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: Jiange LI (jiange.li AT hit.edu.cn), Simeng WANG (simeng.wang AT hit.edu.cn)

**Upcoming talks**

TBA

**Past talks**

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

**March 22, 2023. 16:00 (Beijing time)**

Speaker: Marat Pliev (Southern Institute of Mathematics of Russian Academy of Sciences)

Title: Orthogonally additive operators in vector lattices and around

## Abstract, slides, video

In first part of the talk we discuss basic examples and properties of orthogonally additive operators in vector and Banach lattices emphasizing on connections with other areas of the modern mathematics. In the second part we present some open problems concerning order and topological properties of orthogonally additive operators in Banach and vector lattices.

**March 15, 2023. 16:00 (Beijing time)**

Speaker: Fedor Sukochev (University of New South Wales)

Title: Quantitative estimates for the functional calculus on the Schatten p-classes, 0<p<∞

## Abstract, slides, video

**March 8, 2023. 16:00 (Beijing time)**

Speaker: Hongyin Zhao (University of New South Wales)

Title: Perturbation theory of commuting self-adjoint operators and related topics

## Abstract, slides (part1, part2), video

This talk contains two parts. In the first part, we give an overview of perturbation theory of operators, including Weyl-von Neumann theorem and Kato-Rosenblum theorem, and introduce Voiculescu’s quasicentral modulus. In the second part, we introduce our recent results in the extension of Voiculescu’s results to the case of von Neumann algebras and factors.

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

Speaker: Alexandros Eskenazis (Institut de mathématiques de Jussieu, CNRS)

Title: Learning low-degree functions on the discrete hypercube

## Abstract, slides, video

Let f be an unknown function on the n-dimensional discrete hypercube. How many values of f do we need in order to approximately reconstruct the function? In this talk, we shall discuss the random query model for this fundamental problem from computational learning theory. We will explain a newly discovered connection with a family of polynomial inequalities going back to Littlewood (1930) which will in turn allow us to derive sharper estimates for the query complexity of this model, exponentially improving those which follow from the classical Low-Degree Algorithm of Linial, Mansour and Nisan (1989), while maintaining a running time of the same order. Time permitting, we will also show a matching information-theoretic lower bound. Based on joint works with Paata Ivanisvili (UC Irvine) and Lauritz Streck (Cambridge).

**November 30, 2022. ****16:00 (Beijing time)**

Speaker: Léonard Cadilhac (Sorbonne University)

Title: Marcinkiewicz interpolation for non-commutative maximal norms

## Abstract, slides (partly written by Eric Ricard), video

## In a landmark paper published in 2007, M. Junge and Q. Xu made significant contributions to noncommutative ergodic theory and in particular furthered our understanding of non-commutative maximal norms. One of their key results is a Marcinkiewicz-type interpolation theorem for those norms, for which they provide a long and technical proof. I will present recent work in collaboration with E. Ricard, where we revisit this theorem, providing a more compact proof and improving on the statement. The talk will first discuss non-commutative maximal norms and briefly introduce interpolation, then detail the main ideas and result, and finally present its more technical variants as well as some of the challenges ahead.

**November 23, 2022. ****16:00 (Beijing time)**

Speaker: Boris Kashin (Russian Academy of Sciences)

Title: On some applications and connections of Functional Analysis

## Abstract, slides, video

## In the last two decades, due to the rapid development of computer science, the results from functional analysis (including the geometry of convex bodies and approximation theory in normed spaces) are used rather extensively in practical applications. At the same time, the above mentioned branches of functional analysis and theoretical computer science are getting closer to each other, and methods from each of them turn out to be applicable in the other. In this talk, we will discuss a number of results that illustrate those trends.

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

Speaker: Alexander Frei (University of Copenhagen)

Title: Operator algebras and quantum information:

1) Connes implies Tsirelson and 2) Robust self-testing (hot topic)

## Abstract, slides, video

We give a simple proof of Connes implies Tsirelson, and further advertise a hot topic in quantum information: optimal states and robust self-testing. We showcase here how operator algebraic techniques can be quite fruitful.

For this we begin with by recalling quantum strategies in the context of non-local games, and their description in terms of the state space on the full group algebra of certain free groups. With this description at hand, we then directly obtain the main result via an elementary lifting result by Kim, Paulsen and Schafhauser:

*The Connes embedding problem implies the synchronous Tsirelson conjecture.*

As such the entire proof is elementary, and bypasses all versions of Kirchberg's QWEP conjecture and thelike, as well as any reformulation such as in terms of the micro state conjecture. Moreover, it should be (likely) easier to construct minimal nonlocal games as counterexamples for the synchronous Tsirelson conjecture (which is equivalent to the full Tsirelson conjecture but in a non-trivial way) and so also nonamenable traces for above groups, in other words non-Connes embeddable operator algebras.

After this we continue (as much as time permits) with an advertisement for a topic in quantum information (a hot one):

*Device-independent certification of quantum states – more precisely ROBUST SELF-TESTING – *

which has tremendous importance for the coming era of practical quantum computingand we showcase how operator algebraic techniques can be quite fruitful here.More precisely, we illustrate these techniques on the following two prominent classes of nonlocal games:

1) The tilted CHSH game.

We showcase here how to compute the quantum value using operator algebraic techniques, and how to use the same to derive uniqueness for entire optimal states, including all higher moments as opposed to correlations defined on two-moments only, where the latter compares to traditional self-testing.Moreover, we report in this example on previously unknown phase transitions on the uniqueness of optimal states when varying the parameters for the tilted CHSH game.

2) The Mermin–Peres magic square and magic pentagram game. As before, we also note here uniqueness of optimal states, which in these two examples is a basically familiar result.

The first part is based on preprint: https://arxiv.org/abs/2209.07940

The second part on self-testing (and further robust self-testing) is based on https://arxiv.org/abs/2210.03716 and upcoming joint work with Azin Shahiri.

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

Speaker: Piotr Nayar (University of Warsaw)

Title: The Khintchine inequality

## Abstract, slides, video

We shall discuss the celebrated moment comparison for weighted sums of i.i.d. symmetric Bernoulli random variables, proved in 1923 by Khinchine. This inequality influenced several areas of mathematics, including probability theory, Banach spaces theory and convex geometry. We will focus on the problem of finding optimal constants and present the most elegant proofs, both old and new.

**October 13, 2022. 20:30 (Beijing time) **

Speaker: Jingyin Huang (Ohio State University)

Title: Measure equivalence superrigidity for some generalized Higman groups

## Abstract, slides, video

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.

**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)