This seminar aims to present recent progress around functional analysis. The topics will cover the theory of functional analysis and related fields in a large sense, for example operator algebras, Banach space and operator space theory, classical and noncommutative harmonic analysis, quantum probability, quantum information, topological quantum groups and so on.

The seminar is held on Wednesdays in a hybrid form; online access will be provided for speakers and audiences who cannot come to Harbin in person.

Please follow **this link** to subscribe to our mailing list and receive the information.

Contact: simeng.wang AT hit.edu.cn.

**Upcoming talks**

**October 13 (Thursday), 2022. 20:30 (Beijing time) **!Please note the unusal date and time!

Speaker: Jingyin Huang (Ohio State University)

Title: Measure equivalence superrigidity for some generalized Higman groups

## Abstract

In the 1950s, Higman introduced the first class of examples of infinite finitely presented groups without any non-trivial finite quotient. We study Higman groups from the viewpoint of measure equivalence - a notion introduced by Gromov as a measurable counterpart to quasi-isometry. For most Higman groups and some generalizations, we prove a strong measure equivalence rigidity theorem. In this talk, I'll sketch the proof, discuss some of the consequences, and compare to some other measure equivalence rigidity/flexibility results in the literature. This is joint work with Camille Horbez.

**October 19, 2022. ****16:00 (Beijing time)**

Speaker: Piotr Nayar (University of Warsaw)

Title: TBA

**November 16, 2022. ****16:00 (Beijing time)**

Speaker: Alexander Frei (University of Copenhagen)

Title: Connes implies Tsirelson: from group operator algebras to quantum information theory

**December 7, 2022. ****16:00 (Beijing time)**

Speaker: Alexandros Eskenazis (Sorbonne Université and University of Cambridge)

Title: TBA

**Past talks**

(Playlist of recordings: https://space.bilibili.com/1171904096/channel/collectiondetail?sid=668900)

**September 28, 2022. 13:30 (Beijing time)**

Speaker: Benoît Collins (Kyoto University)

Title: Properties of random tensor matrices with some applications to Quantum Information Theory

## Abstract, video

We will describe a research project in collaboration with Luca Lionni and Razvan Gurau where we study the joint behavior of iid random elements in tensors of matrix algebras whose law is invariant under local conjugations. We will also mention some results on the norm of tensors of random matrices and explain some applications to QIT.

**September 21, 2022. 16:00 (Beijing time)**

Speaker: Christian Voigt (University of Glasgow)

Title: Infinite quantum permutations

## Abstract, slides, video

In this talk I will discuss an approach to study quantum symmetries of infinite sets and graphs. This leads to discrete quantum groups, in analogy to the quantum symmetry groups of Wang and Banica/Bichon. For finite sets and graphs, the resulting quantum groups can in fact be viewed as discretisations of the former (compact) quantum groups. I will discuss a number of examples, and highlight some intriguing open problems as well.

**August 31, 2022. 16:00 (Beijing time)**

Speaker: Vladimir Al. Osipov (Holon Institute of Technology)

Title: Integrability in random matrix theory and its applications

## Abstract, slides, video

Random matrices are widely used to model quantum systems with chaos and disorder. In such models, the observable is expressed as a quantum operator averaged over an ensemble of random matrices with a given probability measure. In my talk, I demonstrate a general approach, “the random matrix integrable theory”, to the nonperturbative calculation of the random-matrix integrals. With this approach, the internal symmetries of the integration measure, expressed in terms of highly non-trivial nonlinear relations for the original integral (the Toda lattice hierarchy, the Kadomtsev-Petviashvili hierarchy) and the relations following from the deformation of the integration measure (Virasoro constraints), are used to represent the integral as a solution of differential equations, where the differentials are taken over the internal (physical) parameters of the model [1,2]. This method represents a particular implementation of results obtained within a more general theory of $\tau$-functions. In particular, the central theorem of this theory states the existence of the Toda lattice and Kadomtsev-Petviashvili hierarchies for the typical random-matrix integrals.

The particular implementation of the integrable theory will be discussed in the example of the physical problem of quantum transport in chaotic cavities [3,4]. A brief introduction to the physics of the problem and the advantage of the integrable theory method for calculation of the conductance cumulants, and of the shot-noise-conductance joint cumulants are going to be presented. In particular, we demonstrate how the conductance cumulant generation function can be expressed in terms of the solution of the Painleve V transcendent equation. In addition, the results of the integrable theory implementation to the averaged random-matrix characteristic polynomials [1], and also for the problem of the power spectrum of the eigenlevel sequences in the quantum chaotic system [2,5] will be discussed.

[1] V.Al.Osipov, E.Kanzieper, “Correlations of RMT characteristic polynomials and integrability: Random Hermitian matrices”, *Annals of Physics***325 **(2010) 2251

[2] R.Riser, V.Al.Osipov, E.Kanzieper, “Nonperturbative theory of power spectrum in complex systems”, *Annals of Physics***413 **(2020) 168065

[3] V.Al.Osipov, E.Kanzieper, “Integrable theory of quantum transport in chaotic cavities”, *Phys.Rev.Let.***101 **(2008) 176804

[4] V.Al.Osipov, E.Kanzieper, “Statistics of thermal to shot noise crossover in chaotic cavities”, *J.Phys.A:Math.Theor.***42 **(2009) 475101

[5] R.Riser, V.Al.Osipov, E.Kanzieper, “Power-spectrum of long eigenlevel sequences in quantum chaology”, *Phys.Rev.Let.***118 **(2017) 204101

**July 6, 2022. 16:00 (Beijing time)**

Speaker: Vladimir Manuilov (Moscow State University)

Title: On Hochshild cohomology of uniform Roe algebras with coefficients in uniform Roe bimodules

## Abstract, slides, video

Uniform Roe algebras play an important role in noncommutative geometry. It was shown recently by M. Lorentz and R. Willett that all bounded derivations of the uniform Roe algebras of metric spaces of bounded geometry are inner. Here we calculate the space of outer derivations of the uniform Roe algebras with coefficients in uniform Roe bimodules related to various metrics on the two copies of the given space. We also give some results on the higher Hochschild cohomology with coefficients in uniform Roe bimodules.

**June 22, 2022. 19:30 (Beijing time)**

Speaker: Ping Zhong (University of Wyoming)

Title: The Brown measure of the sum of a free random variable and an elliptic deformation of Voiculescu's circular element

## Abstract, slides, video

The circular element is the most important example of non-normal random variable used in free probability, and its Brown measure is the uniform measure in the unit disk. The circular element has connection to asymptotics of non-normal random matrices with i.i.d. entries. We obtain a formula for the Brown measure of the addition of an arbitrary free random variable and circular element *c*, which is known to be the limit empirical spectral distribution of deformed i.i.d. random matrices.

Generalizing the case of circular and semi-circular elements, we also consider , a family of elliptic deformations of , that is -free from . Possible degeneracy then prevents a direct calculation of the Brown measure of . We instead show that the whole family of Brown measures of operators are the push-forward measures of the Brown measure of under a family of self-maps of the complex plane, which could possibly be singular. The main results offer potential applications to various deformed random matrix models. This work generalizes earlier results of Bordenave-Caputo-Chafai, Hall-Ho, and a joint work with Ho.

**June 15, 2022. 19:30 (Beijing time)**

Speaker: Jorge Castillejos (National Autonomous University of Mexico)

Title: The Toms-Winter regularity conjecture

## Abstract, slides, video

The classification programme of simple nuclear C*-algebras asserts one can classify such C*-algebras in terms of an inviariant constructed out of K-theory and tracial information. However, in order to be able to use these classifications results, one must verify a topological type regularity property first. The Toms-Winter conjecture predicts that this regularity condition is equivalent to other type of regularity properties which might be easier to verify. In this talk, I will present an overview of this conjecture.

**June 8, 2022. 19:30 (Beijing time)**

Speaker: Roland Speicher (Saarland University)

Title: A dual and a conjugate system for q-Gaussians for all q

## Abstract, slides, video

*q*-Gaussian random variables, for some fixed real with , are of the form , where the are operators satisfying the *q*-relations . Understanding the properties of the non-commutative distributions of those deformations of classical multivariate Gaussian distributions as well as their associated operator algebras -- in particular, whether and how they depend on *q* -- has been of central interest in the last 30 years. I will give an introduction and survey on those *q*-relations and in particular report also some recent progress (from joint work with A. Miyagawa) on the existence of dual systems and conjugate systems for the *q*-Gaussians. Special focus is on the fact that those results are for the whole interval (-1,+1), and not just for some restricted set of *q*.

**June 1, 2022. 16:00 (Beijing time). Mingde Building, B201-1**

Speaker: Fedor Sukochev (University of New South Wales)

Title: Quantum differentiation and integration for the quantum plane

## Abstract, slides, video

We explain recent results concerning (quantum) differentials and integrals on the noncommutative (Moyal) plane. We give full characterisation of elements on the noncommutative (Moyal) plane such that their quantum derivatives belong to the weak Schatten class , which means that these derivatives are d-times integrable in the quantum integration sense. We then calculate the quantum integration of these derivatives by adapting Connes' integration formulae to the noncommutative (Moyal) plane. This is a joint work with E. McDonald and X. Xiong.

**May 25, 2022. 16:00 (Beijing time). Mingde Building, B201-1**

Speaker: Ke Li (Harbin Institute of Technology)

Title: Reliability Function of Quantum Information Decoupling

## Abstract

Quantum information decoupling is a fundamental information processing task, which also serves as a crucial tool in a diversity of topics in quantum physics. I will talk about our recent results on its reliability function, that is, the best exponential rate under which perfect decoupling is asymptotically approached. We have obtained the exact formula when the decoupling cost is below a critical value. In the situation of high cost, we provide upper and lower bounds. These results are given in terms of the sandwiched R\'enyi divergence, providing it with a new type of operational meaning. (Based on joint work with Yongsheng Yao, arXiv:2111.06343)

**May 18, 2022. 16:00 (Beijing time)**

Speaker: Ignacio Vergara (Leonhard Euler International Mathematical Institute in Saint Petersburg)

Title: Around Cowling's conjecture

## Abstract, slides, video

Two very important concepts arose from Haagerup's highly influential work on the reduced C*-algebra of the free group: weak amenability and the Haagerup property. It is an open problem to determine whether weak amenability with Cowling-Haagerup constant 1 implies the Haagerup property. This is often referred to as Cowling's conjecture. In this talk, I will give an overview of this question, and I will discuss a recent progress in this direction: Every countable weakly amenable group with Cowling-Haagerup constant 1 admits a proper cocycle for a uniformly bounded representation on a subspace of an L_{1} space.

**May 11, 2022. 16:00 (Beijing time)**

Speaker: Mateusz Wasilewski (IMPAN Warsaw)

Title: On the isomorphism class of q-Gaussian C*-algebras for infinite variables

## Abstract, slides, video

Bożejko and Speicher introduced q-Gaussian variables to produce examples of generalized Brownian motions. The resulting von Neumann algebras -- the q-Gaussian algebras -- can be viewed as deformations of the free group factors. It is a very natural question whether these von Neumann algebras are actually isomorphic to the free group factors. Guionnet and Shlyakhtenko introduced the free monotone transport techniques and provided a partial answer: if the number of variables is finite and the parameter q is very small then we do get an isomorphism. There are no results known for infinitely many variables and in my talk I plan to describe a related result about q-Gaussian C*-algebras -- in the infinite case they are not isomorphic to their free counterparts. The von Neumann algebraic case remains open.

Joint work with Matthijs Borst, Martijn Caspers and Mario Klisse.

**May 4, 2022. 16:00 (Beijing time)**

Speaker: Joseph Lehec (Université Paris Dauphine)

Title: The Kannan-Lovász-Simonovits conjecture up to polylog

## Abstract, slides, video

The Kannan-Lovász-Simonovits conjecture asserts that high dimensional log-concave probability measures satisfy a certain universal concentration property. In a recent joint work with Bo’az Klartag we prove that this conjecture holds true up to a factor that is polylogarithmic in the dimension. In this talk I’ll mostly speak about the context around this conjecture, in particular I’ll present some of its numerous consequences. If time abides I’ll say a few words about our proof towards the end of the talk.

**April 20, 2022. 16:00 (Beijing time) **

Speaker: Cyril Houdayer (University of Paris-Saclay)

Title: Noncommutative ergodic theory of lattices in higher rank simple algebraic groups

## Abstract, slides

In this talk, I will present a noncommutative Nevo-Zimmer theorem for actions of lattices in higher rank simple algebraic groups on von Neumann algebras. This extends to the realm of algebraic groups defined over arbitrary local fields the noncommutative Nevo-Zimmer theorem we obtained with Rémi Boutonnet in 2019 for real Lie groups.

I will discuss various applications of the above theorem to topological dynamics, unitary representations and operator algebras. I will also present a noncommutative analogue of Margulis’ factor theorem and discuss its relevance regarding Connes’ rigidity conjecture for group von Neumann algebras of higher rank lattices.

This is based on joint work with Uri Bader and Rémi Boutonnet (arXiv:2112.01337)