Probabilistic Operator Algebras Seminar

(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.

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(IRMA, University of Strasbourg)

I will discuss the generalization of free probability theory to the tensor setting. In a joint work with Luca Lionni (arXiv:2605.01887) we extend the approaches of Nechita and Park, and of Collins, Gurau, and Lionni, to give formulae for higher order free cumulants of product of mixed and pure tensors. I will discuss different notions of unitary invariance and their consequences on the definition of tensorial free cumulants. These tensorial free cumulants satisfy product relations reminiscent of those verified in the random matrix case. I will discuss a particular example of interest: Gaussian tensors with random covariances.

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(ISTA)

Polya-Schur problems, named after a classical result of Polya and Schur, aim to characterize linear operators T on polynomials which preserve the property of all the zeros being in a specified domain. For univariate polynomials with real roots, this problem was fully resolved by Borcea and Branden (2009), over a century after some of the earliest progress of Hermite and Laguerre. A natural extension is to then consider how the roots of T[p] are distributed, now that the question where
they are distributed is resolved. For operators of the form T = f(\partial_a) for some f in the Laguerre–Polya class, i.e. those described by the Polya–Benz theorem, we will see that applying T amounts to free convolving with a free stable law and taking a convolution power of the empirical zero measure in the large degree limit. This generalizes and unites the recent progress on repeated differentiation and the backwards heat flow. In fact, by allowing T to vary in the degree any free infinitely divisible distribution can be realized this way. Time permitting, we will discuss the key steps of the proof and versions of this result for the free multiplicative convolution and free rectangular convolution.

Based on joint work with Jonas Jalowy (arXiv:2605.31356)

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