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
January 26: Dimitri Shlyakhtenko: Strong 1-boundedness via non-microstates free entropy techniques
(UCLA)
In this joint work with Ben Major, we show that for a tracial von Neumann algebra M, a variety of decomposition properties (such as having a Cartan subalgebra, a non-trivial tensor product decomposition, or property Gamma) imply the existence of an amenable subalgebra N (in M, or possibly in its ultrapower) and a generating set X for M, such that the algebra generated by the
semicircular perturbation of the set X actually contains M itself. Using properties of Voiculescu’s non-microstates free entropy and a recent result of Jekel and Pi, this gives a short proof of strong 1-boundedness (in the sense of Jung) for such algebras.
February 2: David Jekel and Brent Nelson: Strong convergence of general Gaussian matrices to operator-valued semicirculars
(University of Copenhagen, Michigan State University)
In joint work with Yoonkyeong Lee and Jennifer Pi, we establish a framework for weak and strong convergence of matrix models to operator-valued semicircular systems parametrized by operator-valued covariance matrices \eta = (\eta_{i,j})_{i,j \in I}. Non-commutative polynomials are replaced by covariance polynomials that can involve iterated applications of \eta_{i,j}, leading to the notion of covariance laws. We give sufficient conditions for weak and strong convergence of general Gaussian random matrices and deterministic matrices to a B-valued semicircular family and generators of the base algebra B. In particular, we obtain operator-valued strong convergence for continuously weighted Gaussian Wigner matrices, such as Gaussian band matrices with a continuous cutoff, and we construct natural strongly convergent matrix models for interpolated free group factors.
February 9: Martin Peev: Renormalising Singular PDEs Driven by q-Gausssian Noises
(Imperial College London)
When attempting to construct QFTs that include Fermions using the methods of Stochastic Quantisation, one is naturally forced to consider noncommutative stochastic PDEs. I shall show how to formulate SPDEs driven by noncommutative noises in terms of algebra-valued singular PDEs. Furthermore, I will describe how one can renormalise singular products appearing in such equations, using the example of free probability and q-Gaussian noises. Finally, I will show how one can derive operator insertion estimates, which are crucial in solving singular SPDEs, such as the \Phi_2^4-equation, by changing the topology on the algebra of q-Gaussian. This talk is based on joint work with Ajay Chandra and Martin Hairer.
February 16: Jonathan Husson: Large deviations for the largest eigenvalue of Kronecker matrices
(University Clermont Auvergne)
In many applications of random matrices (in ecology, spin glasses or machine learning for instance), knowing when the extremal eigenvalues of such matrices are atypical is of paramount importance to understand the qualitative behavior of the system we model. We can reformulate this question using the framework of large deviations and ask for a given model: what are the large deviations of the spectrum? Though the solution of this problem was initially known only for orthogonal/unitarily invariant models (such as GUE/GOE), in the last decade there has been numerous advances in this area for more general random matrices. This talk will be about such an advance for the large deviations of the largest eigenvalue of Kronecker random matrices, that is random matrices defined by block where each block are linear combinations of GOE/GUE matrices (therefore allowing for non-trivial correlations between entries).
This talk is based on a joint work with Jana Reker and Alice Guionnet arxiv:2512.15953.
March 2: Kewei Pan: Boolean entropy
(University of Toulouse)
In this talk, I will introduce a notion of Boolean entropy, which is parallel to the classic entropy and free entropy. Via the Large deviation principle for for certain random matrix models, we are able to give an explicit expression for the Boolean entropy in one dimension, and we can extend it to the multivariate case following the scheme of Voiculescu’s definition for free entropy. Moreover, it turns out that this quantity satisfies similar properties to the entropy in classical and free settings. As a consequence, these results contribute to constructing the framework of universality of noncommutative
probability theory.