2026-08-17: Dimitri Shlyakhtenko: A free analog of Bobkov’s isoperimetry inequality
(UCLA)
We prove a one-variable functional inequality which is the free probability analog of Bobkov’s isoperimetry inequality. The inequality involves the L1 norm of the difference quotient of a function f and can be viewed as a non-local isoperimetric inequality. We also prove related inequalities for subsets of an interval as well as subsets of roots of Hermite polynomials. This paper is also an experiment in AI-based exploration of free analogs of classical probability statements.
2026-08-24: Enli Chen: Strongly convergent matrix models for q-Gaussian families
(TU Delft)
Let (s_1^{(q)}, \dots, s_r^{(q)}) be a finite q-Gaussian family. We construct explicit finite-dimensional random matrix models that converge strongly to this family for |q| < \sqrt{2} – 1 . Thus, for every noncommutative polynomial evaluated at the random matrix models, both its normalized trace and its operator norm
converge to the corresponding quantities for the q-Gaussian family in probability. The construction combines two ingredients. First, we consider normalized sums of graph-product semicircular variables indexed by Erdos-Renyi random graphs with an additional Clifford twist when q < 0. We prove quantitative operator-norm estimates for the resulting approximate q-Toeplitz relations, as well as an asymptotically sharp degree-one Khintchine inequality. In particular, the normalized
semicircular sum has limiting norm \frac{2}{\sqrt{1-q}} , which is the norm of the standard q-Gaussian variable. Using ultraproduct methods, we upgrade these estimates to complete multivariable strong convergence. Second, we use a refined tensor-GUE strong-convergence theorem to transfer these operator models to finite-dimensional random matrices. A key feature is uniformity for bounded-degree polynomials with matrix coefficients whose dimensions may grow beyond the size of the random matrices. As operator-algebraic consequences, in the above range of q, the associated q-Gaussian C^*-algebras are MF and, when there are at least two generators, their Brown-Douglas-Fillmore extension semigroups are not groups. This talk is based on recent joint work with Martijn Caspers.
2026-08-31: Therese Basa Landry: Compact quantum metric spaces from free probability
(UC Santa Barbara)
We initiate the study of quantum metric space structures for operator algebras arising from free probability, namely q-Gaussians and free Gibbs laws. Length functions on groups have been well-studied as a means of producing spectral triples which give rise to compact quantum metric spaces. For our first main result, we define compact quantum metric spaces for q-Gaussians using length-like functions by th e same mathod as developed by Ozawa and Rieffel for spectral triples coming from length functions and applied by others to hyperbolic groups, quantum groups of rapid decay, free products, and free graph algebras. Motivated by free transport results for free Gibbs laws, we describe a universal way of defining Lip-norms in terms of a generating set, which behaves well under changes of coordinates. As our construction can be applied to any given self-adjoint tuple (x_1,\dots,x_d) generating a unital C*-algebra A, our methods may be of wider interest for constructing quantum metric spaces. This is joint work with David Jekel.
2026-09-14: Tim Austin: Entropy and large deviations for random unitary representations
(University of Warwick)
This talk will be an introduction to “almost periodic entropy”. This quantity is defined for positive definite functions on a countable group, or more generally for positive functionals on a separable C*-algebra. It is an analog of Lewis Bowen’s “sofic entropy” from ergodic theory. This analogy extends to many of its properties but some important differences also emerge. I will not assume any prior knowledge about sofic entropy. After setting up the basic definition, I will focus on the special case of finitely generated free groups, about which the most is known. For free groups, results include a large deviations principle in a fairly strong topology for uniformly random representations. This, in turn, offers a new proof of the Collins-Male theorem on strong convergence of independent tuples of random unitary matrices, and a large deviations principle for operator norms to accompany that theorem.
2026-09-21: Dima Shlyakhtenko: On the \mathrm{II}_1 factors of Fuchsian groups
(UCLA)
We show that von Neumann algebras of fundamental groups of closed orientable surfaces of genus g \geq 2 are free group factors on 2g-1 generators. The key technical ingredient involves a proof that the element w= ABA^{-1}B^{-1} of the free group F_2 = \langle A, B \rangle is freely complemented in the group factor: L(F_2) = \mathrm{W}*(w) * \mathrm{W}^*(v) for some Haar unitary v \in L(F_2) that is freely independent from w. Combined with previous results, we conclude that for an arbitrary finitely generated torsion-free non-elementary discrete subgroup \Gamma \subset PSL_2(\mathbb{R}), L() is a free group factor settling a conjecture
of de la Harpe and Voiculescu. This result was obtained using OpenAI’s ChatGPT Pro 6.0