9:00-10:00 Dan-Virgil Voiculescu (UC Berkeley)

Title: Around the exact formula for the quasicentral modulus

Abstract: Extensions to fractals and to the hybrid setting of the exact formula for the quasicentral modulus of n-tuples of commuting selfadjoint operators will be discussed. (The quasicentral modulus plays a key role in the multivariable generalizations of the Kato-Rosenblum and Weyl-von Neumann Kuroda theorems.) 

10:30-10:55 Ching Wei Ho (Academia Sinica)

Title: The deformed single ring theorem

Abstract: The Brown measure of an R-diagonal operator is computed by Haagerup and Larsen and by Haagerup and Schultz. Guionnet, Krishnapur, and Zeitouni proved that, under technical assumptions, the eigenvalue distribution of a ``single ring'' random matrix model, whose limiting distribution is an R-diagonal operator, converges to the Brown measure of the R-diagonal operator. In this talk, I will speak on my recent joint work with Ping Zhong. Zhong and I prove that, under technical assumptions, the eigenvalue distribution of the sum of a deterministic matrix and a single ring random matrix model converges to the Brown measure of its limiting operator in distribution.

11:00-11:25 Ping Zhong (University of Wyoming)

Title: Upgrading free convolution to non-normal random variables 

Abstract: Brown measure is a sort of spectral measure for free random variables, not necessarily normal. I will report some recent progress on the Brown measure of the sum X+Y of two free random variables X and Y, where Y has certain symmetry and somewhat explicit R-transform. The procedure relies on Hermitian reduction and subordination functions. The analytic results (due to Voiculescu, Biane, Belinschi and Bercovici) for usual free convolution on the real line are very useful in this approach. The Brown measure results can predict the limit eigenvalue distribution of various full rank deformed random matrix models. The talk is based on joint works with Bercovici, Belinschi and Yin.  

11:30- 11:55 Cong Zhou (University of Mississippi)

Title: Hincin's theorem for additive free convolutions of tracial R-diagonal *-distributions

Abstract: (PDF version) Hincin proved that any limit law associated with a triangular array of uniformly infinitesimal random variables is infinitely divisible. Analogous results for the additive and multiplicative free convolution were proved by Bercovici, Belinschi and Pata. We prove an analogous result for the $\boxplus_{RD}$ convolution of measures defined on the positive half-line. This is the convolution arising from the addition of *-free R-diagonal elements of a tracial, noncommutative probability space.

14:00-15:00 László Kérchy (University of Szeged)

Title: Weak similarity relations between Hilbert space operators

Abstract: (PDF version) Reducing the Hyperinvariant Subspace Problem to a special class of operators, Foias and Pearcy introduced in [1] the concept of ampliation quasisimilarity of the operators A and B, which means quasisimilarity of the inflations A(∞) and B(∞). Later Bercovici, Jung, Ko and Pearcy initiated study of the (seemingly) more general relation of pluquasisimilarity, when these inflations can be injected and can be densely mapped into each other. This relation still preserves existence of proper hyperinvariant subspaces. It is a natural open question whether the relations of ampliation quasisimilarity and pluquasisimilarity coincide. We address the more general problem under what conditions is it true that if A can be injected into B and A can be densely mapped into B, then A is a quasiaffine transform of B. Related questions concerning pluquasisimilarity are also studied. Our talk is based on the papers [3] and [4].

[1] C. Foias and C. Pearcy, On the hyperinvariant subspace problem, J. Funct. Anal., 219 (2005), 134–142.

[2] H. Bercovici, I. B. Jung, E. Ko and C. Pearcy, Generalizations of the relation of quasisimilarity for operators, Acta Sci. Math. (Szeged), 85 (2019), 681–691.

[3] L. Kérchy, Pluquasisimilar Hilbert space operators, Acta Sci. Math. (Szeged), 86 (2020), 503–520.

[4] Maria F. Gamal’ and L. Kérchy, Quasiaffine transforms of Hilbert space operators, to appear in Acta Sci. Math. (Szeged). 

15:30-15:55 Edward Timko (Georgia Institute of Technology)

Title: Some multi-variate adaptations of results from univariate operator theory

Abstract: We review some multi-variate adaptations of results from univariate function and operator theory. We begin with a discussion on constraints and model operators, and then present effects of constraints on structure and spectra. These results will be contrasted with known results from univariate operator theory, with the latter also providing suggestions for future investigations. This is based on joint work with R. Clouatre at the University of Manitoba.

16:00-16:25 Christopher Felder (Indiana University Bloomington)

Title: A quick word on inner functions

Abstract: Classical inner functions have a long history and play a critical role in answering operator and function theoretic questions (most prominently in the Hardy space on the unit disk). We will compare and contrast some generalizations of inner-ness in other settings. We will then choose a definition that seems to work best from an operator theoretic viewpoint, and discuss connections to some interesting open problems. 

16:30-16:55 Vittorino Pata (Politecnico di Milano)

Title: On the Semigroup of Linear Viscoelasticity

Abstract: We consider the abstract integrodifferential equation modeling the dynamics of linearly viscoelastic solids. The equation is known to generate a strongly continuous semigroup of linear contractions on a certain phase space, whose asymptotic properties have been the object of extensive studies in the last decades. Nevertheless, some relevant questions still remain open, with particular reference to the decay rate of the semigroup compared to the decay of the memory kernel, and to the structure of the spectrum of the infinitesimal generator of S(t). In this talk we intends to provide some answers.  

9:00-10:00 Alexandru Nica (University of Waterloo)

Title: A central limit theorem for star-generators of the infinite symmetric group, which relates to traceless CCR-GUE matrices

Abstract: (PDF version) In a 1995 paper, Biane proved a limit theorem concerning the sequence of star-generators  (1,2), (1,3),...,(1,n),... of the infinite symmetric group $S_{\infty}$, where Wigner's semicircle law plays the role of limit law. I will present a result showing what this limit theorem becomes when we fix a finite dimension d and some weights w_1, ... , w_d of sum 1, and we use an expectation functional on $S_{\infty}$ that is naturally defined by this data.  In this case, the limit law turns out to be the law of a d-times-d 'traceless CCR-GUE' matrix, an analogue of the traceless GUE where the off-diagonal entries satisfy certain canonical commutation relations dictated by the weights w_i.  The special case when all the w_i's are equal to 1/d yields the law of a bona fide traceless GUE matrix, and we retrieve a result of Koestler-Nica from 2021, which in turn retrieves (for $d \to \infty$) the original result of Biane.

This is joint work with Jacob Campbell and Claus Koestler, International J. Math 2022 (also available as arXiv:2203.01763).

10:30-10:55 Natasha Blitvic (Queen Mary University of London)

Title: Free(ing) combinatorics

Abstract: Free probability is full of beautiful combinatorics, which have largely played a service role. In this talk, we will flip the script and consider ways in which noncommutative probability may instead help us shed insight into hard open questions in combinatorics. Based on joint works with Einar Steingrímsson (Strathclyde) and Slim Kammoun (Toulouse/Lyon).

11:00-11:25 Hao-Wei Huang (National Tsing Hua University)

Title: Additivity violations of random quantum channels of non-white Wishart types

Abstract: Most core problems in quantum information theory have elementary formulations but still resist solutions, one of which is the additivity conjecture of the minimum output entropy of quantum channels. All previously known results, including extensive numerical work, are consistent with the conjecture until it was shown to be false by Hasting and successive works by others. In this talk, we will briefly introduce the history and developments regarding this problem, present our random quantum channels composed of non-white Wishart ensembles and explore their additivity violations. The noted additivity violations occurring in our constructed random quantum channels are acquired by utilizing random matrix theory.