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: 16. May 2022
Speaker: Nikolaos Chalmoukis
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
Title: The Fejér-Riesz theorem and Schur complements
Date: 9. May 2022
Speaker: Sebastian Toth
Titel: Uniqueness of the multiplier functional calculus for pure K-contractions
Date: 2. May 2022
Speaker: Rachid Zarouf, Aix-Marseille Université
Title: A constructive approach to Schäffer's conjecture
Date: 25. April 2022
Speaker: Alberto Dayan, Norwegian University of Science and Technology
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
Title: New examples of quantum graphs
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
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
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.
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
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
Title: A graph with quantum symmetry and finite quantum automorphism group
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
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.
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
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.
Department of Mathematics
Postfach 15 11 50
Campus building E 2 4