We show that a graph product of tracial von Neumann algebras is strongly $1$-bounded if the first $\ell^2$-Betti number vanishes for an associated dense $*$-subalgebra.  Graph products of tracial von Neumann algebras were studied by Caspers and Fima, and generalize Green's graph product of groups.  Given groups $G_v$ for each vertex of a graph $\Gamma$, the graph product is the free product modulo the relations that $G_v$ and $G_w$ commute when $v \sim w$; for von Neumann algebras, graph products are described by a certain moment relation.  In our paper, the crux of the argument is a generalization to tracial von Neumann algebras of the statement that soficity of groups is preserved by graph products.  We replace soficity for groups with a more general notion of algebraic soficity for a $*$-algebra $A$, which means the existence of certain approximations for the generators of $A$ by matrices with algebraic integer entries and approximately constant diagonal.  We show algebraic soficity is preserved under graph products through a random permutation construction, inspired by previous work of Charlesworth and Collins as well as Au-C{\'e}bron-Dahlqvist-Gabriel-Male.  In particular, this gives a new probabilistic proof of Ciobanu-Holt-Rees's result that soficity of groups is preserved by graph products.

This is based on joint work with Rolando de Santiago, Ben Hayes, Srivatsav Kunnawalkam Elayavalli, Brent Nelson.

We will overview variants of CUR decompositions for tensors. These are low-rank tensor approximations in which the constituent tensors or factor matrices are subtensors of the original data tensors. We will discuss variants of Tucker decompositions and those based on t-products in this framework. Characterizations of exact decompositions are given, and approximation bounds are shown for data tensors contaminated with Gaussian noise via perturbation arguments.  Experiments are shown for image compression and time permitting we will discuss applications to robust PCA.

My goal will be to motivate why looking at distribution algebras associated to p-adic lie groups is natural in the context of number theory. More specifically I will try to briefly outline their importance in the p-adic Langlands program. And then I will give a simple example of an overconvergent distribution algebra of certain kinds of  p-adic groups with an eye towards illuminating techniques used in my work Dagger groups and p-adic distribution algebras (joint w/ Matthias Strauch and Claus Sorensen).

Epigenetic cell memory, the inheritance of gene expression patterns across subsequent cell divisions, is a critical property of multi-cellular organisms. Simulation studies have shown how stochastic dynamics and time-scale differences between establishment and erasure processes in chromatin modifications (such as histone modifications and DNA methylation) can have a critical effect on epigenetic cell memory. 

In this talk, we describe a mathematical framework to rigorously validate and extend beyond these computational findings. Viewing our stochastic model of a chromatin modification circuit as a singularly perturbed, finite state, continuous time Markov chain, we extend beyond existing theory in order to characterize the leading coefficients in the series expansions of stationary distributions and mean first passage times. In particular, we characterize the limiting stationary distribution in terms of a reduced Markov chain, provide an algorithm to determine the orders of the poles of mean first passage times, and describe a comparison theorem which can be used to explore how changing erasure rates affects system behavior. These theoretical tools not only allow us to set a rigorous mathematical basis for the computational findings of prior work, highlighting the effect of chromatin modification dynamics on epigenetic cell memory, but they can also be applied to other singularly perturbed Markov chains especially those associated with chemical reaction networks.

Based on joint work with Simone Bruno, Felipe Campos, Yi Fu and Domitilla Del Vecchio.

There is a deep analogy between, on the one hand, matrices and their trace, and on the other hand, random variables and their expectation.  This idea motivates "quantum" or non-commutative probability theory. Tracial von Neumann algebras are infinite-dimensional analogs of matrix algebras and the normalized trace, and there are several ways to construct von Neumann algebras that represent suitable "limits" of matrix algebras, either through inductive limits, random matrix models, or ultraproducts.  I will give an introduction to this topic and discuss the ultraproduct of matrix algebras and its automorphisms or symmetries. This study incorporates ideas from model theory as well as probability and optimal transport theory.

The Cayley-Bacharach theorem says that if two plane cubics intersect in exactly 9 points, then any third cubic passing through eight of these must pass through the ninth. We'll give a weird, elementary but cute proof which shows something a tiny bit stronger. The prerequisites will be not nil but nilpotent, limited to Bezout's theorem which I'll state carefully in the form I need. This proof came from Math 262A, which apparently got it from Terence Tao's blog.