## Seminar Talks

List of seminar talks (pdf) since 1999.

#### Advanced Topics in Discrete Mathematics

**Title:**Information Dimension of Random Variables and Stochastic Processes

**Speaker:**Dr. Bernhard Geiger (Signal Processing and Speech Communication Laboratory, TU Graz)

**Date:**Freitag 26.1.2018, 11:00 (Kaffee ab 10:30)

**Room:**Seminrraum 2 für Geometrie, Kopernikusgasse 24/4

**Abstract:**

In view of the interplay of the DK `Discrete Mathematics' with the Field of Expertise `Information, Communication & Computing' of TU Graz, the last talk of this semester's seminar comes from a sister institue within the FoE. There are clear relations with the topics of the DK.

Abstract. Information dimension of random variables was introduced by Alfred Renyi in 1959. Only recently, information dimension was shown to be relevant in various areas in information theory. For example, in 2010, Wu and Verdu showed that information dimension is a fundamental limit for lossless analog compression. Recently, Geiger and Koch generalized information dimension from random variables to stochastic processes. They showed connections to the rate-distortion dimension and to the bandwidth of the process. Specifically, if the process is Gaussian, then the information dimension equals the Lebesgue measure of the support of the process' power spectral density. This suggests that information dimension plays a fundamental role in sampling theory.

The first part of the talk reviews the definition and basic properties of information dimension for random variables. The second part treats the information dimension of stochastic processes and sketches the proof that information dimension is linked to the process' bandwidth.

#### Strukturtheorie-Seminar

**Title:**Schreier graphs of spinal groups

**Speaker:**Aitor Perez (Univ. Genf)

**Date:**Donnerstag, 25.1.2018, 11 Uhr c.t.

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

**Abstract:**

Spinal groups form a family of branch groups acting on d-regular rooted trees containing many interesting examples, including Grigorchuk's family of groups of intermediate growth. Because of its natural action on the tree, it is interesting to know how their Schreier graphs look like. In this talk we will define the spinal family, provide some examples, describe the Schreier graphs in general and find some of the properties they exhibit, like how many ends do they have, how do they partition into isomorphism classes and under which conditions is the action linearly repetitive or Boshernitzan, properties related to symbolic dynamics.

#### Strukturtheorie-Seminar

**Title:**Spectral estimates for infinite quantum graphs

**Speaker:**Dr. Norma Nicolussi (Univ. Wien)

**Date:**Donnerstag, 18.1.2018, 11:15

**Room:**Seminar room AE02, Steyrergasse 30, ground floor

**Abstract:**

We investigate the bottom of the spectra of infinite quantum graphs, i.e., Kirchhoff Laplacians on metric graphs having infinitely many edges and vertices. We introduce a new definition of the isoperimetric constant for quantum graphs and then prove the Cheeger-type estimate. Moreover, we establish a connection with the combinatorial isoperimetric constant, which enables us to prove a number of criteria for a quantum graph to be uniformly positive or to have purely discrete spectrum. If time permits, we'll demonstrate our findings by considering trees, antitrees and Cayley graphs.

#### Strukturtheorie-Seminar

**Title:**Infinite quantum graphs

**Speaker:**Prof. Aleksey Kostenko (Univ. Wien + Laibach)

**Date:**Donnerstag, 18.1.2018, 10:30

**Room:**Seminar room AE02, Steyrergasse 30, ground floor

**Abstract:**

The notion of quantum graph refers to a graph considered as a one-dimensional simplicial complex and equipped with a differential operator (``Hamiltonian'').

From the mathematical point of view, quantum graphs are interesting because they are a good model to study properties of quantum systems depending on geometry and topology of the configuration space. They exhibit a mixed dimensionality being locally one-dimensional but globally multi-dimensional of many different types.

We will review the basic spectral properties of infinite quantum graphs (graphs having infinitely many vertices and edges). In particular, we will discuss recently discovered, fruitful connections between quantum graphs and discrete Laplacians on graphs.

Based on a joint work with P. Exner, M. Malamud and H. Neidhardt