Schedule for: 25w5440 - Free Probability, Random Matrices and Finite-Dimensional Approximations

Lecture 1 Core concepts and examples

Free entropy

he topological free entropy is a version of the microstates free entropy adapted for C*-algebras. Here the norm replaces the trace in the definition of microstates. I will also discuss some of the open problems in this area.

Lecture 2: Smooth expansions and the polynomial method

Free entropy dimension

Lecture 3: Proof of Peterson-Thom

1-bounded entropy

Additive and multiplicative finite free convolutions are operations on the set of polynomials with real roots, whose empirical distributions are known to approximate free additive and multiplicative convolutions, as the degree $d$ converges to infinity. We will study the infinitesimal distribution in the limit, namely the fluctuations up to order $1/d$. Using the moment-cumulant formula at the infinitesimal level, we will describe how the infinitesimal distributions changes under the additive and multiplicative finite free convolutions. We will mention some examples and applications, including an explicit formula to compute the effect of repeated differentiation on the infinitesimal distribution. Joint work with Octavio Arizmendi and Josué Vázquez-Becerra (arXiv:2505.01705).

In this talk we investigate the anticoarse space of the von Neumann algebras generated by the kernel and the domain of a closable derivation. We show that when a tuple $(x_i)_{i\in I}$ admits a conjugate system, then for any proper subset $J \subset I$ with $|J| \geq 2$ the inclusion $W*(x_j :j \in j) \subset W*(x_i: i \in I)$ is irreducible, infinite index and non- regular. As an application, we show that irreducible subfactors of a separable II1 factor are generic.

I will discuss both weak and strong convergence of matrix models to operator-valued semicirculars. This requires setting a framework for an analog of non-commutative polynomials for the operator-valued setting. This is joint work with David Jekel, Yoonkyeong Lee, and Brent Nelson.

Many lattices come equipped with a "rank" metric which is of combinatorial importance. In this talk we will explain how methods of first order logic in the category of metric structures can be used to understand limiting objects for various classes of metric lattices. We propose a natural class of limiting objects for finite partition lattices and describe their analytic and combinatorial properties. This is joint work with Jose Contreras Mantilla.

In 2015, Marcus, Spielman, and Srivastava realized that expected characteristic polynomials of sums and products of randomly rotated matrices behave like finite versions of Voiculescu’s free convolution operations. In 2022, I obtained a similar result for commutators of such random matrices, in parallel with the work of Nica and Speicher on commutators of freely independent variables. The key technical tool for this result is Weingarten calculus, which reduces unitary matrix integrals to algebraic combinatorics. I will give an overview of this technique, with an emphasis on the exact formulas involving characters of symmetric groups, and I will show how it can be applied to questions in finite free probability. Then, I will show how my result on commutators naturally leads to considerations about hypergeometric polynomials, in the vein of recent work by Martinez-Finkelshtein, Morales, and Perales. Based on arXiv:2209.00523 and arXiv:2502.00254 (joint w/ Rafael Morales and Daniel Perales).

In the first part of the talk, we will present a finite analogue of Voiculescu’s S-transform, developed in joint work with Fujie, Perales, and Ueda. This notion arises from studying the behavior of finite free cumulants under differentiation of polynomials. As expected, the finite S-transform converges to Voiculescu’s classical S-transform in the large-degree limit, while preserving key properties such as multiplicativity and monotonicity. A central ingredient in the proof of this convergence is a relation between the S-transform and the inverse moments of free additive convolution powers. In the second part of the talk, I will discuss recent work with Hasebe and Kitagawa, addressing a fundamental problem in the theory of free probability: the construction of the S-transform for arbitrary measures. Interestingly, a key idea in this construction comes from finite free probability.

Given a locally compact group G, any left Haar measure uniquely determines a faithful normal semifinite weight on the associated group von Neumann algebra. This weight, called the Plancherel weight, is tracial if and only if G is unimodular, and for countable discrete groups it is the usual tracial state. In the setting of non-unimodular groups, the modular automorphism group of the necessarily non-tracial Plancherel weight is explicitly determined by the so-called modular function of G. In this talk, we will introduce the class of "almost unimodular groups" for which the Plancherel weight is almost periodic, a notion due to Connes from 1972. We will give some examples of such groups, including some whose group von Neumann algebras are type III factors. This is joint work with Brent Nelson.

We show that, for many choices of finite tuples of generators $\mathbf{X}=(x_1,\dots,x_d)$ of a tracial von Neumann algebra $(M,\tau)$ satisfying certain decomposition properties (non-primeness, possessing a Cartan subalgebra, or property $\Gamma$), one can find a diffuse, hyperfinite subalgebra in $W^*(\mathbf{X})^\omega$ (often in $W^*(\mathbf{X})$ itself), such that $W^*(N,\mathbf{X}+\sqrt{t}\mathbf{S})=W^*(N,\mathbf{X},\mathbf{S})$ for all $t>0$. (Here $\mathbf{S}$ is a free semicircular family, free from $\{\mathbf{X}\cup N\}$). This gives a short 'non-microstates' proof of strong 1-boundedness for such algebras. This is joint work with Dimitri Shlyakhtenko.

We propose to study a natural version of Connes' Rigidity Conjecture that involves property (T) groups with infinite center. Utilizing techniques at the intersection of von Neumann algebras and geometric group theory, we establish several cases where this conjecture holds. In particular, we provide the first example of a W*-superrigid property (T) group with infinite center. In the course of proving our main results, we also generalize the main W*-superrigidity result from [CIOS21] to twisted group factors. This is joint work with Ionut Chifan, Denis Osin and Hui Tan.

The goal of this work is to obtain a free probability framework that describes large n behavior of the both Wasserstein distance and the entropy of invariant multi-matrix ensembles. In particular, I discuss the problem of whether the free entropy is concave along geodesics in the Wasserstein distance. The naive approach would be to obtain matrix approximations that have asymptotically the correct entropy and Wasserstein distance. However, it turns out that in the framework of non-commutative laws, certain $X$ and $Y$, the same random matrix models cannot asymptotically realize the desired entropy and Wasserstein distance simultaneously, due to their not accounting for additional information about how $X$ interacts with the ambient algebra. This additional information can be captured by the type in continuous model theory, which allows for partial progress on the question of concavity along geodesics as well as a general theory of optimal couplings and momentum measures in the multivariate setting.

We will discuss completions of C$^*$-algebras with respect to the uniform trace norm which we call tracially complete C$^*$-algebras. These algebras originated from work on the structure and classification of nuclear C$^*$-algebras. This class of algebras generalizes both tracial von Neumann algebras and Ozawa's W$**$-bundles. This is joint work with Carrión, Evington, Gabe, Schafhauser, Tikuisis and White.