About
The Probabilistic Operator Algebras Seminar (POAS) is an online seminar organized by Dan-Virgil Voiculescu that features talks on recent advances in free probability, operator algebras, random matrices, and related topics. It meets usually weekly from 9:00-10:30am Pacific time. This site is managed by David Jekel. Please contact [email protected] for more information or to be added to the email list.
Upcoming Talks
May 4: Magdalena Musat and Mikael Rørdam: Tensor products of convex compact sets and entanglement in C*-algebras
(University of Copenhagen)
The Namioka-Phelps tensor product of two convex compact sets associates two new compact convex sets: the minimal and the maximal tensor product where the former is contained in the latter. They show that the two tensor products agree if one of the two convex compact sets is a Choquet simplex. It remains an open problem, known as Barker’s conjecture, if the converse also holds. Barker’s conjecture was recently verified by Aubrun-Lami-Palazuelos-Plavala in the finite dimensional case (and was verified by Namioka and Phelps when one of the two convex compact sets is the square).
We show that Barker’s conjecture holds when the compact convex sets are state spaces of \mathrm{C}^*– algebras, and we describe the two Namioka-Phelps tensor products. The minimal tensor product is precisely the set of entangled states in the (minimal) tensor product of the \mathrm{C}^*-algebras, and one can describe the maximal tensor product in terms of positive maps. We identify the trace space of
the tensor product of \mathrm{C}^*-algebras as the Namioka-Phelps tensor product of the trace spaces and use this to say when the trace simplex of a tensor product of \mathrm{C}^*-algebras is the Poulsen simplex.
May 11: Otte Heinävaara: Convolution comparison measures
(Caltech)
We introduce a new functional comparison between classical and free convolutions the expectation of function f with respect to classical convolution (of two compactly supported probability measures) is larger than the expectation with respect to free convolution, as long as the fourth derivative of f is non-negative. The comparison is based on the existence of convolution comparison measures, novel measures on the plane whose positivity depends on a peculiar identity involving Hermitian matrices.
May 18: Félix Parraud: Asymptotic expansion for transport maps between laws of multimatrix models
(Queen’s University)
The so-called multimatrix models are tuples of random matrices of dimension N with joint density (with respect to the Lebesgue measure) proportional to e^{N^2V} for some function V . More precisely, it is known that as long as V is close enough from the quadratic potential one can find a “transport map” T^N such that, given X^N a tuple of independent GUE random matrices, the law of T^N(X^N) is the one of our multimatrix model. Heuristically, this implies that “up to a change of variables”, our models are the same. However, the construction of TN, which is given by stochastic calculus, usually yields an object harder to study than our multimatrix model itself. The aim of this talk is therefore to study its asymptotic behaviour by giving an asymptotic expansion of this transport map. This also yields some corollaries such that the strong convergence of our multimatrix models. In a few words, our strategy consist in building a suitable space of differentiable functions such that solutions of stochastic differential equations can be viewed as functions of an infinite family of GUE random matrices. Besides, this space of functions is also built in a way that is compatible with the the theory of asymptotic expansion for functions evaluated in GUE matrices developed in previous papers.
This talk is based on joint work arXiv:2604.03213 with David Jekel and Evangelos A. Nikitopoulos.
May 25: Friedrich Götze: Cramer-Wold bounds for Kantorovich and Zolotarev distances
(University of Bielefeld)
Upper bounds for the Kantorovich and Zolotarev distances for probability measures on multidimensional Euclidean spaces are given in terms of similar distances between their one dimensional projections which are often easier accessible. This quantifies for instance the Cramer-Wold continuity theorem of weak convergence for probability measures. We derive bounds for the multivariate Kantorovich transport distance for as well as for the Zolotarev distances using Fourier analysis in Euclidean spaces.
This is based on joint work with Sergey Bobkov in arXiv:2506.17745 and arXiv:2412.10276.
June 1: David Jekel: Free information geometry and the model theory of noncommutative stochastic processes
(University of Copenhagen)
We study optimal transport theory in the free probabilistic setting motivated by the large-n theory of random tuples of matrices. We define a new version of free entropy \chi_{\operatorname{chron}}^{\mathcal{U}}, which is concave along geodesics in the corresponding Wasserstein space. Moreover the heat evolution satisfies the evolution variational inequality, which means that the heat evolution is the Wasserstein gradient flow for entropy in the metric sense. It also has further desirable properties such as chain rule for iterated conditioning, and an expression in terms of stochastic control problems. This entropy is defined using microstate spaces of matrix approximations with respect to an expanded class of test
functions called chronological formulas, which are constructed so as to be closed under taking partial suprema and infima and application of a free heat semigroup. These formulas are part of a novel framework for studying noncommutative filtrations and stochastic processes as metric structures in the sense of continuous model theory.