(University of Copenhagen)

We seek an analog for the quantum permutation group S_n^+ of the normalized Hamming distance for permutations. We define three distances on the tracial state space of C(S_n^+) that generalize the L^1-Wasserstein distance of probability measures on S_n equipped with the normalized Hamming metric, for which we demonstrate basic metric properties, subadditivity under convolution, and density of the Lipschitz elements in the \mathrm{C}^*-algebra. Two of those metrics are based on a similar construction to Biane and Voiculescu’s free Wasserstein distance.

This is joint work with Anshu and Therese Landry.

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(University Aix Marseille)

Within the framework of representation theory of Lie groups of type SU(N) we shall recall the pictograph approach relating Littlewood-Richardson (LR) coefficients to honeycombs and other dual presentations, in particular O-blades standing for Ocneanu blades. We shall also present a pictographic approach of Kostka coefficients using degenerated O-blades. If time allows, we shall see how the asymptotic behavior of LR coefficients is related to the volume function of the hive polytope, which can be obtained in terms of the Fourier transform of a convolution product of Harish-Chandra orbital measures, and see how such considerations can be generalized to study intertwiners for other types of Lie groups

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(KU Leuven)

Building on the classical noncommutative entropy for automorphisms of nuclear \mathrm{C}^*-algebras, I will introduce a formally different notion of entropy which uses the more refined cpc approximations given by finite nuclear dimension or finite decomposition rank. I will compare these formally different notions and relate them to a measure-theoretic notion of entropy for traces.

This is joint work with Bhishan Jacelon

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(IISER Bhopal)

A random matrix is said to be a missing data matrix if a few entries are zero other entries are random variables. In this talk, we discuss the convergence and asymptotic freeness of missing data matrices of iid, elliptic and covariance random matrices. Specifically, it is known that independent iid, elliptic, and covariance matrices converge to freely independent circular, elliptic, and Marchenko Pastur variables, respectively. We provide the necessary and sufficient conditions on deterministic matrices for which these results hold true for independent missing data matrices of these three types of random matrices. This framework is commonly used in various applied fields, such as biology, neuroscience, and network data analysis.

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(CIMAT Guanajato)

Hoskins and Steinerberger (2022) showed that repeated differentiation of a random polynomial with i.i.d. mean-zero variance-one roots, followed by suitable rescaling converges to a Hermite polynomial. In this talk, I will present joint work with Andrew Campbell and Katsunori Fujie, where we extend their result using the framework of finite free probability. First we establish central limit theorems describing the fluctuations of both the polynomials and their roots around the deterministic Hermite limits. Second, we relax the finite variance assumption on the roots, uncovering a broader phenomenon in which Hermite polynomials are replaced by random Appell sequences naturally associated to finite free probability and infinitely divisible distributions. Throughout, finite free cumulants serve as the key tool enabling concise proofs.

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(Warsaw University of Technology)

The study of perpetuities– fixed points of affine-type distributional equations– has a long history in classical probability, with connections to random walks, branching processes, and heavy-tailed distributions. In this talk, we will present a free probabilistic analogue of this theory, where the fixed-point equation is interpreted in terms of free independence. The first part of the talk will focus on the free multiplicative random walk underlying the problem. We will discuss how the asymptotics of moments under free multiplicative convolution govern the behavior of potential solutions, and explain how these tools allow us to establish the existence of free perpetuities. The second part of the talk will turn to a finer analysis of the distributional tails of free perpetuities. Using subordination techniques for free convolution, we will describe precise asymptotic estimates for one-sided and symmetric cases. These estimates reveal a power-law decay in the critical case, reminiscent of Kesten’s theorem in the classical setting, but with genuinely free-probabilistic features.

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(Notre Dame University)

Given complex value b, the b-polar derivative of a polynomial can be understood as a generalization (or deformation) of the usual derivative. From this point of view, usual differentiation corresponds to 1-polar differentiation. Given a real number b and a sequence of real rooted polynomials with a fixed asymptotic root distribution, we study the root distribution after repeated b-polar differentiation. Our first main result shows that limiting distribution can be understood as the result of fractional free convolution and pushforward maps along Mobius transforms for distributions. Based on the limiting distribution, we define a two-parameter family of operations on measures F^t_b . Here b is the real value we used to polar differentiate, while t keeps track of the proportion of derivatives we take (with respect to the degree). Our second main result provides a non-trivial commutation relation between F^t_b and F^s_a. The proof uses the simple fact that polar differentiation (with respect to distinct values) commutes. Finally, we will explain some other interesting properties of the new operation F^t_b: a semigroup property when fixing b; a notion of polar free infinite divisibility; Belinschi-Nica type semigroups; and examples of well-behaved distributions, such as Marchenko-Pastur and Cauchy.

This is a joint work with Zhiyuan Yang (arXiv:2508.18575).

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(Paderborn University)

This talk explores the limiting empirical zero distributions of polynomials as their degree tends to infinity. In particular, we discuss the role of the coefficients, the implications for (finite) free probability and how zeros evolve under certain differential flows. We shall focus on real-rooted polynomials and present a user-friendly approach to determine real limit zero distributions via the “exponential profile” of the coefficients. Apart from applications to classical polynomial ensembles, this enables us to study the effect of repeated differentiation, the heat flow and finite free convolutions. For instance, this approach provides a simple proof of finite free convolutions converging to free convolutions, even for non-compactly supported distributions.

Complementing these insights on real zeros, I will give an overview on complex limit distributions of certain (random) polynomials and the evolution of their complex zeros under differential flows. For instance, the limiting zero distribution of the so-called Weyl polynomials undergoing the heat flow evolves from the circular law into the elliptic law until it collapses to the Wigner semicircle law. Such an interpolation can be viewed in terms of Brown measures (of R-diagonal operators), but also opens the door to various other viewponts, such as optimal transport, and PDE’s. This part will be accompanied by illustrative simulations leading to unexplored phenomena.

The results are based on joint works with Brian Hall, Ching Wei Ho, Antonia Hofert, Zakhar Kabluchko and Alexander Marynych.

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(Chalmers University of Technology)

I will talk about cone \mathrm{C}^*-algebras. We will discuss their “lifting properties” and applications, in particular a unified approach to some known results (e.g. quasidiagonality and quasidiagonality of amenable traces) and some new results about cones. If time permits, we will discuss homotopy invariance of several \mathrm{C}^*-algebraic properties.

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(University of Oxford)

Based on joint work with Davide Macera and Joe Thomas.

The first non-zero eigenvalue, or spectral gap of the Laplacian on a closed hyperbolic surface encodes important geometric and dynamical information about the surface. We study the size of the spectral gap for random large genus hyperbolic surfaces sampled according to the Weil-Petersson probability measure. We show that there is a c > 0 such that a random surface of genus g has spectral gap at least 1/4 – O(g^{-c}) with high probability. Our approach adapts the polynomial method for the strong convergence of random matrices, introduced by Chen, Garza-Vargas, Tropp and van Handel, and its generalization to the strong convergence of surface groups by Magee, Puder and van Handel, to the Laplacian on Weil-Petersson random hyperbolic surfaces.

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(University of Notre Dame)

Biane’s free multiplicative Brownian motion b_t is the large-N limit of the Brownian motion in the general linear group GL(N;\mathbb{C}) and can be viewed as the solution to a free stochastic differential equation driven by a circular Brownian motion. I will consider random walk approximations to b_t which are discrete approximations to the solution of the SDE. These approximations have the form of a product of steps, each of which is the identity plus a multiple of a circular element. We are able to compute the Brown measure of the model with a fixed number of steps using the linearization method. We are then able to let the number of steps tend to infinity and recover the previously computed Brown measure of b_t itself. A key step in the argument is a new freeness result for block elements. In general, matrices with freely independent entries are not freely independent in the ordinary sense but only in the “operator valued” sense . But we show that in some interesting examples, we do obtain freeness in the ordinary sense. We also show that for a fixed number of steps, the empirical eigenvalue distribution of the corresponding matrix model converges to the Brown measure of the free model.

This is joint work with Bruce Driver, Ching Wei Ho, Todd Kemp, Yuriy Nemish, Evangelos Nikitopolous, and Félix Parraud.

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(University of Virginia)

I will talk about joint work with Kunnawalkam Elayavalli, Robert, and Patchell. In it, we introduce and study a natural notion of selflessness for inclusions of C*-probability spaces, which in particular necessitates all intermediate C*-algebras to be selfless in the sense of Robert. We identify natural sources of selfless inclusions in the realms of \mathcal{Z}-stable and free product C*-algebras. As an application of this, we prove selflessness for a new family of C*-probability spaces outside the regime of free products and group C*-algebras. These include the reduced free unitary compact quantum groups.

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(University of Wrocław)

We introduce a two-parameter deformation of the classical Poisson distribution from the viewpoint of noncommutative probability theory, by defining a (q,t)-Poisson type operator (random variable) on the (q,t)-Fock space. From the analogous viewpoint of the classical Poisson limit theorem in probability theory, we are naturally led to a family of orthogonal polynomials, which we call the (q,t)-Charlier polynomials. These generalize the q-Charlier polynomials of Saitoh Yoshida and reflect deeper combinatorial symmetries through the additional deformation parameter t .A central feature of this paper is the derivation of a combinatorial moment formula of the (q,t)-Poisson type operator and the (q,t)-Poisson distribution. This is accomplished by means of a card arrangement technique, which encodes set partitions together with crossing and nesting statistics. The resulting expression naturally exhibits a duality between these statistics, arising from a structure rooted in generalized Fibonacci numbers. Our approach provides a concrete framework where methods in combinatorics and theory of orthogonal polynomials are used to investigate the probabilistic properties arising from the (q,t)-deformation.

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(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.

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(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.

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(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.

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(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.

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(Queens University Kingston)

A universal rule for computing mixed moments of independent and unitarily invariant random matrices gives in the large N limit, the rule for free independence. In the forty years since Voiculescu gave this rule many extensions and generalizations have been found. One extension, found by
Belinschi and Shlyakhtenko, is called infinitesimal freeness. This has been shown to model the case where the random matrices have different scales, however there are cases where the theory doesn’t cover some standard examples, in particular the Gaussian orthogonal ensemble. Guillaume Cébron and I have found the extension of infinitesimal freeness to the case of orthogonal invariance. In addition to covering the orthogonal case, real infinitesimal freeness also turns out to be the model of freeness needed for the subleading term of finite freeness, which has recently been found by Arizmendi, Perales and Vazquez Becerra.

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(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.

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(UCLA)

We review the classical Mean Field Games theory and the Mean Field Games master equation, to motivate the study of optimal control problems where the observables are non-commuting self-adjoint operators. Under certain convexity assumptions, we show that the value of the optimal control problems in the non-commutative setting, describes the large-n limit of control problems on tuples of self-adjoint matrices. The classical master equation is a non-local equation of hyperbolic type whose studies were completed only under stringent assumptions on the data. Well-posedness of a “free master equation” remains an unexplored direction of research.

This talk is based on works in collaboration with Jekel-Nam-Palmer and Mou-Meszaros-Zhang.

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(Tokyo University)

Tensor categories have been playing more and more important roles in various fields of mathematical physics. In this talk, I will present their recent emergence in two-dimensional topological order and compare physical results using 4-tensors and technique of connections in subfactor theory. It is clear that they are similar, but I will give their precise relations, particularly through studies of the zipper condition in mathematical physics, the usual pentagon relation in tensor categories and the flat part of a connection in subfactor theory.

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(UCSD)

Using a new, non-K-theoretic approach involving embedding spaces in II1 factors with plenty of freely independent Haar unitaries, we prove that the reduced free group C*-algebras for different numbers of free generators are pairwise non-isomorphic. This recovers the seminal result of Pimsner and Voiculescu with a short new proof.

This talk is based on joint work with Srivatsav Kunnawalkam Elayavalli.

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(University of Maryland)

A group G is said to be a matricial field (MF) if it admits a “strongly converging” sequence of approximate homomorphisms into matrices, i.e. norms of polynomials converge to the corresponding value in the left regular representation. G is then said to be purely MF (PMF) if this sequence of maps into matrices can be chosen as actual homomorphisms. G is further said to be purely finite field (PFF) if the image of each homomorphism is finite. These questions have phenomenal applications to the study of \mathrm{C}^* and von Neumann algebras, spectral geometry, random walks and random graphs, spectral gaps of hyperbolic manifolds, minimal surface theory and Yau’s conjectures and even applied mathematics including but not limited to signal processing. By developing a new operator algebraic approach to the MF problems, we are able to prove the following result bringing several new examples into the fold. Suppose G is a MF (resp., PMF, PFF ) group and H is separable (i.e. H = \bigcap_{i=1}^N H_i where H_i < G are finite index subgroups) and K is a residually finite MF (resp., PMF, PFF) group. If either G or K is exact, then the amalgamated free product G *_H (H \times K)
is MF (resp., PMF, PFF). Our work has several applications. Firstly, as a consequence of MF, the Brown-Douglas-Fillmore semigroups of many new reduced C*-algebras are not groups. Secondly, we obtain that arbitrary graph products of residually finite exact MF (resp.,PMF, PFF) groups are MF (resp., PMF, PFF), yielding a significant generalization of the breakthrough work of Magee-Thomas. Thirdly our work resolves the open problem of proving PFF for 3-manifold groups, more generally all RAAGs. Prior to our paper, PFF results remained unknown even in the simple subcase of free products. These results are of further significance since PFF is th property that is used in Antoine Song’s approach towards the existence of minimal surfaces in spheres of negative curvature.

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