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
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.
June 8: Charles-Philippe Diez: Obata’s rigidity theorem in free probability
(Technion)
In this talk, we will discuss a rigidity phenomenon for Voiculescu’s free Poincaré inequality under a non-commutative curvature-dimension condition. In the classical setting, on a n-dimensional Riemannian manifold, Lichnerowicz’s estimate gives a sharp lower bound \lambda_1 \geq n for the first nonzero eigenvalue of the Laplace-Beltrami operator under a uniform positive Ricci curvature bound katex]\operatorname{Rig}_g \geq (n-1)g[/katex], and Obata’s theorem identifies the round sphere as the unique equality case. Cheng and Zhou later established an infinite-dimensional analogue: on a weighted Riemannian manifold satisfying CD(K;1) with K > 0, equality in the spectral gap occurs only when a one-dimensional Gaussian space splits off. This result was subsequently extended to the non-smooth RCD framework by Gigli , Ketterer, Kuwada and Ohta. We will explain how an analogous mechanism appears in free probability, in the “Ricci-flat” setting of free difference quotients. We first establish a free Brascamp-Lieb (Hessian-Poincaré) inequality, showing that under a suitable curvature condition the free Poincaré constant is bounded by the inverse of the convexity parameter, in full analogy with the classical log-concave case. Under the same condition, any extremizer (achieving equality) of the free Poincaré equality must be an affine function of the generators, as in the Ornstein-Uhlenbeck situation. As a consequence, we show that the underlying tracial von Neumann algebra necessarily splits off an abelian, freely complemented semicircular component, which, by Popa’s results, is maximal amenable. In higher rank, under additional assumptions, the extremal directions form a free semicircular family, yielding a free product with a free group factor. This provides a free analogue of the classical rigidity phenomenon and highlights new connections between commutation relations, semigroup techniques, non-commutative Dirichlet forms, and free Bakry-Émery theory.