## Seminar Talks

List of seminar talks (pdf) since 1999.

#### Strukturtheorie-Seminar

**Title:**Linear representations of non-commutative rational functions, free probability theory, and large random matrices

**Speaker:**Tobias Mai (Universität des Saarlandes)

**Date:**01.06.2017, 11:00c.t.

**Room:**Seminarraum AE06, Steyrergasse 30, Erdgeschoss

**Abstract:**

The concept of linear representations (aka realizations or linearizations) provides some very powerful tool to deal with non-commutative rational functions, namely elements in the universal field of fractions for the ring of non-commutative polynomials (in finitely many variables). While these methods are mostly of purely algebraic origin, they are also nicely compatible with the analytic machinery of (operator-valued) free probability. This theory and the underlying notion of free independence were invented around 1985 by D. Voiculescu, originally for operator-algebraic purposes. It can be seen as a highly non-commutative analogue of classical probability theory and has deep connections to many other fields of mathematics, especially to random matrix theory. In my talk, I will explain how this fascinating interplay leads to explicit algorithms for the computation of distributions and Brown measures, respectively, of evaluations of non-commutative rational functions in freely independent random variables. As we will see, this can be used to determine the asymptotic eigenvalue distribution of certain random matrix models. Furthermore, some concrete examples will show that these algorithms are easily accessible for numerical computations. This is based on joint works with S. Belinschi, J. W. Helton, and R. Speicher.

#### Strukturtheorie-Seminar

**Title:**Ricci curvature for Markov chains via dynamic optimal transport

**Speaker:**Dr. Jan Maas (Institute of Science and Technology Austria - ISTA)

**Date:**Donnerstag, 11.5.2017, 11 Uhr c.t.

**Room:**Seminarraum AE06, Steyrergasse 30, Erdgeschoss

**Abstract:**

In the past decade there has been a lot of progress in analysis on metric measure spaces based on ideas from optimal transport. We discuss how some of these ideas can be developed for Markov chains on discrete spaces, using a discrete analogue of the Kantorovich-Wasserstein metric. In particular we present a discrete notion of Ricci curvature based on geodesic convexity of the entropy, which allows us to obtain discrete functional inequalities, such as spectral gap and logarithmic Sobolev inequalities. We also discuss recent applications to interacting particle systems. This is based on joint works with Matthias Erbar (Bonn), Prasad Tetali (Georgia Tech), and Max Fathi (Toulouse).

#### Strukturtheorie-Seminar

**Title:**Multipartite rational functions: the universal skew field of fractions of a tensor product of free algebras

**Speaker:**Jurij Volčič (University of Auckland)

**Date:**Dienstag 28.02.2017, 10:15

**Room:**Seminarraum AE02 (STEG006), Steyrergasse 30, EG

**Abstract:**

A commutative ring embeds into a field if and only if it has no zero divisors; moreover, in this case it admits a unique field of fractions. On the other hand, the problem of localization of noncommutative rings and embeddings into skew fields (that is, division rings) is much more complex. For example, there exists noncommutative rings without zero divisors that do not admit embeddings into a skew field, and rings with several non-isomorphic ``skew fields of fractions''. This lead Paul Moritz Cohn to introduce the notion of the universal skew field of fractions to the general theory of skew fields in the 70's. However, almost all known examples of rings admitting universal skew fields of fractions belong to a relatively narrow family of Sylvester domains. One of the exceptions is the tensor product of free algebras. With the help of matrix evaluations we will construct the skew field of multipartite rational functions, which turns out to be the desired universal skew field of fractions. We will also explain its role in the difference-differential calculus in free analysis.

#### Strukturtheorie-Seminar

**Title:**Direct product of automorphism groups of digraphs

**Speaker:**Lukasz Wojakowski (Uniwersitet Wrocławski)

**Date:**Donnerstag 16.02.2017, 10:00 c.t.

**Room:**AE02 (STEG006), Steyrergasse 30, EG

**Abstract:**

The problem of representability of a permutation group $A$ as the full automorphism group of a digraph $G = (V, E)$ was first studied for regular permutation groups by many authors, the solution of the problem for undirected graphs was first completed by Godsil in 1979. For digraphs, L. Babai in 1980 proved that, except for the groups $S_2^2$, $S_2^3$ , $S_2^4$, $C_3^2$ and the eight element quaternion group $Q$, each regular permutation group is the automorphism group of a digraph. Later on, the direct product of automorphism groups of graphs was studied by Grech. It was shown that, except for an infinite family of groups $S_n \times S_n$, $n\ge $2, and three other groups $D_4 \times S_2$, $D_4\times D_4$, and $S_4 \times S_2 \times S_2$, the direct product of automorphism groups of two graphs is, itself, an automorphism group of a graph. We study the direct product of automorphism groups of digraphs. We show that, except for the infinite family of permutation groups $S_n \times S_n , n \ge 2$ and four other permutation groups $D_4 \times S_2$, $D_4 \times D_4$, $S_4 \times S_2 \times S_2$, and $C_3 \times C_3$, the direct product of automorphism groups of two digraphs is itself the automorphism group of a digraph.

#### Strukturtheorie-Seminar

**Title:**Heat kernel bounds for isotropy-like Laplacians

**Speaker:**Prof. Alexander Bendikov (Univ. Wroclaw)

**Date:**Thursday, 16. February 2017, 11:00 c.t.

**Room:**Seminar room AE06, Steyrergasse 30, ground floor

**Abstract:**

#### Strukturtheorie-Seminar

**Title:**Free infinite divisibility of $R$-diagonal elements.

**Speaker:**Kamil Szpojankowski (Politechnika Warszawska)

**Date:**Dienstag 14.02.2017, 14:15

**Room:**AE02 (STEG006), Steyrergasse 30, EG

**Abstract:**

#### Strukturtheorie-Seminar

**Title:**Heat content asymptotics for Levy processes

**Speaker:**Dr. Wojciech Cygan (TU Graz + Univ. Wroclaw)

**Date:**Donnerstag, 12.1.2017, 11 Uhr c.t.

**Room:**Seminarraum AE02, Steyrergasse 30, Erdgeschoss

**Abstract:**

I will recall and discuss a notion of heat content related to Levy processes in Euclidean space. To start with, I will present instructive examples including Brownian motion and stable processes, and next I will focus on the study of

the small time behaviour of the heat content for a rich class of Levy processes. The talk is based on the joint work with Dr. Tomasz Grzywny (Wroclaw University of Science and Technology).

#### Special Colloquium in Applied Stochastics

**Title:**A stochastic model of eye lens growth

**Speaker:**Prof. Hrvoje Šikić (Univ. Zagreb)

**Date:**Wednesday, 11 January 2017, 16:00 c.t.

**Room:**Seminar Room AE06, Steyerergasse 30, ground floor

**Abstract:**

The biological lens in the eye of a mammal focuses light on the retina. Its shape and size is crucial for that purpose. We base our work on abundance of data collected at Washington University in St Louis, mostly on mice. We provide the first ever growth model of the mouse eye and succeed in capturing a variety of behavior regarding the size of the lens, number of cells in the anterior capsule of the lens (epithelium) and the dynamics of the cell movement between the various zones of the epithelium. The lens grows through the entire life and exhibits significantly different behavior throughout life. Our model is based on branching processes with immigration and emigration.

(This is joint work with Steven Bassnett and members of his lab at Washington University. Research supported by NIH grant R01 EYO9852 and a Marie Curie FP7-PEOPLE-2013-IOF-622890 MoLeGro Fellowship.)