I'll discuss some of the history of the use of random matrices for studying the structure of operator algebras, starting with Voiculescu's notion free independence. We'll see that the original notions of convergence of random matrix models to certain infinite-dimensional operators is actually fairly weak, and discuss the more recent "strong convergence" phenomenon and its applications to C*-algebras. Finally, I'll touch upon some ongoing work, joint with A. Shiner and S. White, for continuing to use random matrix tools to prove structural properties of C*-algebras.

Matrix-product state (MPS) skeletons are connected networks of local one-dimensional quantum lattice models with ground states admitting an MPS representation with finite bond dimension. In this talk, I will discuss how such skeletons underlie certain families of models obeying the Onsager algebra, and how these simple ground states provide a route to explicitly computing correlation functions.

In this talk I will give a gentle introduction to some aspects of the theory of hypergeometric functions as a natural language for addressing various integrals appearing in quantum field theory (QFT). In particular I will focus on the so-called intersection pairings as well as the differential equations satisfied by the integrals, and I will show how these aspects of the mathematical theory can find a natural interpretation in concrete QFT applications. I will mostly focus on Feynman integrals as paradigmatic example, where the language will shed new light on our most powerful method for computing Feynman integrals as well as their non-local symmetries. I will then give an outlook how these methods could allow us to also learn about integrals appearing in other places in field and string theory, such as Coulomb branch amplitudes, celestial holography and AdS (supergravity and string) amplitudes.

Even in a totally discrete space $X$ you need to know how to move between distinct points. A path $P_{x,y}$ between two points $x,y \in X$ is a sequence of points in $X$ that starts with $x$ and ends with $y$. A path system is a collection of paths $P_{x,y}$, one per each pair of distinct points $x, y$ in $X$. We restrict ourselves to the undirected case where $P_{y,x}$ is $P_{x,y}$ in reverse.

Strictly metrical path systems are ubiquitous. They are defined as follows: There is some spanning, connected graph $(X, E)$ with positive edge weights $w(e)$ for all $e\in E$ and $P_{x,y}$ is the unique $w$-shortest $xy$ path. A metrical path system is defined likewise, but $w$-shortest paths need not be unique. Even more generally, a path system is called consistent  (no $w$ is involved here) if it satisfies the condition that when point $z$ is in $P_{x,y}$, then $P_{x,y}$ is $P_{x,z}$ concatenated with $P_{z,y}$. These three categories of path systems are quite different from each other and in our work we find quantitative ways to capture these differences.

Joint work with Daniel Cizma.