In many contexts a correspondence has been found between the classical compact groups and certain boundary conditions -- $U(n)$ corresponding to periodic, $USp(2n)$ corresponding to Dirichlet, $SO(2n)$ corresponding to Neumann and $SO(2n+1)$ corresponding to Zaremba. In this talk, I will try to elucidate this correspondence in Lie theoretic terms and in the process relate random matrix theory to Yang-Mills theory, free fermions and modular forms.

Recently, it has been more clearly understood that the N=4 superconformal index leads to the microstate counting of the BPS black holes in AdS_5 X S^5. The leading N^2 behavior of the free energy was shown in various ways to match that of the known BPS black hole in the gravity side, but this correspondence has not been realized at the level of the saddle point analysis of the full matrix model for the N=4 index. Here, I will try to clarify how such saddles corresponding to the BPS black holes arise as 'areal' distributions. The talk is based on https://arxiv.org/abs/2111.10720 with Sunjin Choi, Seok Kim, and Eunwoo Lee; https://arxiv.org/abs/2103.01401 with Sunjin Choi and Seok Kim.

Small angle x-ray scattering is one of the most flexible and readily available experimental methods for obtaining information on the structure of proteins in solution. In the advent of powerful predictive methods such as the alphaFold and rossettaFold algorithms, this information has become increasingly in demand, owing to the need to characterise the more flexible and varying components of proteins which resist characterisation by these and more standard experimental techniques. To deal with structures about little of which is known a parsimonious method of representing the tertiary fold of a protein backbone as a discrete curve has been developed. It represents the fundamental local Ramachandran constraints through a pair of parameters and is able to generate millions of potentially realistic protein geometries in a short space of time. The data obtained from these methods provides a treasure trove of information on the potential range of topological structures available to proteins, which is much more constrained that that available to self-avoiding walks, but still far more complex than currently understood from existing data. I will introduce this method and its considerations then attempt to pose some questions I think topological data analysis might help answer. Along the way I will ask why roadies might also help give us some insight….

We use Cech closure spaces, also known as pretopological spaces, to develop a uniform framework that encompasses the discrete homology of metric spaces, the singular homology of topological spaces, and the homology of (directed) clique complexes, along with their respective homotopy theories. We obtain nine homology and six homotopy theories of closure spaces. We show how metric spaces and more general structures such as weighted directed graphs produce filtered closure spaces. For filtered closure spaces, our homology theories produce persistence modules. We extend the definition of Gromov-Hausdorff distance to filtered closure spaces and use it to prove that our persistence modules and their persistence diagrams are stable. We also extend the definitions Vietoris-Rips and Cech complexes to closure spaces and prove that their persistent homology is stable.

This is joint work with Nikola Milicevic.

Topological data analysis is starting to establish itself as a powerful and effective framework in machine learning , supporting the analysis of neural networks, but also driving the development of novel algorithms that incorporate topological characteristics. As a problem class, graph representation learning is of particular interest here, since graphs are inherently amenable to a topological description in terms of their connected components and cycles. This talk will provide
an overview of how to address graph learning tasks using machine learning techniques, with a specific focus on how to make such techniques 'topology-aware.' We will discuss how to learn filtrations for graphs and how to incorporate topological information into modern graph neural networks, resulting in provably more expressive algorithms. This talk aims to be accessible to an audience of TDA enthusiasts; prior knowledge of machine learning is helpful but not required.

The Persistent Homology Transform, and the Euler Characteristic Transform are topological analogs of the Radon transform that can be used in statsistical shape analysis. In this talk I will consider an interesting variant called the Extended Persistent Homology Transform (XPHT) which replaces the normal persistent homology with extended persistent homology. We are particularly interested in the application of the XPHT to binary images. This paper outlines an algorithm for efficient calculation of the XPHT exploting relationships between the PHT of the boundary curves to the XPHT of the foreground.

Topological data analysis (TDA) is a field with tools to quantify the shape of data in a manner that is concise and robust using concepts from algebraic topology. Persistent homology, one of the most popular tools in TDA, has proven useful in applications to time series data, detecting shape that changes over time and quantifying features like periodicity. In this talk, I will present two applications using tools from TDA to study time series data: the first using zigzag persistence, a generalization of persistent homology, to study bifurcations in dynamical systems and the second, using the shape of weighted, directed networks to distinguish periodic and chaotic behavior in time series data.