### Talks in 2020

#### Strukturtheorie-Seminar

**Title:**Three recreational tales

**Speaker:**Dr. Florian Lehner (Thomas) (TU Graz)

**Date:**Thursday, 29 October 2020, 11:30

**Room:**Lecture room P2 (Physics building) + webex

**Abstract:**

Mathematical puzzles often lead to interesting and challenging research questions. In this talk I will outline three instances of problems rooted in recreational mathematics each of which has inspired a substantial amount of research: pursuit-evasion games, asymmetric graph colouring, and graph reconstruction. For each of these problems I will highlight open research questions, survey some recent results, and (depending on time) provide rough proof sketches.

The talk is aimed at a non-specialist audience, no previous knowledge will be required.

Thursday, 29 Oct, 2020 11:15 CET

Webex meeting number: 137 992 7734

Password: HEeNZ3Vhp42

https://tugraz.webex.com/tugraz/j.php?MTID=m888653d1ba2a8be86ed61f2fa8528e03

The talk starts at 11:30.

The lecture room can host up to 51 persons.

#### Seminar für Kombinatorik und Optimierung

**Title:**Decompositions of Hypergraphs

**Speaker:**Felix Joos (Universität Heidelberg)

**Date:**Friday 23rd October 14:15

**Room:**Online meeting (Webex)

**Abstract:**

I'll present various results on approximate decompositions of

hypergraphs. In all of the results, the key technique are random

processes that yield the desired results if combined suitably.

Meeting link:

\[

\text{ https://tugraz.webex.com/tugraz/j.php?MTID=m1cd0904285a119237aa9a7ce985ad803}

\]

Meeting number:

137 149 1265

Password:

JYc3B3dunG2

#### Strukturtheorie-Seminar

**Title:**WEIGHTED GRAPHS OVER ORDERED FIELDS

**Speaker:**Dr. Anna Muranova (TU Graz)

**Date:**Thursday, 22 October 2020, 13:00 on time

**Room:**Seminar room AE06 + webex

**Abstract:**

A weighted graph is a graph where each edge is given a weight. In the classical theory of weighted graphs, this weight is a positive real number. We consider more general weights, namely, positive elements of any ordered field. We prove the existence and uniqueness of the solution of a Dirichlet problem on finite graphs and investigate some properties of infinite graphs. Classical weighted

graphs are related to electrical networks with resistors. In a similar way, one can relate weighted graphs over the field of rational functions with electrical networks with coils, capacitors, and resistors. The ordered field of rational functions is the most known non-Archimedean field. Its Cauchy completion is the Levi-Civita field R. Therefore, we consider some known infinite electrical networks (e.g. Feynman ladder) over R.

No pre-knowledge on ordered fields is assumed in this talk.

Please note the atypical time. The talk is also accessible via webex mode.

Meeting number: 137 044 3093

Password: M2jDZr53emd

https://tugraz.webex.com/tugraz/j.php?MTID=mddc8c646cc16cfa1c71847fc3f5341b8

Webex meeting hosted by Christian Lindorfer, beginning at 12:45 (middle European time); the talk starts at 13:00

#### Vortrag

**Title:**Hopf Algebras and Diagonal Harmonics

**Speaker:**Cesar Ceballos (Inst. f. Geometrie, TU Graz)

**Date:**16.10.2010, 11:00

**Room:**HS G, Kopernikusgasee 24, bzw. Videokonferenz

**Abstract:**

The theory of Hopf algebras is a fundamental area in

mathematics which originated in the 1940’s and 1950’s motivated by work of

Hopf on algebraic topology and of Diedonné on algebraic groups. Diagonal

harmonics, on the other hand, is a more recent and apparently unrelated

area initiated by Garsia and Haiman in the early 1990’s, which has

remarkable connections to Macdonald polynomials, algebraic geometry,

representation theory, knot theory, and mathematical physics.

In this talk, I will give an insight to these fascinating areas, mainly

through a series of examples and without many technicalities. The main

purpose is to present some unexpected connections arising in the study of a

Hopf algebra structure on pipe dreams, certain discrete objects that

provide a combinatorial understanding of Schubert polynomials.

The talk is addressed to a general mathematical audience and no previous

knowledge of Hopf algebras or diagonal harmonics will be assumed.

#### Strukturtheorie-Seminar

**Title:**Operator algebras for random walks

**Speaker:**Dr. Adam Dor-On (University of Copenhagen)

**Date:**Thursday, 15 October 2020, 10:30

**Room:**webex meeting

**Abstract:**

We present some newly emerging connections between the theory of random walks and operator algebras. More specifically, to each random walk P we can associate several operator algebras that capture various kinds of behaviors of P. This yields new insight on random walks and provides operator algebraic interpretations for the strong ratio limit property of random walks. No knowledge on operator algebras will be assumed in the talk.

webex meeting

Thursday, 15 Oct, 2020 10:15 Rome, Stockholm, Vienna

Meeting number: 137 600 7164

Password: f3AbF427pUs

https://tugraz.webex.com/tugraz/j.php?MTID=m2ab4cfe9ac172dc045d6dd6512a479da

[tugraz.webex.com]

(Start time is 10:15, the talk will start at 10:30)

#### Zahlentheoretisches Kolloquium

**Title:**On the maximum of inclomplete Kloosterman sums

**Speaker:**Dr. Dante Bonolis (IST Wien)

**Date:**08.10.2020, 14:30 Uhr

**Room:**HS E, Kopernikusgasse 24, 1. OG

**Abstract:**

\begin{document}

Let $t:\mathbb{F}_{p}\rightarrow\mathbb{C}$ be a complex valued function on $\mathbb{F}_{p}$. A classical problem in analytic number theory is bounding the maximum

\[

M(t):=\max_{0\leq H<p}\Big|\frac{1}{\sqrt{p}}\sum_{0\leq n < H}t(n)\Big|

\]

of the absolute value of the incomplete sums $\frac{1}{\sqrt{p}}\sum_{0\leq n < H}t(n)$. In this very general context one of the most important results is the P\'olya-Vinogradov bound

\[

M(t)\leq \left\|\hat{t}\right\|_{\infty}\log 3p,

\]

where $\hat{t}:\mathbb{F}_{p}\rightarrow\mathbb{C}$ is the normalized Fourier transform of $t$. In this talk, we provide a lower bound for certain incomplete Kloosterman sums, namely we prove that there exists a subset of $a\in\mathbb{F}_{p}^{\times}$ such that

\[

M( e((ax+\overline{x})/p))\geq \left(\frac{2}{\pi}+o(1)\right)\log\log p,

\]

as $p\rightarrow \infty$. We prove this by studying the growth of the moments of $\{M(e((ax+\overline{x})/p))\}_{a\in\mathbb{F}_{p}^{\times}}$. This is a joint work with Pascal Autissier and Youness Lamzouri.

\end{document}

#### Seminar für Kombinatorik und Optimierung

**Title:**Cutvertices in random planar maps

**Speaker:**Benedikt Stufler (TU Wien)

**Date:**Friday 16th October 14:15

**Room:**Online meeting (Webex)

**Abstract:**

We study the number of cutvertices in a random planar map as the number of edges tends to infinity. Interestingly, the combinatorics behind this seemingly simple problem are quite involved.

(Joint work with Marc Noy and Michael Drmota)

Meeting link:

\[

\text{ https://tugraz.webex.com/tugraz/j.php?MTID=m1cd0904285a119237aa9a7ce985ad803}

\]

Meeting number:

137 149 1265

Password:

JYc3B3dunG2

#### Seminar für Kombinatorik und Optimierung

**Title:**Non-concentration of the chromatic number and the Zig-zag conjecture

**Speaker:**Annika Heckel (Department Mathematik, LMU München)

**Date:**Friday 2nd October 14:15

**Room:**Online meeting (Webex)

**Abstract:**

The chromatic number of a graph is the minimum number of colours we need to colour all vertices so that adjacent vertices receive different colours.

What can we say about the chromatic number of a random graph $G(n,p)$? One main direction of past research has been the likely value of this random variable, i.e. proving that certain upper and lower bounds hold with high probability (whp). The other main direction of research has been the following question: how sharp is the concentration of the chromatic number? In other words, what is the length of the shortest interval (or rather sequence of intervals) which contains the chromatic number whp?

The starting point is a classic result of Shamir and Spencer who showed that the chromatic number of $G(n,p)$ is whp contained in some sequence of intervals of length at most about $n^{1/2}$. For sparse random graphs, this can be improved dramatically: Alon and Krivelevich proved that the chromatic number of $G(n,p)$ is two-point concentrated whenever $p < n^{-1/2 - \epsilon}$.

In view of strong concentration results, Bollobás and Erdős asked the opposite question: can we find any examples where the chromatic number is not very narrowly concentrated? Specifically, can we show that the chromatic number of $G(n, 1/2)$ is not whp concentrated on 100 integers?

In this talk, I will present a recent result showing that, at least for some values $n$, the chromatic number of $G(n, 1/2)$ is not concentrated on fewer than $n^{1/2- o(1)}$ consecutive values, almost matching Shamir and Spencer's upper bound. I will also discuss and give evidence for a recent conjecture on the correct concentration interval length, which seems to depend on n.

\newpage

Meeting link:

\[

\text{ https://tugraz.webex.com/tugraz/j.php?MTID=m1cd0904285a119237aa9a7ce985ad803}

\]

Meeting number:

137 149 1265

Password:

JYc3B3dunG2

#### Mathematisches Kolloquium

**Title:**Monotone chains in multiplicative sets

**Speaker:**Dr. Oleksiy Klurman (Max-Planck-Institut für Mathematik, Bonn)

**Date:**12.08.2020, 14:30 Uhr

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

**Abstract:**

It is believed that given a typical multiplicative function $f:\mb{N} to\mb{R} $ and admissible integers $a_1... a_k$ each possible arrangement $f(n+a_1)<f(n+a_2)<....<f(n+a_k)$ occurs for infinitely many n. This problem is widely open in general.

In this talk, we describe a new approach to deal with questions of this type and present various applications.

#### Zahlentheoretisches Kolloquium

**Title:**Dirichlet Series with Periodic Coefficients and their Value-Distribution near the Critical Line

**Speaker:**Athanasios Sourmelidis (TU Graz)

**Date:**17.7.2020, 13:30

**Room:**HS BE01, Steyrergasse 30

**Abstract:**

The class of Dirichlet series associated with a periodic arithmetical function $f$ includes the Riemann zeta-function as well as Dirichlet $L$-functions to residue class characters.

We study the value-distribution of these Dirichlet series in a neighbourhood of the critical line (which is the abscissa of symmetry of the related Riemann-type functional equation).

In particular, for a fixed complex number $a\neq 0$, we consider for even or odd periodic $f$ the values taken at the $a$-points of the $\Delta$-factor of the functional equation, prove the existence of the mean-value of these points, show the uniform distribution of their ordinates, and obtain a related discrete universality theorem. This is joint work with Jörn Steuding and Ade Irma Suriajaya.

Remark: Athanasios Sourmelidis is a new member of the Institute of Analysis and Number Theory, where he will work as a PostDoc researcher.

#### Seminar für Kombinatorik und Optimierung

**Title:**Keeping a graph connected via nonpreemptive edge scheduling (Maximizing the minimum load time in a graphic matroid)

**Speaker:**Lasse Wulf (Institut für Diskrete Mathematik, TU Graz)

**Date:**2.7.2020, 16:15 (talk), 15:50-16:15 informal chat

**Room:**online meeting (Webex)

**Abstract:**

Consider the following process over time: Given a graph $G = (V,E)$ with positive integral edge weights $w(e)$, we choose for each edge e exactly one time point $t(e) \in [0, \infty)$. This causes $e$ to be active during the

interval $[t(e), t(e) + w(e)]$. We now ask how we can choose the times $t(e)$ such that the subgraph of active edges is spanning for a maximal amount of time. The problem is related to the classical problems of spanning

tree packing and Menger's problem. It can also be seen as a generalization of maximizing the minimum load time

in nonpreemptive scheduling. In this talk, we show that the problem is NP-complete, even if $G=K_{2,n}$ or if

$w(e) \in \{1, ... , 6\}$. Furthermore, if P $\ne$ NP, the problem can not be approximated in polynomial time by a factor

better than 7/6. On the other hand, if both the treewidth of the input graph and the edge weights are bounded

by a constant, we give a linear time algorithm.

{

Webex meeting info

Meeting number (access code): 137 382 3722

Meeting password: 8diN3Pd9q34

https://tugraz.webex.com/tugraz/j.php?MTID=m01485e57708c11c02cfe8673af6dcd11}

}

#### Strukturtheorie-Seminar

**Title:**Planar Cayley graphs and Kleinian groups

**Speaker:**Prof. Agelos Georgakopoulos (University of Warwick)

**Date:**Wednesday, July 1st, 2020, 16:15 (virtual coffee break beginning 15:45)

**Room:**Webex meeting

**Abstract:**

Kleinian groups are a classical topic. I will give an overview

and explain their relationship to groups having planar Cayley graphs.

Moreover, I will show that if a finitely generated group G acts faithfully

and properly discontinuously by homeomorphisms on a planar surface, then G

admits such an action that is in addition co-compact.

Link to preprint: https://arxiv.org/pdf/1905.06669}

Webex meeting number: 137 170 6111

Password: iUMCF5Pb6M5

Meeting-Link:

https://tugraz.webex.com/tugraz-de/j.php?MTID=m31a117d32de0fca9218c0731e7d60b47}

#### Seminar für Kombinatorik und Optimierung

**Title:**Some combinatorial aspects of pop-stack sorting

**Speaker:**Andrei Asinowski (Institut für Mathematik, Universität Klagenfurt)

**Date:**25.6.2020, 15:00 (talk), informal chat starts at 14:50

**Room:**online meeting (Webex)

**Abstract:**

Pop-stack sorting is a natural procedure for sorting permutations, where at

each iteration all maximal descending strings are reversed (for example

T(526314) = 251364). It can be seen as a variation of the classical stack

sorting. I will present some structural, enumerative, and algorithmic results

and conjectures related to the pop-stack sorting, including links to finite

automata, lattice paths and random permutations.

Joint work with Cyril Banderier (University of Paris North) and Benjamin Hackl (University of Klagenfurt)

{

Webex meeting info

Meeting number (access code): 137 215 5408

Meeting password: dT6yWAuqu64

https://tugraz.webex.com/webappng/sites/tugraz/meeting/info/487199bb6fd74508b490443deb606df7}

#### UPDATE: Strukturtheorie-Seminar

**Title:**Cayley-Abels graphs of totally disconnected locally compact groups

**Speaker:**Prof. Rögnvaldur Möller + Dr. Waltraud Lederle (University of Iceland + Université Catholique de Louvain)

**Date:**Wednesday 10 June 2020, 16:15 (virtual coffee break beginning 15:45)

**Room:**Webex meeting

**Abstract:**

The Cayley-Abels graph of a compactly generated totally disconnected locally compact group is an analogue of the ordinary Cayley graph for a finitely generated group. Given a compactly generated totally disconnected locally compact group G, what is the lowest possible valency of a Cayley-Abels graph? How does the valency relate to other properties of the group? Is there something special about graphs that have this lowest possible valency?

Joint work with Arnbjörg Soffia Arnadottir (Waterloo, Canada)

Webex meeting number: 321 855 198

Password: ZNpTHv9PT27

tugraz.webex.com/tugraz/j.php?MTID=m77665a58f92ac664ce28fc666b6d1f5d

Join by video system:

Dial 321855198@tugraz.webex.com

#### Probevortrag Habilitationsverfahren

**Title:**Der Fundamentalsatz der Algebra

**Speaker:**Oliver Cooley (Institut für Diskrete Mathematik, TU Graz)

**Date:**27.05.2020, 16:00

**Room:**WebEx Meeting

**Abstract:**

Der Link fuer den Vortrag wird rechtzeitig bekanntgegeben.

#### Strukturtheorie-Seminar

**Title:**Poisson type limit theorems for a noncommutative independence associated with positive symmetric cones

**Speaker:**Lahcen Oussi (Intitut für Diskrete Mathematik, TU Graz)

**Date:**Mittwoch, 13.5.2020, 16:15

**Room:**https://tugraz.webex.com/meet/lahcen.oussi

**Abstract:**

We present an analogue of the classical Law of Small Numbers, formulated for a noncommutative independence (the bm-independence), where the random variables are indexed by elements of positive symmetric cones in Euclidean spaces, including $\mathbb{R}^{d}_{+}$, the Lorentz cone in Minkowski spacetime and positive definite real symmetric matrices. The geometry of the cones plays an important role in the study as the volume characteristic sequences of each cone, related to the growth of volumes of intervals in the cone, appears in our final formulas. Also the combinatorics of ordered partitions is crucial for our study as one of the main tools for performed computations.

#### Mathematisches Kolloquium

**Title:**Hypergraph containers with applications in discrete geometry

**Speaker:**Dr. Oliver Roche-Newton (RICAM, Linz, Austria)

**Date:**17.04.2020, 14:00 Uhr

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

**Abstract:**

In recent years, the newly developed theory of hypergraph containers has resulted in several remarkable results in graph theory and extremal combinatorics. Essentially, this theory gives detailed information about the independent sets of hypergraphs, provided that the edges are distributed reasonably well. I will discuss recent joint work with Audie Warren, in which these tools were applied to problems in discrete geometry. In particular, an upper bound for the number of subsets of the finite plane with no collinear triples is given.

#### Mathematisches Kolloquium

**Title:**Minimality of the rock-salt structure and Universal Optimality for multi-component lattice systems

**Speaker:**Dr. Laurent Betermin (Universität Wien)

**Date:**20.03.2020, 14:00 Uhr

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

**Abstract:**

The mathematical justification of crystallization phenomena is usually a challenging problem and very few results exist about the optimality of ionic crystals. In this talk, I will present new analytical and numerical results obtained with Markus Faulhuber (University of Vienna) and Hans Knüpfer (University of Heidelberg) about the optimality of the rock-salt structure among lattices and charges distributions. These results are based on optimality results for special lattice functions arising in Number Theory: the Epstein zeta functions and the lattice theta functions. Many open problems will be presented, including the Universal Optimality for alternating species that we have already obtained in dimension 2 for the triangular lattice with Markus Faulhuber.

#### Colloquium of the Institute of Discrete Mathematics

**Title:**Infinite bridges for tree-valued Markov chains

**Speaker:**Prof. Anton WAKOLBINGER (Universität Frankfurt)

**Date:**Wednesday, 11 March 2020, 16:15

**Room:**Lecture room BE01, Steyrergasse 30, ground floor

**Abstract:**

For a few examples of tree-valued Markov chains (Rémy’s tree growth chain, the radix sort chain and the PATRICIA chain), we discuss representations of their Doob-Martin boundary that are obtained from their extremal “bridges to infinity”

(and thus from a conditioning on their remote future).

The talk is based on joint work with Steve Evans (Berkeley) and Rudolf Grübel (Hannover).

EGW, Doob-Martin boundary of Rémy's tree growth chain, Ann. Probab. 45b (2017), 225-277.

EW1, Radix sort trees in the large, Electron. Commun. Probab. 22 (2017).

EW2, PATRICIA bridges. In: Genealogies of Interacting Particle Systems, eds.

M. Birkner, R. Sun and J. Swart, pp. 233-267, World Scientific 2020.

#### Selection procedure for assistant position

**Title:**Short talks by several candidates

**Speaker:**K. Heuer / K. Kolesko / A. Muranova / C. Alves / F. Tonti ()

**Date:**29 + 30. + 31.1.2020, afternoon

**Room:**

**Abstract:**

Assistant position (with PhD) - selection procedure

20 minutes' scientific talks (titles below) followed by

20 Minuten Lehrvortrag (Deutsch), Thema

``Die Eulersche Zahl e''

**Wednesday, 29.1.2020**, seminar room A306 (Steyrergasse 30, 3rd floor):

14:00 Karl HEUER (Berlin): Forcing Hamiltonicity in locally finite graphs via forbidden induced subgraphs

15:30 Konrad KOLESKO (Innsbruck): Limit theorems in branching processes

**Thursday, 30.1.**, seminar room AE06 (Steyrergasse 30, ground floor):

13:30 Anna MURANOVA (Bielefeld): On the notion of effective impedance for networks

15:00 Caio ALVES (Leipzig): Decoupling inequalities in loop percolation

**Friday, 31.1.**, seminar room AE02 (Steyrergasse 30, ground floor):

13:30 Fabio TONTI (Vienna): A new proof of Thoma's theorem

#### Zahlentheoretisches Kolloquium

**Title:**How to quantify the randomness of a real-valued sequence?

**Speaker:**Niclas Technau (Tel Aviv University)

**Date:**28.1.2020, 13:30

**Room:**SR Analysis-Zahlentheorie, Kopernikusgasse 24, 2nd floor

**Abstract:**

This talk concerns local spacing statistics

which are used to quantify the randomness

of a given real-valued sequence.

We survey past and recent developments,

focusing on pair correlation statistics,

and explain the connection to (arithmetic) quantum chaos.

Furthermore, we report on joint work with Zeev Rudnick

about lacunary sequences.

#### Vorstellungsvortrag im Rahmen des Habilitationsverfahrens

**Title:**Linear differential equations and difference algebraic groups

**Speaker:**Dr. Michael Wibmer (TU Graz)

**Date:**24.01.2020, 14:00 Uhr

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

**Abstract:**

Classical differential Galois theory associates a linear algebraic group to a linear differential equation. This group measures the algebraic relations among the solutions. In joint work with L. Di Vizio and Ch. Hardouin I developed a Galois theory that measures the difference algebraic relations among the solutions of a linear differential equation. In this Galois theory the Galois groups are linear difference algebraic groups, i.e., subgroups of the general linear group defined by algebraic difference equations in the matrix entries. This Galois theory is helpful for understanding the behavior of the solutions under a transformation of the independent variable and also applies to linear differential equations depending on a parameter.

Because of their role in the study of linear differential equations it is desirable to have a comprehensive structure theory for linear difference algebraic groups. In this talk we will discuss some progress in this direction.

#### Vortrag

**Title:**Statistical methods and risk modeling in insurance companies

**Speaker:**Sanela Omerovic (FMA, Finanzmarktaufsicht Österreich)

**Date:**Freitag, 24. Jänner 2020, 15:15 Uhr

**Room:**SR für Statistik (NT03098), Kopernikusgasse 24/III

**Abstract:**

Abstract:

The implementation of Solvency II has introduced a set of risk-based principles for the solvency capital requirement (SCR) of insurance companies. Undertakings are therefore required to calculate their SCR by means of a holistic balance sheet approach. These requirements entail the use of sophisticated mathematical models. This presentation addresses different statistical methods and basic legal aspects for risk modeling in insurance companies.

#### Mathematisches Kolloquium

**Title:**Suprema in spectral spaces and density

**Speaker:**Carmelo A. Finocchiaro (University of Catania)

**Date:**17.01.2020, 15 Uhr c.t.

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

**Abstract:**

Let $X$ be a spectral space. As it is well known, the spectral topology of $X$ can be refined to another topology, the so called constructible topology}, which remains spectral and becomes Hausdorff. Since any spectral space is Kolmogoroff, $X$ admits a canonical partial order $\leq$, usually called the specialization order}, defined by setting, for any $x,y\in X$, $x\leq y$ if $y\in \overline{\{x\}}$. We will provide conditions for a subset $Y$ of the partially ordered set $(X,\leq)$ in order that the supremum of $Y$ (in $X$) exists and belongs to the closure of $Y$ in the constructible topology. Algebraic applications of such topological results will concern density properties of some spaces of rings and ideals. Moreover, we will provide topological characterizations of distinguished classes of domains, in terms of certain properties of their ideals. Joint work with D. Spirito.

#### Mathematisches Kolloquium

**Title:**BOUNDED EXPONENTIAL SUMS

**Speaker:**Reynold Fregoli (Royal Holloway University of London)

**Date:**17.01.2020, 14:00 Uhr

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

**Abstract:**

Let A $\subset$ $\mathbb{N}$, $\alpha$ $\in$ (0, 1), and for x $\in$ $\mathbb{R}$ let e(x) := e$^2$^\pi$$^i$$^x$. We set

S$_A$($\alpha$,N) := $\sum$ n$\in$A$ n$\le$N e(n$\alpha$)

Recently, Lambert A’Campo proposed the following question: is there an infinite non-cofinite set A $\subset$ $\mathbb{N}$ such that for all $\alpha$ $\in$ (0, 1) the sum S$_A$($\alpha$,N) has bounded modulus as N $\rightarrow$ +$\infty$? In this talk I will give an idea of why such sets do not exist. To show this, I use a theorem by Duffin and Schaeffer on complex power series. The above result can also be extended to prove that if the sum S$_A$($\alpha$,N) is bounded in modulus on an arbitrarily small interval and on the set of rational points, then the set A has to be either finite or cofinite. On the other hand, it can be shown that there are infinite non-cofinite sets A such that $\mid$S$_A$($\alpha$,N)$\mid$ is bounded for all $\alpha$ $\in$ $E$ $\subset$ (0, 1), where $E$ has full Hausdorff dimension and $\mathbb{Q}$ $\cap$ (0, 1) $\subset$ $E$.

#### Geometrie-Seminar

**Title:**Using multi-cover persistent homology as a fingerprint for periodic crystals

**Speaker:**Theresa Heiss (ISTA)

**Date:**Thursday 16.1.2020, 10:30

**Room:**Seminarraum 2, Kopernikusgasse 24/IV

**Abstract:**

As the atoms in periodic crystals are arranged periodically,

such a crystal can be modeled by a periodic point set, i.e. by the union

of several translates of a lattice. Two periodic point sets are considered

equivalent if there is a rigid motion from one to the other. A periodic

point set can be represented by a finite cutout s.t. copying this cutout

infinitely often in all directions yields the periodic point set. The fact

that these cutouts are not unique creates problems when working with them.

Therefore, material scientists would like to work with a complete,

continuous invariant instead. We conjecture that a variant of persistent

homology, namely the sequence of $k$-fold cover persistence diagrams for all

positive integers $k$, is such a complete, continuous invariant for

equivalence classes of periodic point sets

#### Mathematisches Kolloquium

**Title:**On the Skolem Problem for parametric families of linear recurrence sequences and some G.C.D. problems

**Speaker:**Alina Ostafe (The University of New South Wales)

**Date:**16.01.2020, 14:00 Uhr

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

**Abstract:**

In this talk we discuss a parametric version of the Skolem Problem about decidability of the existence of a zero in a linear recurrence sequence.

We then connect this problem to studying the greatest common divisor of two linear recurrence sequences of polynomials.

#### Seminar für Kombinatorik und Optimierung

**Title:**Cycles in random planar graphs

**Speaker:**Michael Missethan (TU Graz)

**Date:**Friday 10th January 14:15

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

**Abstract:**

In this talk we discuss recent results on largest and shortest cycles in random planar graphs. More precisely, we consider the following questions in random planar graphs. What is the order of the longest cycle? Is the longest cycle in the largest component? Is there even a threshold phenomenon such that all cycles longer than a certain value belong to the largest component and all shorter cycles lie outside? We use structural properties of a random planar graph (e.g. order of the complex part, core and kernel) to find answers to these questions. But also a version of the P\'{o}lya urn model plays a crucial role in our approach to obtain bounds on the cycle lengths.

This talk is based on joint work with Mihyun Kang.