider directed embeddings, where each face is bounded by a directed walk in the digraph. Much less is known about maximum genus in this setting. Previous work by other people provided the answer in the very special case of regular tournaments, and in some cases of directed 4-regular graphs the answer can be found using an algorithm for the representable delta-matroid parity problem. We describe some recent work, joint with Joanna Ellis-Monaghan of the University of Amsterdam, where we have solved the maximum directed genus problem in some reasonably general situations.

**“The loss of maximality in Hilbert squares”**

**Rares Rasdeaconu, Vanderbilt University**

The talk will be an introduction to the Smith-Thom maximality of real algebraic manifolds. An unexpected loss of maximality for the Hilbert square of real algebraic manifolds exhibited in a joint work with V. Kharlamov (University of Strasbourg) will be discussed. Time permitting, I will outline several open problems.

## Talk by Darren Creutz and Mike Neamtu

Darren Creutz, Vanderbilt University

Mike Neamtu, Vanderbilt University

**Darren Creutz, Symbolic Dynamics: Connecting short-term and long-term complexity in dynamical systems
**

Given a dynamical system and a (reasonable) partition of the space, there isa natural map from the system to a symbolic system: assign each element of the partition a distinct ‘letter’, look at the orbit of a given point and ‘read off the infinite word’ by writing the letter of the element of the partition at each time. Symbolic dynamics is the study of subshifts — closed, shift-invariant subspaces of \mathcal{A}^{\mathbb{Z}} where\mathcal{A} is the ‘alphabet’ — exactly what one obtains from a dynamical system and a partition.

To study the complexity of a subshift (and by extension the complexity of the underlying system it arose from), there are both quantitative and qualitative notions. Quantitatively, there is the complexity function p(q) = the number of distinct words of length q appearing in any of the infinite words in the subshift. Qualitatively, there are various notions of mixing and asymptotic independence.

I will present my work (some joint with R. Pavlov and S. Rodock) on word complexity cutoffs for various qualititative mixing properties and (briefly) explore some of the consequences for ‘low complexity’ systems.

**Mike Neamtu, ****Convergence of Matrix Products, Subdivision, Refinement Equations, and Cascade Networks**

In this talk, we will discuss the following question: Given two square matrices of the same dimension, we consider their arbitrary repeated products and ask under which conditions these products converge to a continuous matrix function. This topic arises from various areas of approximation theory, including subdivision methods for generating curves and surfaces in computer-aided geometric design, refinement equations in multi-resolution analysis, and cascade algorithms. The talk is based in part on joint work with my former student, Diana Sordillo.

## Curve graphs and totally geodesic subvarieties of moduli spaces of Riemann surfaces- Location – TBD

Alex Wright- University of Michigan

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

## Detection and characterization of chirps and oscillating singularities in data: multivariate multifractal techniques

Stephane Jaffard- University Paris Est Creteil

Many types of signals display a very oscillatory behavior in the neighborhood of singularities. It is for example the case for gravitational waves, fully developed turbulence, or brain data. A major issue is to detect such behaviors (referred to as “oscillating singularities” or “chirps”) which are the signature of important physical phenomena. We will show how a multivariate multifractal analysis based on wavelet methods allows to meet these challenges.

## Talk by Jean-Francois Paquet

Jean-Francois Paquet, Vanderbilt University

Abstract tba