# Research Seminar Noncommutative and Functional Analysis

The research seminar Noncommutative and Functional Analysis is run by Roland Speicher, Michael Hartz and Moritz Weber. It consists in talks on latest research and graduation projects in Functional Analysis, Complex Analysis, Free Probability, Quantum Groups and Quantum Information.

The seminar takes place on Mondays, from 16:15, in lecture hall IV, building E2 4, or online. Usually, the talks are 60 minutes for research talks and 45 minutes for expository talks of graduation theses („Bachelorseminarvortrag/Masterseminarvortrag“) excluding time for discussions.

## List of all Talks

Date: 06.02.2023

Speaker: Alan Sola, Stockholm University

Time: 16:15

Title: Local theory of stable polynomials and bounded rational functions

Abstract: We will discuss the boundary behavior of bounded rational functions in several variables from several perspectives, including existence of non-tangential limits and higher non-tangential regularity. The results we obtain in two variables rely on local descriptions of stable polynomials, and motivate a conjecture (since resolved by J. Kollar) regarding the characterization of bounded rational functions in the bidisk with a given stable denominator.

This reports on joint work with K. Bickel, G. Knese, and J.E. Pascoe.Date: Tuesday, 31. Januar 2023

Time: 17:15-18:15

Place: HS IV

Speaker: Jacob Campbell (Waterloo, Canada)

Title: Commutators in finite free probability

Abstract: In free probability, a fundamental result of Voiculescu is that random unitary matrices are asymptotically free. A representative special case is the fact that sums A + U B U* and products A U B U* of large randomly rotated matrices approximate free additive and multiplicative convolution. In 2015, Marcus, Spielman, and Srivastava realized that in the non-asymptotic setting, one can recover ``finite" analogues of these free convolutions by looking at the expected characteristic polynomials of A + U B U* or A U B U*. After reviewing these ideas, I will show how techniques from combinatorial representation theory can help to understand finite free convolutions, focusing on the problem (which I recently solved in arXiv:2209.00523) of describing the commutator of randomly rotated matrices in this context. The main techniques are Weingarten calculus and the Goulden-Jackson immanant formula. Time permitting, I will discuss some combinatorial questions which are raised by comparison with the commutator in free probability.Date: Thursday, 26 Januar 2023

Time: 16:15-17:15

Place: SR 9

Speaker: Jinzhao Wang (Stanford)

Title: Free probability in quantum gravity

Abstract: I will survey two models in quantum gravity that showcase the effectiveness of free probability. They are the Penington-Shenker-Stanford-Yang (PSSY) model and the Double Scaled SYK model. Unlike other physical applications that mostly relate to free probability via random matrices. Here the links are drawn combinatorially and algebraically. This connection allows us to formulate and address questions that are otherwise difficult. In particular, I will emphasize on what do we gain by using the free probabilistic toolkit.Date: 16.01.2023

Speaker: Freek Witteveen

Time: 16:15

Title: Random tensor network states and free probability theory

Abstract: Random quantum states are an important tool in quantum information theory. In this talk I will discuss random tensor network states, which have found application both in quantum information theory and as a toy model for holographic quantum gravity. We introduce a refined model with arbitrary link states. I will explain how the entanglement properties of such tensor network states depend on the graph structure and on the link states, focusing on a connection to free probability theory. This talk is based on arXiv:2206.10482 which is joint work with Newton Cheng, Cecilia Lancien, Geoff Penington and Michael Walter.Date: 07.11.2022

Speaker: Maximilian Tornes, Universität des Saarlandes

Time: 17:15

Title: Weighted composition operators on unitarily invariant spaces on B_d

AbstractDate: 11.07.2022

Speaker: Marwa Banna (NYU Abu Dhabi)

Time: 16:15

Title: Berry-Esseen Bounds for Operator-valued Free Limit Theorems

Abstract:

The development of free probability theory has drawn much inspiration from its deep and far reaching analogy with classical probability theory. The same holds for its operator-valued extension, where the fundamental notion of free independence is generalized to free independence with amalgamation as a kind of conditional version of the former. Its development naturally led to operator-valued free analogues of key and fundamental limiting theorems such as the operator-valued free Central Limit Theorem due to Voiculescu and the asymptotic distributions of matrices with operator-valued entries.

In this talk, we show Berry-Esseen bounds for such limit theorems. The estimates are on the level of operator-valued Cauchy transforms and the L\'evy distance. We also address the multivariate setting for which we consider linear matrix pencils and noncommutative polynomials as test functions. The estimates are in terms of operator-valued moments and yield the first quantitative bounds on the Lévy distance for the operator-valued free CLT. This also yields quantitative estimates on joint noncommutative distributions of operator-valued matrices having a general covariance profile.

This is a joint work with Tobias Mai.Date: 16. May 2022

Speaker: Nikolaos Chalmoukis

Time: 16:15

Title: Semigroups of composition operators on spaces of analytic functions.

Abstract: We will discuss the maximal subspace of strong continuity of a semigroup of composition operators acting on the space of analytic functions of bounded mean oscillation in the unit disc. The minimality of this space is related to a well known theorem of Sarason about the space of analytic functions of vanishing mean oscillation. In the case of elliptic semigroups we give a complete characterization in terms of the Koenigs function of the semigroups that can replace rotations in Sarason's Theorem. This answers to the affirmative a conjecture of Blasco et al. Similar results are also obtained for the Bloch space.

This is a joint work with V. Daskalogiannis.Speaker: Lisa Karst

Time: 17:15

Title: The Fejér-Riesz theorem and Schur complements

Abstract

Date: 9. May 2022

Speaker: Sebastian Toth

Time: 16:15

Titel: Uniqueness of the multiplier functional calculus for pure K-contractions

AbstractDate: 2. May 2022

Speaker: Rachid Zarouf, Aix-Marseille Université

Time: 16:15

Title: A constructive approach to Schäffer's conjecture

AbstractDate: 25. April 2022

Speaker: Alberto Dayan, Norwegian University of Science and Technology

Time: 16:15

Title: Dobinski Sets, Function Theory and Sets of Null Capacity

Abstract: The first half of the talk will focus on the construction of some Dobinski sets, which can be thought of as exceptional subsets of the unit interval made of points that are very well approximated via dyadic rationals. We will determine their logarithmic Hausdorff dimension and their logarithmic capacity. The second half of the talk will try to give a brief overview of how such exceptional sets can be used in function theory. In particular, we will see how sets of capacity zero are related to some open problems for the Dirichlet space on the unit disc, and time permitting we discuss some ongoing research in that direction.Date: 31. January 2022

Speaker: Daniel Gromada

Time: 16:15

Title: New examples of quantum graphs

Place: Zoom

Abstract: Quantum graphs are the analogues of classical graphs in the world of non-commutative geometry. Their definition is very new and very little is known about them so far. Not only that: the current literature is also lacking some concrete non-trivial examples of quantum graphs to begin with. In this talk, we are going to summarize three different approaches for the definition of a quantum graph. Then we will present some ways how to construct concrete examples. We show how quantum graphs over a fixed quantum space can be classified, we show an example of a quantum graph which is not quantum isomorphic to any classical graph, and we show a certain twisting procedure for classical Cayley graphs of abelian groups. This talk is based on a recent preprint arXiv:2109.13618.Date: 24. January 2022

Speaker: Nicolas Faroß

Time: 16:15

Title: Spatial Pair Partitions and Applications to Finite Quantum Spaces

Place: MS Teams

Abstract: In 2016, Cébron-Weber introduced spatial partition quantum groups as a generalization of easy quantum groups. These are compact matrix quantum groups whose intertwiners are indexed by categories of three-dimensional partitions.

We study the quantum group associated to the category spatial pair partitions on two levels and show that it is isomorphic to the projective orthogonal groups. Further, we generalize combinatorial methods for partitions to the setting of spatial partition. This allows us to find an explicit description of a category of spatial partitions linked to quantum symmetries of finite quantum spaces.Date: 17. January 2022

Speakers:Alexander Wendel

Time: 16:15

Title: Quantum Channels and Entangled States associated to Easy Quantum Groups

Place: MS Teams

Abstract: In 2017 Brannan and Collins constructed highly-entangled spaces and states from the representation theory of the orthogonal quantum groups. A key role in their construction is played by the famous Jones-Wenzl projections. These projections are usually defined via a certain commutation property and are projections onto irreducible representations of the orthogonal quantum group.

Shortly before (2014/2016) Freslon and Weber gave a combinatorial characterisation of the irreducible representations and fusion rules of easy quantum groups.

We are going to combine these two papers to investigate howfar one can carry over the results of Brannan and Collins to other easy quantum groups. Especially we are going to show that the projections onto irreducibles satisfy a similar characterisation as the Jones-Wenzl projections and are going to characterise by this the image of these projections. We are then going to transfer some results on entangled spaces to the symmetric quantum group and indicate how to carry out these constructions for other easy quantum groups.Arne Berrens

Time: 17:00

Title: Von Neumann algebras and zero sets of Bergman spaces

Place: MS Teams

Abstract: The leading question in this talk will be under which condition there exists a Bergman space function vanishing on a given set. The Bergman space is the space of holomorphic functions on the unit disc that are square integrable. Using von Neumann algebras, we get new insights into the structure of the weighted Bergman spaces. Vaughan Jones used Fuchsian groups that act on the Bergman space as well as on the upper half plane. By studying the group von Neumann algebra they generate, he got a necessary and sufficient condition for the existence of a vanishing function on the orbit. To get to this result, one has to use mainly the theory of von Neumann dimension as well as the theory of reproducing kernel Hilbert spaces.

This approach is not constructive, and we only get the existence of this function and no further information on what this function might look like. We will also look further into the connection of von Neumann algebras and Bergman spaces.

Date: 10. January 2022

Speaker: Steven Klein

Time: 16:15

Title: Shlyakhtenko's non-microstates approach to strongly 1-boundedness

Place: MS Teams

Abstract: Building on Voiculescu's microstates approach to the analogue of entropy in free probability theory, Jung introduced in 2007 some property of von Neumann algebras called "strongly 1-boundedness" which has found many interesting applications. Recently, in 2020, Shlyakhtenko developed an approach to strongly 1-boundedness using non-microstates techniques. In particular, he proved estimates for the non-microstates free entropy of operators satisfying algebraic relations and used this to give an alternative proof of a criterion for strongly 1-boundedness obtained originally by Jung.

In my talk, I will give a brief introduction to Voiculescu's concepts of free entropy and will discuss their relations to strongly 1-boundedness with focus on Shlyakhtenko's non-microstates approach.Date: 15. December 2021

Speaker: Simon Schmidt, QMATH, Copenhagen

Time: 11:00

Title: A graph with quantum symmetry and finite quantum automorphism group

Place: Zoom

Abstract: This talk concerns quantum automorphism groups of graphs, a generalization of automorphism groups of graphs in the framework of compact matrix quantum groups. We will focus on certain colored graphs constructed from linear constraint systems. In particular, we will give an explicit connection of the solution group of the linear constraint system and the quantum automorphism group of the corresponding colored graph. Using this connection and a decoloring procedure, we will present an example of a graph with quantum symmetry and finite quantum automorphism group. This talk is based on joint work with David Roberson.Date: 13. December 2021

Speakers:Matias Klimpel

Time: 16:15

Title: Representations of Graph C*-algebras

Place: MS Teams

Abstract: Graph C*-algebras were introduced in 1998 as a generalization of Cuntz-Krieger algebras introduced by Cuntz and Krieger in 1980, which in turn arose as a more generalized version of the Cuntz Algebra O_n introduced by Cuntz in 1977. As we will see, the class of graph C*-algebras is quite large and as such a useful one to understand.

In this talk we will present these objects and their relations as well as visualize them on a host of examples. In particular, we classify all graph C*-algebras associated to finite graphs without cycles, give an algorithm to construct a non-trivial representation for a graph C*-algebra associated to a row-finite graph and present two import uniqueness theorems for graph C*-algebras, namely the gauge-invariant uniquess theorem and the Cuntz-Krieger uniqueness theorem.Dean Zenner

Time: 17:00

Title: Hypergraph C*-algebras

Place: MS Teams

Abstract: In this talk I will introduce the concept of hypergraph C*-algebras. The concept is based on a new definition that was conveyed to me by Simon Schmidt and Moritz Weber.

Our main goal is to show that hypergraph C*-algebras define a generalization of graph C*-algebras. In contrary to graph C*-algebras, we will see that hypergraph C*-algebras do not need to be nuclear. They actually form a stricly larger class than the class of graph C*-algebras. Besides that, we will have a look at some interesting examples that I investigated.

At the end of the talk I will speak about a way to 'hyperize' graph C*-algebras.

Date: 29 November 2021

Time: 16:15

Speaker: Marc Hermes

Title: Peano Arithmetic in Constructive Type Theory and Tennenbaum's Theorem

Place: MS Teams

Abstract: Gödel's first incompleteness theorem entails that the first-order theory of Peano arithmetic (PA) and its consistent extensions admit a wealth of independent statements. By the completeness theorem then, PA cannot be categorical, meaning it does not posses a unique model up to isomorphism.

A theorem by Stanley Tennenbaum however tells us that if we restrict our attention to computable models, first-order PA is categorical with regards to this class of models.

The goal of this talk is to present the first-order theory of PA inside of a constructive type theory and to revisit and study Tennenbaum's theorem in this constructive setting.

We will start out the talk with an introduction to the maybe unfamiliar world of constructive mathematics and how this setting influences the possibilities we have in the investigation of mathematical questions in general. We then come back to the particular case of Tennenbaum’s theorem, where the setting allows for a synthetic viewpoint of computability, by consistently assuming that every function on the natural numbers is computable (i.e. Church's thesis), making it possible to abstract from many details in computability arguments. We will then finish with a few words on the computer-verified proof of the theorem.

## Postal address

Saarland University

Department of Mathematics

Postfach 15 11 50

66041 Saarbrücken

Germany

## Physical address

Saarland University

Campus building E 2 4

66123 Saarbrücken

Germany