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).

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.

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.

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.

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.

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

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.

Jean-Francois Paquet, Vanderbilt University

Special relativity meets fluid dynamics: the transformative role of nucleus colliders

Quark-gluon plasma is an exotic phase of nuclear matter that can be produced by colliding large nuclei at velocities close to the speed of light. This plasma is the smallest and hottest liquid ever produced, extending the size of a nucleus but reaching temperatures higher than those found in the most extreme astrophysical events. The explosive expansion of this plasma can be described with a version of viscous fluid dynamics that accounts for the effect of special relativity. I will discuss the scientific community’s efforts to use some of the world’s largest particle colliders to understand the exceptional properties of quark-gluon plasma and, in turn, push the boundaries of our theoretical understanding of relativistic viscous fluid dynamics.

Dick Canary, University of Michigan

An invitation to Anosov representations

Fuchsian groups arise naturally as groups of covering transformations of hyperbolic surfaces. One may view them as images of discrete faithful representation of free groups and surface groups into PSL(2,R). The study of hyperbolic surfaces and deformation spaces of Fuchsian groups is a rich and classical subject. One may naturally generalize this to the study of groups of covering transformations of hyperbolic manifolds and this also has a beautiful, well-developed theory especially in dimension three.

Introduced in 2006, the theory of Anosov representations into semi-simple Lie groups (e.g. PSL(d,R)) has emerged as a higher rank analogue of the theory of Fuchsian groups. Our talk will begin by recalling some of the classical facts about Fuchsian groups. We will then gently introduce the subject of Anosov representations and their emerging theory. We will focus on the subject of representations into PSL(3,R) whose images arise as covering transformations of convex projective structures on surfaces.

Lorenzo Gavassino, Vanderbilt University

Robert McCann, University of Toronto

The monopolist’s free boundary problem in the plane: an excursion into the economic value of private information

The principal-agent problem is an important paradigm in economic theory for studying the value of private information: the nonlinear pricing problem faced by a monopolist is one example; others include optimal taxation and auction design. For multidimensional spaces of consumers (i.e. agents) and products, Rochet and Chone (1998) reformulated this problem as a concave maximization over the set of convex functions, by assuming agent preferences are bilinear in the product and agent parameters. This optimization corresponds mathematically to a convexity-constrained obstacle problem. The solution is divided into multiple regions, according to the rank of the Hessian of the optimizer.
Apart from four possible pathologies,  if the monopolists costs grow quadratically with the product type we show that a smooth free boundary delineates the region where it becomes efficient to customize products for individual buyers.  We give the first complete solution of the problem on square domains, and discover new transitions from unbunched to targeted and from targeted to blunt bunching as market conditions become more and more favorable to the seller.

Leonardo Abbrescia, Georgia Tech

A Lorentzian manifold-with-boundary where causality breaks down due to shock singularities

We present a novel example of a Lorentzian manifold-with-boundary featuring a dramatic degeneracy in its deterministic and causal properties known as “causal bubbles” along its boundary. These issues arise because the regularity of the Lorentzian metric is below Lipschitz and fit within the larger framework of low regularity Lorentzian geometry. Although manifolds with causal bubbles were recently introduced in 2012 as a mathematical curiosity, our example comes from studying the fundamental equations of fluid mechanics and shock singularities which arise therein. No prior knowledge of Lorentzian geometry or fluid mechanics will be assumed for this talk.

Steve Trettel, University of San Francisco

What do 3-manifolds Look Like?

Thurston’s Geometrization Theorem, later proved by Perelman, describes 3-manifolds in terms of geometric building blocks. Each manifold decomposes into pieces modeled on one of just eight homogeneous 3-dimensional spaces, known as the Thurston geometries.

In this talk, we explore the question: ‘What does a 3-manifold look like?’ through the lens of geometrization. After discussing how one might compute the “inside view” one would see in such spaces, we will use software written in collaboration with Remi Coulon, Sabetta Matsumoto, and Henry Segerman to explore some manifolds built from the Thurston geometries. Finally, we will see how these geometric pieces can be fit together to form a picture of general 3-manifolds.

Ronnie Pavlov, University of Denver

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Mark de Cataldo, Stony Brook University

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