### Talks in 2018

#### Seminar Applied Analysis and Computational Mathematics

**Title:**Spectral gaps and discrete magnetic Laplacians

**Speaker:**Prof. Dr. Olaf Post (Universität Trier)

**Date:**21.3.2018, 16:15

**Room:**A306

**Abstract:**

The aim of this talk is to give a simple geometric condition

that guarantees the existence of spectral gaps of the discrete Laplacian

on periodic graphs. For proving this, we analyse the discrete magnetic

Laplacian on the finite quotient and interpret the vector potential as a

Floquet parameter. We develop a procedure of virtualising edges and

vertices that produces matrices whose eigenvalues (written in ascending

order and counting multiplicities) specify the bracketing intervals

where the spectrum of the Laplacian is localised.

#### FWF START Seminar

**Title:**From Interval Exchange Transformations to low-discrepancy

**Speaker:**Christian Weiß (Hochschule Ruhr West)

**Date:**20.3.2018, 10.30

**Room:**Seminarraum Analysis-Zahlentheorie, Kopernikusgasse 24, 2.OG

**Abstract:**

Kronecker sequences build an important class of one dimensional low-discrepancy sequences. They can also be realized as orbits of circle rotations, which are in in turn the simplest class of interval exchange transformations (IETs). Hence, it is natural to ask whether there exist more general IETs which also yield low-discrepancy sequences. In the case of n=3 intervals, the criteria of circle rotation can easily be carried over. For more than three intervals the dynamic of IETs is much more complex. In this talk, it is shown how to obtain low-discprancy sequences involving certain IETs with an arbitrary number of intervals.

#### KOLLOQUIUMSVORTRAG im Vorfeld eines Habilitationsantrags

**Title:**Spectral and asymptotic properties of periodic media

**Speaker:**Andrii KHRABUSTOVSKYI (Institut für Angewandte Mathematik)

**Date:**Donnerstag, 15.3.2018, 14:00 Uhr

**Room:**TU Graz, Seminarraum AE02, Steyrergasse 30, EG

**Abstract:**

It is well-known that the spectrum of self-adjoint periodic differential operators has the form of a locally finite union of compact intervals called bands. In general the bands may overlap. A bounded open interval (a,b) is called a gap in the spectrum σ(H) of the operator H if (a,b)∩σ(H)=Ø with a,b belonging to σ(H).

The presence of gaps in the spectrum is not guaranteed: for example, the spectrum of the Laplacian in ℝn has no gaps. Therefore the natural problem is a construction of periodic operators with non-void spectral gaps. The importance of this problem is caused by various applications, for example in physics of photonic crystals.

Another interesting question arising in this area is how to control the location of spectral gaps via a suitable choice of coefficients of the underlying operators or/and via a suitable choice of geometric parameters of the medium. In the talk we give an overview of the results, where this problem is studied for various classes of periodic differential operators. In a nutshell, our goal is to construct an operator (from some given class of periodic operators) such that its spectral gaps are close (in some natural sense) to predefined intervals.

#### Vortrag im Seminar für Kombinatorik und Optimierung

**Title:**Lambda-terms enumeration and statistics

**Speaker:**Dani\`ele Gardy (Universit\'e de Versailles Saint-Quentin)

**Date:**Dienstag 6.3.2018, 14:15

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

**Abstract:**

Lambda-calculus is a well-known model of computation, whose statistical properties have been recently under investigation. From a syntactic point of view, the terms of lambda-calculus (lambda-terms) are interesting combinatorial structures: the underlying structure is a Motzkin tree to which are added edges from some unary nodes towards leaves, or equivalently whose nodes are colored, according to some specific set of rules. While trees are usually easy to enumerate, this enriched version is actually a class of directed graphs, and their combinatorial study gives rise to many interesting problems.

While the enumeration and statistical analysis of unrestricted lambda-terms still remains an open problem, different models, either redefining the notion of size of the lambda-term or restricting the set of lambda-terms under investigation (such as terms of restricted height or restricted node arity), have proved more amenable to study. We present enumeration and statistical behaviour for these models, using tools from analytic combinatorics.

(No previous knowledge either of lambda-terms or of analytic combinatorics is assumed.)

#### 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.

#### Zahlentheoretisches Kolloquium

**Title:**$f$-vectors of simplicial and simple polytopes

**Speaker:**Dr. Roswitha Rissner (TU Graz)

**Date:**Freitag, 19. 1. 2018, 14:15 Uhr

**Room:**Seminarraum Analysis-Zahlentheorie (NT02008), Kopernikusgasse 24/II

**Abstract:**

The $f$-vector of a $d$-dimensional (convex) polytope $P$ is defined as

\begin{equation*}

f(P) = (f_0(P), f_1(P), \ldots, f_d(P))

\end{equation*}

where $f_i(P)$ is the number of $i$-dimensional faces of $P$.

The question is whether a given vector of non-negative numbers is the $f$-vector of a polytope.

For $d\ge 4$, finding a complete characterization of $f$-vectors is an open problem.

However, the so-called $g$-Theorem gives a description for $f$-vectors of simplicial and simple polytopes.

The purpose of this talk is to give a summary of the lectures of a summer school at MSRI on the $g$-Theorem and related topics.

#### 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