### Talks in 2019

#### Zahlentheoretisches Kolloquium

Title: Galois groups of differential equations
Speaker: Dr. Michael Wibmer (TU Graz)
Date: Dienstag, 2. Juli 2019, 14:00 Uhr
Room: Seminarraum Analysis-Zahlentheorie (NT02008), Kopernikusgasse 24/II
Abstract:

Abstract: The Galois group of a homogeneous linear differential equation is a linear algebraic group that governs the symmetries among the solutions. I will explain progress towards understanding these Galois groups in the case when the linear differential equations have rational function coefficients. Joint work with Anette Bachmayr, Julia Hartmann and David Harbater.

#### Zahlentheoretisches Kolloquium

Title: An overview on Arboreal Galois representations
Speaker: Dr. Andrea Ferraguti (Max Planck Institut, Bonn)
Date: Dienstag, 18. 6. 2019, 13:30 Uhr
Room: Seminarraum Analysis-Zahlentheorie (NT02008), Kopernikusgasse 24/II
Abstract:

Abstract: Arboreal Galois representations are central objects in modern arithmetic dynamics. They are defined as continuous homomorphisms, associated to rational maps over algebraic varieties, from the absolute Galois group of a field to the automorphism group of a special graph, and they are considered to be the dynamical avatars of Galois representations attached to Tate modules of abelian varieties. Due to their nature, they combine in a beautiful way several combinatorial, arithmetic and group-theoretic information. In this talk I will introduce them, showing peculiar examples and the most important conjectures around the topic. Afterwards, I will explain the recent developments due to my research: our proof of Jones' conjecture (joint with G. Micheli) and our work around the inverse problem (joint with D. Casazza and C. Pagano).

#### Mathematisches Kolloquium

Title: Machine learning in Finance
Speaker: Prof. Dr. Josef Teichmann (ETH Zürich)
Date: Freitag, 14. 6. 2019, 14:00 Uhr c.t.
Room: HS BE01, Steyrergasse 30, EG, TU Graz
Abstract:

Abstract: We show several instances of machine learning technology in
Finance like deep hedging, deep portfolio optimization, deep
calibration or deep simulation. In return several stochastic methods
from mathematical finance might shed some new light on machine
learning methods: we prove a version of Chow's theorem to underline
that randomness matters in training networks and we apply the
Johnson-Lindenstrauss Lemma to construct tractable approximations of signatures.

#### Number Theory Seminar

Title: Large Oscillations of the Argument of the Riemann Zeta-function
Speaker: Kamalakshya Mahatab (NTNU Trondheim)
Date: Wednesday, 12.6.2019, 13:00
Room: Seminarraum Analysis-Zahlentheorie, Kopernikusgasse 24, 2.OG
Abstract:

Let S(t) be the argument of the Riemann zeta function on the critical line. This function plays an important role to estimate the number of zeros of Riemann zeta function in the critical strip up to a height t. In this talk we will estimate large positive and negative values of S(t) using the resonance method.

#### Intensive course on Advanced Analytic Combinatorics

Title: Advanced methods in analytic combinatorics
Speaker: Wenjie Fang (TU Graz)
Date: Dienstag 11.6., 14:15-16:00 und Dienstag 18.6., 14:15-16:00
Room: Seminarraum AE06, Steyrergasse 30, Erdgeschoss
Abstract:

Analytic combinatorics is the study of asymptotic enumeration of combinatorial objects through analytical aspects of their corresponding generating functions, especially their singularities. In this series of lectures, we will give an introduction to more advanced methods in this domain, such as saddle point method and Mellin transform, which are applicable to some problems that are out of reach for standard methods such as transfer theorems. This introduction will be given in the context of asymptotic enumeration of variants of integer partitions and plane partitions, including recent work of the lecturer in collaboration with Hsien-Kwei Hwang and Mihyun Kang.

#### Seminar Angewandte Analysis und Numerische Mathematik

Title: On self-adjoint boundary conditions for singular Sturm-Liouville operators
Speaker: Prof. Dr. Fritz Gesztesy (Baylor University, Waco, Texas)
Date: 6.6.2019, 14:15 Uhr
Room: AE 02
Abstract:

The classical boundary values for regular Sturm-Liouville operators associated with a three-coefficient differential expression on a compact interval $[a,b]$, is extended in a natural manner to the case where the differential expression is singular on an arbitrary open interval $(a,b)$ of the real line under the assumption that the associated minimal operator is bounded from below. The notion of (non)principal solutions of the associated differential equation plays a key role in this analysis.

We briefly discuss the singular Weyl-Titchmarsh-Kodaira m-function and illustrate the theory with the special case of Bessel and Legendre operators.

This is based on joint work with Lance Littlejohn and R. Nichols.

#### Vortrag im Seminar für Kombinatorik und Optimierung

Title: On the exponents of extremal numbers
Speaker: Jaehoon Kim (University of Warwick)
Date: Dienstag 4.6.2019, 14:15
Room: Seminarraum AE06, Steyrergasse 30, Erdgeschoss
Abstract:

The extremal number ex$(n,F)$ of a graph $F$ is the maximum number of edges in an $n$-vertex graph not containing $F$ as a subgraph. A real number $r \in [1,2]$ is realisable if there exists a graph $F$ with ex$(n,F) = \Theta(n^r)$. Erd\H{o}s and Simonovits conjectured that every rational number in $[1,2]$ is realisable. We show that $2 - \frac{a}{b}$ is realisable for any integers $a,b \geq 1$ with $b>a$ and $b = \pm 1$ (mod $a$). This includes all previously known realisable numbers. This is joint work with Dong Yeap Kang and Hong Liu.

#### Number Theory Seminar

Title: On Markov numbers
Speaker: Buket Eren Gökmen (Galatasaray University, Istanbul)
Date: Tuesday, 28.5.2019, 11:15
Room: Seminarraum Analysis-Zahlentheorie, Kopernikusgasse 24, 2.OG
Abstract:

The Markov numbers are the solutions $(x,y,z) \in \mathbb{Z}^3_{+}$  to the Markov equation $x^2+y^2+z^2=3xyz$. Markov (1879) showed that all possible solutions are generated from $(1,1,1)$ by a simple algorithm. Does this algorithm generate each solution in a unique way? More precisely, Frobenius $(1913)$ asked whether it is true that for all $z> 0$, there exists at most one pair $(x, y)$ such that $x <y <z$ and $(x, y, z)$ is a solution. This conjecture remains open to this day, despite the simplicity of its statement.

Markov numbers arise in many different contexts such as binary quadratic forms, hyperbolic geometry, combinatorics etc. with beautiful interconnections. The purpose of this talk is to present a part of the Markov theory that is built around an intriguing conjecture and Markov's theorem which combines approximation of irrationals and Diophantine equations in a totally unexpected way. In the end, we will introduce an involution of the real line called Jimm induced by the outer automorphism of the extended modular group $\mathrm{PGL(2,\mathbb{Z})}$ that may be relevant to the subject.

#### Vortrag

Title: Capture-recapture for population size estimation based upon zero-truncated count distributions with one-inflation
Speaker: Dankmar Böhning (Statistical Sciences Research Institute, University of Southampton/UK)
Date: 28.05.2019, 17:15 Uhr
Room: SR für Statistik (NT03098), Kopernikusgasse 24/III
Abstract:

Abstract: Population size estimation by means of capture-recapture methods using zero-truncated count distributions has become popular. The estimator of Chao is likewise frequently used as it is asymptotically unbiased if the model holds and provides a lower bound in the case of population heterogeneity. However, if one-inflation occurs Chao’s estimator can seriously overestimate as it builds largely on the count of ones, the singletons, in the sample. The talk highlights how one-inflation can be detected and how it can be dealt with, and ultimately provides a more reasonable population size estimator. Two examples will motivate and illustrate one-inflated modelling: the size of a dice-snake population in Graz (Austria) as well as the size of the flare star cluster in the Pleiades.

#### Student Workshop

Title: Approximation Theory and Applications
Speaker: ()
Date: 24.5.2019, 14:00
Room: Seminarraum 2, Kopernikusgasse 24/4
Abstract:

{\bf Maria Charina} (Univ. Wien): Reelle Nullstellen von Polynomen und Origami

{\bf Dennis Elbrächter} (Univ. Wien): Universal sparsity of deep neural networks

{\bf Svenja Hüning} (TU Graz): Convergence of subdivision processes in nonlinear geometries

{\bf Thomas Lang} (Univ. Passau): Segmentation of CT Scans using Support Vector Machines

#### Seminarvortrag

Title: Best Estimate Berechnung und Validierung in der Lebensversicherung
Speaker: Simon Hochgerner (FMA - Finanzmarktaufsicht Österreich)
Date: 24.05.2019, 14:15
Room: SR für Statistik (NT03098), Kopernikusgasse 24/III
Abstract:

Seit Inkrafttreten von Solvency II per 1.1.2016 sind Versicherungsunternehmen verpflichtet, den Wert der eingegangenen Verpflichtungen marktkonsistent und unter Berücksichtigung realistischer Annahmen zu bestimmen ("Best Estimate").
Speziell für die klassische Lebensversicherung führen diese Bedingungen zu besonderen Herausforderungen, da es bei diesen Produkten eine enge Verflechtung von Aktiv-, Passivseite und Managementregeln gibt.
Im Rahmen des Vortrags werden wir auf einige Probleme im Zusammenhang mit der Best Estimate Berechnung eingehen und die wichtigsten Validierungsschritte vorstellen.

#### Vorstellungsvortrag im Rahmen eines Habilitationsverfahrens

Title: High-dimensional connectedness: cores and components
Speaker: Oliver Cooley (TU Graz, Institut für Diskrete Mathematik)
Date: Freitag 24.5.2019, 11:00
Room: Seminarraum 2, Institut für Geometrie, Kopernikusgasse 24/IV
Abstract:

The talk will provide an overview of some of my recent research topics, with a common theme of generalising the standard graph notions of connectedness and components to higher-dimensional structures.

These include the $k$-core of a graph, i.e. the unique largest subgraph of minimum degree at least $k$, which we analyse by means of a message-passing algorithm introduced in physics literature. We show how an understanding of this local algorithm helps us to determine the global structure of the $k$-core and its interaction with other vertices.

We also consider $j$-tuple-connected components in $k$-uniform hypergraphs, a notion of connectedness related to $j$-tight paths. We observe some phase transition phenomena analogous to famous and classical graph results, but also discuss why the hypergraph case is richer and more complex.

#### Seminar Angewandte Analysis und Numerische Mathematik

Title: Maxwell, Dirac and their connection via Picard
Speaker: Dr. Marcus Waurick (University of Strathclyde, Glasgow)
Date: 22.5.2019, 11:00 Uhr
Room: A 111
Abstract:

We will consider Maxwell's equations and recall the construction of the related Picard's extended Maxwell system, which proves to be useful in spectral theory and homogenisation. We will provide some information of the extended Maxwell system on compact embeddings and an analysis of the kernel and its relation to the geometry of the underlying domain. Furthermore, we shall show that the extended Maxwell operator is strongly related to the Dirac operator.

The talk is based on joint work with Rainer Picard and Sascha Trostorff; see also [Picard, R.; Trostorff, S.; Waurick, M. On a Connection between the Maxwell System, the Extended Maxwell System, the Dirac Operator and Gravito-Electromagnetism. Math. Meth. Appl. Sci., 40(2): 415-434, 2017].

#### Vortrag im Seminar für Kombinatorik und Optimierung

Title: Non-uniform random simplicial complexes
Speaker: Philipp Sprüssel (TU Graz)
Date: Dienstag 21.5.2019, 14:15
Room: Seminarraum AE06, Steyrergasse 30, Erdgeschoss
Abstract:

Random simplicial complexes have received considerable attention in the last years as a higher-dimensional analogue of random graphs. Two models of binomial'' random simplicial complexes of dimension d have been studied. In both models, the vertex set is $\{1,...,n\}$ and each $d$-simplex is present with some global probability $p=p(n)$ independently. The first model, due to Linial, Meshulam, and Wallach, furthermore contains all simplices of dimension smaller than $d$. By contrast, the other model, recently introduced by Cooley, Del Giudice, Kang, and Sprüssel, only contains those simplices of dimension $1$ up to $d-1$ that are contained in some d-simplex. For both models, higher-order connectivity of the complex can be defined via the vanishing of cohomology groups, and sharp thresholds for these properties have been determined for various choices of coefficients for cohomology.

Both models mentioned above are uniform'' in the sense that the randomness lies only in the choice of the d-simplices. In this talk, we present a non-uniform'' model in which the simplices of all dimensions from $1$ up to $d$ are chosen randomly. In particular, both uniform models are special cases of the non-uniform model. We determine a sharp threshold for the aforementioned notion of higher-order connectedness in the non-uniform model, where the coefficients of the cohomology groups are chosen from any abelian group. This result implies the corresponding results for the uniform models.

This talk is based on joint work with Oliver Cooley, Nicola Del Giudice, and Mihyun Kang.

#### Number Theory Seminar

Title: Khintchine's theorem with extra divergence instead of monotonicity
Speaker: Laima Kaziulyte (TU Graz)
Date: Tuesday, 21.5.2019, 11:15.
Room: Seminarraum Analysis-Zahlentheorie, Kopernikusgasse 24, 2.OG.
Abstract:

New results towards the Duffin-Schaeffer conjecture, which is a fundamental unsolved problem in metric number theory, have been established recently assuming extra divergence. Given a non-negative function $\psi: \mathbb{N}\to\mathbb{R}$ we denote by $W(\psi)$ the set of all $x\in\mathbb{R}$ such that $|nx-a|<\psi(n)$ for infinitely many $a,n$. Analogously, we write $W'(\psi)$ if we additionally require $a,n$ to be coprime.

Aistleitner et al. proved that $W'(\psi)$ is of full Lebesgue measure if there exists an $\varepsilon>0$ such that $\sum_{n=2}^\infty\psi(n)\varphi(n)/(n(\log n)^\varepsilon)=\infty$. This result seems to be the best one can expect from the method used. Assuming the extra divergence $\sum_{n=2}^\infty\psi(n)/(\log n)^\varepsilon=\infty$ we prove that $W(\psi)$ is of full measure. This could also be deduced from the results in Aistleitner et al., but we believe that our proof is of independent interest, since its method is totally different from theirs. As a further application of our method, we prove that a variant of Khintchine's theorem is true without monotonicity, if the support of $\psi$ can be restricted subject to a condition on the set of divisors.

#### Seminarvortrag

Title: Counterpart Default Risk in the Solvency 2 Standard Formula
Speaker: Mihael Perman (University of Ljubljana )
Date: 17.05.2019, 15:15
Room: SR für Analysis - Zahlentheorie, Kopernikusgasse 24/II
Abstract:

Capital requirements in Solvency 2 are assembled from many components. In the
talk we will focus on counter-party default risk. We will start from the mysterious
looking instructions how to compute the contribution to capital requirements and
try to explain the underlying statistical model and the meaning of parameters. The
results will then be compared to simulated results in the real case of a reinsurance
company.

#### Zahlentheoretisches Kolloquium

Title: Local statistics of sqrt(n) mod 1 and related problems
Speaker: Dr. Daniel El-Baz (Max Planck Institut, Bonn)
Date: Freitag, 17. 5. 2019, 14:00 Uhr
Room: Seminarraum Analysis-Zahlentheorie (NT02008), Kopernikusgasse 24/II
Abstract:

Abstract:
It is an elementary exercise to show that the sequence of the
square roots of the positive integers is equidistributed modulo 1. I
will discuss results concerning the fine-scale statistics of this
sequence, such as the determination of its gap distribution by Elkies
and McMullen (using homogeneous dynamics) and its pair correlation in
joint work with Jens Marklof and Ilya Vinogradov (based on the
Elkies-McMullen approach along with some analytic number theoretic
estimates). I will also mention an ongoing project with Carlo Pagano
whose goal is to understand such statistics for the square roots of
subsets of the integers (such as the square-free integers).

#### Konferenz

Title: Austrian Numerical Analysis Day 2019
Speaker: ()
Date: 9.-10.5.2019
Room: TU Graz, Hörsaal BE01, Steyrergasse 30, EG, 8010 Graz
Abstract:

Detailliertes Programm siehe
www.applied.math.tugraz.at/tagungen/anaday19/

#### Vortrag im Seminar für Kombinatorik und Optimierung

Title: Selected Cases of Vehicle Routing – From a Real World Application to a Machine Learning Based Approach
Speaker: Nikolaus Furian (Institut für Maschinenbau- und Betriebsinformatik, TU Graz)
Date: 7.5.2019, 14:15
Room: Seminarraum AE06, Steyrergasse 30, Parterre
Abstract:

We first report on a real world case study of vehicle routing. Patient transits in the Auckland City Hospital are carried out by so called orderlies that transfer patients from and to appointments within the hospital complex. For some transits the assistance of a nurse is required. Ad-hoc dispatching of staff members, nurses and orderlies, to transits has been identified as one major source for delays. We present automated, optimized dispatching algorithms which rely
on a network formulation which is strongly related to an established approach for the VRP with soft time windows. However, the need to synchronize the routes of staff members of different types (nurses and orderlies) adds a whole new layer of complexity to the problem, as routes cannot be assessed independently. We present a set of algorithms with varying complexity, ranging from simple heuristics to the use of critical path methods to combine mixed integer formulations for the separated orderly and nurse problems. To address a transit service's stochasticity, volatility and the resulting need for constant re-optimization, we embed the optimization algorithms in a discrete event simulation to evaluate their performance under realistic circumstances.
Some elements of the underlying structure of the above outlined problem have not been explicitly addressed by the literature on vehicle routing. We present machine learning models and some preliminary results on the predictability of optimal solution structures for a sampled version of the VRP with time windows that can be found in numerous applications. Further, we outline some possibilities to make use of such predicted solution structures within heuristics methods, as well as exact algorithms for vehicle routing.

#### Strukturtheorie-Seminar

Title: Rational Embeddings of Hyperbolic Groups
Speaker: Dr. Francesco Matucci (Università Bicocca Milano)
Date: 7.5.2019, 11 Uhr c.t.
Room: Seminar Room AE06, Steyrerg. 30
Abstract:

For a finitely generated group, the Cayley graph is a metric space encoding the structure of the group. Gromov introduced the notion of a $\delta$-hyperbolic group, a finitely generated group with a negatively curved Cayley graph, that is, for any triangle in the graph with geodesic sides, each side is contained in the $\delta$-neighborhood of the union of the two other sides. Hyperbolic groups are prevalent'' among finitely generated groups.

Grigorchuk, Nekrashevych and Sushchanskii defined the rational group as the full group of homeomorphisms of a Cantor space and which admit precisely finitely many types of local actions'' described by finite state transducers (one of many models of computing machines). This is a rather large group and, by construction, it contains all groups generated by finite state automata (for example, the Grigorchuk group of intermediate word growth).

In this talk I will introduce these groups and some of their properties and explain how to embed a class of hyperbolic groups in the rational group.

Parts of this talk are joint with James Belk, Collin Bleak and James Hyde.

#### Zahlentheoretisches Kolloquium

Title: On abelian cubic polynomials
Speaker: Dr. Stanley Yao Xiao (University of Toronto)
Date: Dienstag, 30. 4. 2019, 14:00 Uhr
Room: Seminarraum Analysis-Zahlentheorie (NT02008), Kopernikusgasse 24/II
Abstract:

Abstract: In this talk I will describe some recent work on counting
monic cubic polynomials with respect to various heights. In particular,
we prove that the number of irreducible, monic, Galois cubic polynomials
$x^3 + a_2 x^2 + a_1 x + a_0$ with $\max\{|a_2|, |a_1|, |a_0|\} \leq X$
and whose $I,J$-invariants are co-prime is $O(X \log X)$.

#### 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.