### Talks in 2019

#### Strukturtheorie-Seminar

**Title:**Central Limit Theorem for the capacity of the range of stable random walks

**Speaker:**Dr. Stjepan Šebek (Univ. Zagreb)

**Date:**Thursday, 25. April 2019, 15:15

**Room:**Seminar room A306, Steyrerg. 30/3

**Abstract:**

In this talk, we will establish a central limit theorem for the capacity of

the range process for a class of d-dimensional symmetric alpha-stable random

walks with the index satisfying $d \geq 3\alpha$. Our approach is based on

controlling the limit behavior of the variance of the capacity of the range

process which then allows us to apply the Lindeberg-Feller theorem.

#### Seminar talk

**Title:**Connectivity of the Flip-Graph of Triangulations

**Speaker:**Emo Welzl (ETH Zurich and IST Austria)

**Date:**10.4.2019, 10:15

**Room:**Seminarroom IST, Inffeldgasse 16b, 2nd floor, room IC02062

**Abstract:**

We investigate the connectivity of the flip-graph of all (full) triangulations of a given

finite point set $P$ in general position in the plane and prove that, for $n:=|P|$ large enough,

both edge- and vertex-connectivity are determined by the minimum degree occurring in the

flip-graph, i.e.\ the minimum number of flippable edges in any triangulation of $P$. It is known

that every triangulation allows at least $(n-4)/2$ edge-flips.

This result is extended to so-called subtriangulations, i.e. the set of all triangulations of

subsets of P which contain all extreme points of $P$, where the flip operation is extended to

bistellar flips (edge-flips, and insertion and removal of an inner vertex of degree three).

Here we prove $(n-3)$-edge-connectedness (for all $P$) and $(n-3)$-vertex-connectedness of $n$ large

enough ($(n-3)$ is tight, since there is always a subtriangulation which allows exactly n-3

bistellar flips). This matches the situation known (through the secondary polytope) for

so-called regular triangulations (i.e. subtriangulations obtained by liftings).

(joint work with Uli Wagner, IST Austria)

#### Zahlentheoretisches Kolloquium

**Title:**On the GCD of n and the n-th Fibonacci number

**Speaker:**Paolo Leonetti (TU Graz)

**Date:**Freitag, 29.3.2019, 14:00

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

**Abstract:**

Let $A$ be the set of all integers of the form $\gcd(n, F_n)$, where $n$ is a positive integer and $F_n$ denotes the $n$-th Fibonacci number. We show that

$\# A(x) \gg x / \log x$ and $\# A(x) =o(x)$ as $x\to \infty$. As a consequence, we obtain that the set of all integers $n$ such that $n$ divides $F_n$ has zero asymptotic density relative to $A$.

Remark: Paolo Leonetti is a new PostDoc researcher at the Institute of Analysis and Number Theory since February 2019.

#### Strukturtheorie-Seminar

**Title:**Jacobi Polynomials and the Discrete Laguerre Operator

**Speaker:**Aleksey Kostenko (Univerza v Ljubljani / Universität Wien)

**Date:**28.3.2019, 14:00 c.t.

**Room:**SR AE02, Steyrergasse 30, ground floor

**Abstract:**

The talk is focused on Bernstein-type estimates for Jacobi polynomials and their applications to various branches in mathematics. This is an old topic but we want to add a new wrinkle by establishing some intriguing connections with dispersive estimates for a certain class of Schrödinger equations whose Hamiltonian is given by the generalized Laguerre operator, i.e., the Jacobi matrix associated with generalized Laguerre polynomials. These operators feature prominently in the recent study of nonlinear waves in (2+1)-dimensional noncommutative scalar field theory since they appear as the linear part in the nonlinear Klein--Gordon and the nonlinear Schrödinger equations investigated in the recent of Chen, Fröhlich and Walcher (2003) and Krueger and Soffer (2015), respectively.

We show that dispersive estimates for the evolution group are connected with Bernstein-type inequalities for Jacobi polynomials. We use known uniform estimates for Jacobi polynomials to establish some new dispersive estimates. In turn, the optimal dispersive decay estimates lead to new Bernstein-type inequalities.

The talk is based on joint work with T. H. Koornwinder (Amsterdam) and G. Teschl (Vienna).

#### Zahlentheoretisches Kolloquium

**Title:**Numerical methods for partial differential equations with random coefficients

**Speaker:**Prof. Dr. Josef Dick (University of New South Wales, Sydney)

**Date:**Freitag, 15. 3. 2019, 14:00 c.t.

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

**Abstract:**

Abstract: Mathematical models often contain uncertainty in parameters and measurements. In this talk we focus on partial differential equations where some parameters are modeled by random variables. The main example comes for the diffusion equation where the diffusion parameters is modeled as a random field which randomly fluctuates around a given mean. We discuss recent progress on numerical methods in quantifying this uncertainty.

#### Vortrag im Seminar für Kombinatorik und Optimierung

**Title:**Real Algebra and Geometry: The Commutative and the Non-Commutative World

**Speaker:**Tim Netzer (Universität Innsbruck)

**Date:**Dienstag 12.3.2019, 14:15

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

**Abstract:**

Real algebra and geometry studies semialgebraic sets, i.e. solution sets to systems of polynomial inequalities. The classical theory provides Positivstellensätze, which are of the same spirit as Nullstellensätze in the case of varieties, i.e. solution sets to systems of polynomial equations. Many interesting (and hard) such results have been developed in the last decades, and surprising applications in optimization and convexity have arisen. A much more recent development is the theory of non-commutative semialgebraic sets. Triggered by questions in electrical engineering, control theory and quantum physics, several exciting results have been proven. But beyond the mentioned applications, the non-commutative theory also sheds light on classical questions.

In this talk, I will give an introduction to both the classical theory and their non-commutative extension, as well as some interesting applications.

#### Number Theory Seminar

**Title:**On the distribution of gaps between consecutive sums of two squares

**Speaker:**Alexander Kalmynin (National Research University Moscow)

**Date:**6.3.2019, 14.00

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

**Abstract:**

Let $s_n$ be the sequence of all positive integers that are sums of two squares, arranged in increasing order. Finding nice estimates for the gap sequence $g_n=s_{n+1}-s_n$ is a classical problem in analytic number theory. In my talk, I will construct certain series involving values of Bessel function that will allow us to prove new results about the power moments of the sequence $g_n$. We will also discuss possible generalizations of these results.

Remark: This talk is part of a joint Austrian-Russian FWF-RSF research project on Diophantine approximation and geometry of numbers.

**Title:**Differential Operators on Graphs and Waveguides

**Speaker:**()

**Date:**25.2.2019 - 1.3.2019

**Room:**Hörsaal BE01, Steyrergasse 30

**Abstract:**

The aim of the conference is to present and discuss recent results on differential operators on metric graphs and domains with waveguide-like geometry. Some of the main topics are spectral and scattering theory, asymptotic analysis and homogenization, point interactions, and random models. For a detailed conference program see

www.math.tugraz.at/diffop2019/program.html

#### Vortrag Habilitationswerberin

**Title:**Cluster growth models and fractals

**Speaker:**Dr. Ecaterina Sava-Huss (TU Graz)

**Date:**Donnerstag, 14.2.2019, 11:00 s.t.

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

**Abstract:**

A significant part of my research deals with understanding the behavior of the following cluster growth models: internal diffusion limited aggregation, the rotor model, and the divisible sandpile model.

These models can be run on any infinite state space, and they are based on particles moving around according to some rule (that can be either random or deterministic) and aggregating. Describing the limit shape of the cluster whjich these particles produce is one of the main questions one would like to answer. For some of the models, the limit shape is hard to understand, and according to simulations, the fractal nature of the sets they produce is, from the mathematical point of view, far away from being understood. I will present several results concerning the limit shape of the clusters. In particular, I will present a limit shape universality result on the Sierpinski gasket graph, and conclude with some future research directions one can pursue within this topic.

#### Strukturtheorie-Seminar (Master-Vortrag)

**Title:**The Wiener index of Schreier graphs of the basilica automaton

**Speaker:**Stefan Hammer (TU Graz)

**Date:**7.2.2019, 11:00 s.t.

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

**Abstract:**

Automata and graphs are associated in many ways. For invertible automata one can

define the automaton group and observe its action on the set of finite words over the

input alphabet. This leads to the construction of Schreier graphs.

The sum of all distances in a graph, called Wiener index, is a graph property of wide interest. Harry Wiener showed that the properties of molecules are related to the Wiener index of chemical structural formulas. In my presentation I am going to introduce all necessary tools and prove an upper bound for the Wiener index of Schreier graphs of the Basilica automaton.

#### Workshop

**Title:**Workshop: East Austria TDA Meeting

**Speaker:**()

**Date:**30.01.2019

**Room:**Seminarraum 2 Geometrie

**Abstract:**

Schedule:

9:30-10:00 Coffee

10:00-10:50: Michael Kerber: Algorithmic advances for multi-parameter persistence

10:50-11:10: Coffee

11:10-12:00: Mickael Buchet: On the structure of indecomposable modules of multi-dimensional commutative grids

12:00-12:30: Arnur Nigmetov: Metric spaces with expensive distances

12:30-14:00: Lunch break

14:00-14:50: Hubert Wagner: Bregman geometry and Information Topology

14:50-15:20: Hannah Schreiber: Discrete Morse Theory for Computing Zigzag Persistence

15:20-15:50: Coffee

15:50-16:40: Herbert Edelsbrunner: Tri-partitions of complexes

16:40-17:30: Georg Osang: Persistence of Multi-Covers

17:30- Discussions

#### Workshop

**Title:**Groups, Automata and Graphs

**Speaker:**https://www.math.tugraz.at/GAG/ ()

**Date:**February 11-12, 2019

**Room:**Seminarroom AE06

**Abstract:**

More information concerning the talks and the speakers can be found on the webpage: https://www.math.tugraz.at/GAG/

If you want to attend the workshop, please let us know in order to organize the coffee breaks.

#### Vortrag

**Title:**Japanese employment system

**Speaker:**Prof. Dr. Hisashi Okamoto (Kobe Gakuin University (Japan))

**Date:**Freitag, 25. 1. 2019, 14:00 c.t.

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

**Abstract:**

The so-called Japanese employment system consists of three unique components: long-term employment, seniority-based wage system and enterprise unions. Partly because the system worked properly, Japan successfully achieved the rapid growth in 1960s.

In the talk, first, we give a brief description of Japanese employment system. Then, we introduce two leading theories explaining the economic rationality of seniority-based wage system. Finally, we show an application of such theories as these to empirical analysis taking one of my studies as an example.The details are as follows.

1. An overview of Japanese employment system

Japanese employment system

has three unique components such as

Long-term employment

Seniority-based wages system

Enterprise unions

worked well especially in 1960s, but has been working poorly since early 1990s ,when the bubble economy collapsed

is undergoing the drastic reform affected by the progress in the reform in corporate governance

2. Focus on seniority-based wage system

There exist two leading theories explaining the rationality of seniority-based wage system as below

Human capital theory ( by G. Becker )

Delayed compensation theory ( by P. Lazear )

The former sheds light on the role of the system in employees’ skill formation, and the latter regards it as an incentive device

In Japan, delayed compensation theory seems more plausible

3. An application of economic theories to empirical analysis

We show how economic theories are applied to empirical analysis taking one of my studies as an example which employed the theories explained above for examining “Employment Ice Age” (the period just after the bubble economy collapsed, when Japanese firms seriously cut back on hiring of new university graduates influenced by negative macroeconomic shocks)

#### Zahlentheoretisches Kolloquium

**Title:**Diophantine approximation with primes from imaginary quadratic number fields

**Speaker:**Marc Technau (TU Graz)

**Date:**11.1.2019, 14:15

**Room:**Seminarraum Analysis und Zahlentheorie

**Abstract:**

A classical variation of Dirichlet's theorem on Diophantine approximation asks for how well a given number $\alpha\in\mathbb{R}\setminus\mathbb{Q}$ can be approximated by fractions with prime denominators. This problem has attracted the attention of a number of researchers, amongst these, Vinogradov, Vaughan, Harman, Jia, and Heath-Brown. Ten years ago, Matomäki achieved unconditionally the result that, for any $\epsilon>0$, there are infinitely many primes $p$ such that $\min_{a\in\mathbb{Z}} \lvert p\alpha-a \rvert < p^{-1/3+\epsilon}$.---A result previously known only on assuming the Generalised Riemann Hypothesis.

In this talk we consider a natural two-dimensional variation of the aforementioned problem. Namely, for an imaginary quadratic number field $\mathbb{K}\subset\mathbb{C}$ and $\alpha\in\mathbb{C}\setminus\mathbb{K}$, we seek to establish the existence of infinitely many prime elements $p$ in the ring of algebraic integers $\mathcal{O}$ of $\mathbb{K}$ such that $p\alpha$ is `close' to some element of $\mathcal{O}$ in a manner to be made precise in the talk.

The main ingredients of our attack on the problem are a smoothed version of a sieve method due to Harman, Poisson summation, and some point distribution results in the setting of $\mathcal{O}$ in the spirit of Vinogradov. In particular, the introduction of smoothing allows for stronger results than those based on truncating certain Fourier series. For technical reasons we only obtain results for those $\mathbb{K}$ with class number one.

The talk is based on joint work in progress with Stephan Baier.}

#### Vortrag im Seminar für Kombinatorik und Optimierung

**Title:**Restricted Assignment Scheduling with Resource Constraints

**Speaker:**Hans Kellerer (Institut für Statistik und Operations Research, Universität Graz)

**Date:**8.1. 2019, 14:15

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

**Abstract:**

We consider parallel machine scheduling with job assignment restrictions,

i.e., each job can only be processed on a certain subset of the machines.

Moreover, each job requires a set of renewable resources. Any resource can

be used by only one job at any time. The objective is to minimize the

makespan. We present approximation algorithms with constant worst-case bound

in the case that each job requires only a fixed number of resources. For

some special cases optimal algorithms with polynomial running time are

given. If any job requires at most one resource and the number of machines

is fixed, we give a PTAS. On the other hand we prove that the problem is

APX-hard, even when there are just three machines and the input is

restricted to unit-time jobs.