Title: bmt-independence: combining notions of independence

Abstract: In this talk we will present a framework where monotonic, Boolean and (classical) tensor independence can coexist. In a sense, we generalize Wysoczańsky's bm- independence by allowing the tensor case to be included and removing the restriction of having a partial order of the algebras. This is a work in progress with Saúl Mendoza Jacobo and Josué Vasquez.

Title:  Limit theorems in finite free probability and their degree limits

Abstract: Since the 2010s, when Marcus, Spielman, and Srivastava solved the Kadison--Singer conjecture and found a connection between its solution and free probability theory, this research area has been called finite free probability. Much progress has been made recently, and of particular interest are finite free cumulants by Octavio and Perales, where free cumulants are the basic tool used as a discretization for the characteristic function in the context of free probability. Just recently, the speaker, Octavio Arizmendi (CIMAT) and Yuki Ueda (Hokkaido Education University) have proved a few limit theorems in finite free probability by a unified approach using finite free cumulants in arXiv:2303.01790. The goal of this talk is to present how interesting combinatorial formulas appear and are used in the proofs.

Title: Uniform Weak Convergence of Random Walks for Additive Processes and its Applications

Abstract: An Additive process is a class of continuous in probability stochastic processes with independent increment. A Brownian motion is an example of an additive process, and Donsker’s theorem is a limit theorem for a Brownian motion. So our aim is to get the limit theorem for additive processes. In this talk, I will define the new convergence “uniform weak convergence”, and then I will give the necessary and sufficient condition for random walks generated by an infinitesimal triangular array to weakly converge an additive process uniformly. Afterwards, we will consider an identical distributed case, an infinitely divisible case and so on. This talk is based on a joint work with Takahiro Hasebe (Hokkaido university). 

Title:  Finite free probability and the limiting root distribution of polynomials after differentiation

Abstract:  We will introduce the finite free additive and multiplicative convolutions of two polynomials and explain how it is related to free probability in the limit. We will also study some elementary combinatorial tools such as the finite free cumulants. We will then use finite free probability to study the effect of differentiating a sequence of polynomials several times and then looking at the resulting limiting root distribution (joint work with Octavio Arizmendi and Jorge Garza-Vargas). Time permitting, we will discuss some new analogies between the finite free and the free world that are part of an ongoing project with Andrei Martinez-Finkelshtein and Rafael Morales.