### Talks in 2020

#### Strukturtheorie-Seminar

**Title:**Markovian linearization of random walks on groups

**Speaker:**Bastien Dubail (ENS Lyon)

**Date:**Thursday, 17.Dec.2020, 11:00 on time

**Room:**Webex virtual meeting 121 760 4652

**Abstract:**

In this talk I will present a new application of the so-called linearization trick, famous for its applications in various fields including but not limited to random matrices and operators algebras, to random walks on groups. It allows to study any finitely supported random walk by studying instead a nearest-neighbor « colored » random walk. Extending known formulas for the drift and entropy to colored walks we can obtain explicit expressions for finite range walks as well.

This work was done in collaboration with Charles Bordenave.

webex meeting, Thursday, 17 Dec, 2020, 10:40 (Rome, Stockholm, Vienna)

hosted by Franz Lehner

Meeting number: 121 760 4652

Password: r34M9ShVMfF

https://tugraz.webex.com/tugraz/j.php?MTID=m13059cfec1f6eee78faf0e3b3c003d67}

meeting to be opened at 10:40, the talk will start at 11:00

#### Zahlentheoretisches Kolloquium

**Title:**Manin's conjecture for surfaces and their symmetric squares

**Speaker:**Julian Lyczak (IST Austria)

**Date:**Freitag, 11.12.2020, 15:15

**Room:**online via webex

**Abstract:**

Manin's conjecture for surfaces and their symmetric squares

One topic in arithmetic geometry is the study of points on a variety over one fixed number field. This talk will be about the study of points over all quadratic extension of the base field simultaneously. For the study of rational points many techniques and conjectures are available. We can also apply these to the study of quadratic points by consider the symmetric square of the variety; any quadratic point on a variety is naturally a rational point on its symmetric square.

During the talk we will count points of bounded height on the symmetric square of some surfaces and compare these results with the results predicted by a class of conjectures first attributed to Manin. Other relevant conjectures we will encounter come from work of Batyrev, Peyre and Tschinkel. I will report on the successes in verifying these conjectures for specific surfaces and failures in trying to do so for a general del Pezzo surface.

This talk is based on joint work with Nils Gubela, and Francesca Balestrieri, Kevin Destagnol, Jennifer Park and Nick Rome.

Meeting number: 174 225 0835

Meeting password: zErKuTnF274

link:

https://tugraz.webex.com/tugraz/j.php?MTID=m96f40d003b6f5e095d22e21698ddb31d

#### Seminar für Kombinatorik und Optimierung

**Title:**Resilience for Hamiltonicity in random hypergraphs

**Speaker:**Olaf Parczyk (London School of Economics)

**Date:**Friday 11th December 14:15

**Room:**Online meeting (Webex)

**Abstract:**

Sudakov and Vu introduced the concept of local resilience of graphs for measuring robustness with respect to satisfying a given property.

A classical result of Dirac states that any subgraph $G$ of the complete graph $K_n$ of minimum degree $\delta(G) \ge \tfrac 12 n$ contains a Hamilton cycle.

In the binomial random graph $G(n,p)$ the threshold for the appearance of a Hamilton cycle is $p= \log n/n$.

Lee and Sudakov generalised Dirac’s result to random graphs by showing that with $p \ge C \log n/n$ asymptotically almost surely any subgraph $G$ of $G(n,p)$ with minimum degree $\delta(G) \ge (\tfrac 12 + \epsilon)n$ contains a Hamilton cycle, where $C$ depends only on $\epsilon >0$.

These kind of resilience problems in random graphs received a lot of attention.

In this talk we discuss a generalisation of the result of Lee and Sudakov to tight Hamilton cycles in random hypergraphs.

This is joint work with Peter Allen and Vincent Pfenninger.

Meeting link:

\[

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

\]

Meeting number:

137 149 1265

Password:

JYc3B3dunG2

#### Special invited colloquium

**Title:**Mathematical Philosophy

**Speaker:**Prof. DDr. Hannes Leitgeb (Universität München)

**Date:**Friday, 11.Dec.2020, 11:00 on time

**Room:**Webex virtual meeting

**Abstract:**

This years' `Discrete Mathematics Day' of the FWF Doctoral Programme `Discrete Mathematics' (TU + KFU Graz + MU Leoben) consists in a special invited colloquium.

Prof. DDr. Hannes Leitgeb is Chair and Head of the Munich Center for Mathematical Philosophy. He holds PhD degrees in Mathematics and Philosophy from the University of Salzburg. After carreer stations in Stanford and Bristol, in 2007 he became Professor of Mathematical Logic and Philosophy of Mathematics at LMU München. He is member of the Academia Europaea and of the Leopoldina (Deutsche Akademie der Naturforscher).

Abstract: Clearly, mathematical methods are of utmost importance in the sciences

and in technology. It is much less known, however, that mathematical methods

have also turned out to be crucial in philosophy. This talk will explain why

this is so, it will present some examples, and it will argue that the future of

philosophy is mathematical (amongst others).

CORRECTION: webex meeting, Friday, 11 Dec, 2020, 10:30 (Rome, Stockholm, Vienna)

hosted by Christian Lindorfer

Meeting number: 174 770 3676 ‚ Password: f7EmTY7PBs3

https://tugraz.webex.com/tugraz/j.php?MTID=m036d2f4e854cea9c3ff0a5833f8728da

meeting to be opened at 10:30, the talk will start at 11:00

#### Strukturtheorie-Seminar

**Title:**Random walks on dense subgroups

**Speaker:**Dr. Hanna Oppelmayer (TU Graz)

**Date:**Thursday, 10.Dec.2020, 11:00 on time

**Room:**Webex meeting

**Abstract:**

We study the relation between the Poisson boundary of certain random

walks on a countable group and on a totally disconnected, locally

compact (tdlc) group which densely contains the countable one. In

particular we will find examples where they agree, and the tdlc group

acts transitively on its Poisson boundary. Transitive actions are

easier to understand and we thus can conclude several structural

properties in these examples, like primeness and reducibility of the

quasi-regular representation (which disproves a conjecture by

Bader-Muchnik).

This talk aims to be introductory, so no pre-knowledge on Poisson

boundaries is assumed.

webex meeting, Thursday, 10 Dec, 2020, 10:40 (Rome, Stockholm, Vienna)

Meeting number: 121 901 3211 ‚ Password: y5uSFEP9yi3

https://tugraz.webex.com/tugraz/j.php?MTID=me5d9bf8f0e91198d098690c30de2e76b

meeting to be opened at 10:40, the talk will start at 11:00

#### Seminar für Kombinatorik und Optimierung

**Title:**Edit distances to graphs with no induced cycles of a set length

**Speaker:**Richard Mycroft (University of Birmingham)

**Date:**Friday 4th December 13:30

**Room:**Online meeting (Webex)

**Abstract:**

Given an instance of the random graph $G_{n, p}$, how many edges must be

edited (i.e. added or removed) to obtain a graph with no induced $C_h$

(cycle of length $h$)? This seems a natural question by itself, but

additional interest is added by a theorem of Balogh and Martin, which

implies that $G_{n, p}$ asymptotically maximises the number of edge

alterations needed over all graphs on $n$ vertices with density close to

$p$.

Martin and Peck answered the above question for not-too-small $p$, in

particular finding the asymptotic maximum number of edits required over

all graphs on $n$ vertices. In this talk I will explain some of the

methods that can be used to address this problem and show how these can

be used to extend the range of $p$ for which the answer is known.

This is joint work with Amarja Kathapurkar.

{\bf Please note the unusual start time. }

Meeting link:

\[

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

\]

Meeting number:

137 149 1265

Password:

JYc3B3dunG2

#### Strukturtheorie-Seminar

**Title:**Patterns in sets of positive density on trees and buildings

**Speaker:**Prof. James Parkinson (University of Sydney)

**Date:**Thursday, 3.Dec.2020, 11:00 on time

**Room:**Webex virtual meeting

**Abstract:**

We prove an analogue for homogeneous trees and affine buildings of a result of Bourgain on geometric Ramsey theory in Euclidean spaces. In particular, we show that certain configurations of vertices are guaranteed to exist in any set of positive upper density in a homogeneous tree or affine building. This is joint work with M. Björklund and A. Fish.

webex meeting, Thursday, 3 Dec, 2020, 10:30 (Rome, Stockholm, Vienna)

Meeting number: 174 752 2504 ‚ Password: GPenhDZy386

https://tugraz.webex.com/tugraz/j.php?MTID=m6077a440f57464ead778055b114740c0

meeting to be opened at 10:30, the talk will start at 11:00

#### Seminar für Kombinatorik und Optimierung

**Title:**Stability from symmetrisation arguments

**Speaker:**Oleg Pikhurko (University of Warwick)

**Date:**Friday 27th November 14:15

**Room:**Online meeting (Webex)

**Abstract:**

We present a sufficient condition for a strong version of a stability property of extremal graph problems that can be solved via Zykov's symmetrisation. Our criterion is stated in terms of a limit version of the problem. We present some of its application to the inducibility problem where one maximises the number of induced copies of a given complete partite graph $F$ in an $n$-vertex graph.

This is joint work with Hong Liu, Maryam Sharifzadeh and Katherine Staden.

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:**A solution to Erdős and Hajnal's odd cycle problem

**Speaker:**Richard Montgomery (University of Birmingham)

**Date:**Friday 20th November 14:15

**Room:**Online meeting (Webex)

**Abstract:**

I will discuss how to construct cycles of many different lengths in graphs, in particular answering the following two problems on odd and even cycles. Erdős and Hajnal asked in 1981 whether the sum of the reciprocals of the odd cycle lengths in a graph diverges as the chromatic number increases, while, in 1984, Erdős asked whether there is a constant C such that every graph with average degree at least C contains a cycle whose length is a power of 2.

This is joint work with Hong Liu.

Meeting link:

\[

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

\]

Meeting number:

137 149 1265

Password:

JYc3B3dunG2

#### Strukturtheorie-Seminar

**Title:**The geometry of branching random walk at the recurrence threshold

**Speaker:**Dr. Tom Hutchcroft (University of Cambridge)

**Date:**Thursday, 19.Nov.2020, 11:00 on time

**Room:**Webex virtual meeting

**Abstract:**

Let G be a Cayley graph of a finitely generated group. Branching

random walk on G is a Markov process in which a cloud of particles evolves

by each particle randomly splitting into a collection of new particles, each

of which immediately performs a random walk step. This process has a

positive probability to survive forever whenever the mean of the offspring

distribution is strictly larger than 1. When G is nonamenable, an

interesting second phase transition occurs when the mean offspring is equal

to the reciprocal of G's spectral radius, rho: when the mean is between 1

and 1/rho the branching random walk survives forever with positive

probability but visits each vertex at most finitely often, while when the

mean is strictly larger than 1/rho every vertex is visited infinitely often

with positive probability.

We study the geometry of the set of vertices visited by the branching random

walk when the offspring mean is exactly 1/rho. We prove that this set is

always tree-like in the sense that it has infinitely many ends and no

isolated ends almost surely, confirming a conjecture of Benjamini and

Müller: this is essentially equivalent to the statement that two independent

branching random walks intersect at most finitely often almost surely. The

proof is based on an interesting combinatorial fact about trees known as the

magic lemma.

webex meeting, Thursday, 19 Nov, 2020 10:30 (Rome, Stockholm, Vienna)

Meeting number: 174 741 1222 , Password: vvFwVVWq522

https://tugraz.webex.com/tugraz/j.php?MTID=me740164504cd7dd8f029561a24e46b8d

meeting to be opened at 10:30, the talk will start at 11:00

#### Lehrprobe im Rahmen des Habilitationsverfahrens

**Title:**Färbungen und die Symmetriegruppe des Würfels

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

**Date:**Freitag, 13.11.2020, 13:00 Uhr

**Room:**Webex Meeting

**Abstract:**

Meeting-Link:

https://tugraz.webex.com/tugraz-de/j.php?MTID=mfb1bbed334c4aaef1c50084da208ddc9

Meeting-Kennnummer:

174 076 1584

Passwort:

CJgM3nZU3m6

#### Vorstellungsvortrag im Rahmen eines Habilitationsverfahrens

**Title:**Tree structure in combinatorial objects

**Speaker:**Joshua Erde (TU Graz, Institut für Diskrete Mathematik)

**Date:**Friday 13th November 11:00

**Room:**Online meeting (Webex)

**Abstract:**

A tree-decomposition is broadly an attempt to display the global structure of an object in a tree-like fashion, by splitting it into smaller substructures, whose relative structure can be modelled as a tree. The use of tree-decompositions in graph theory was pioneered by Robertson and Seymour as part of their work on Wagner's conjecture, and since then these ideas have been generalised in many different settings to other types of tree-decompositions of graphs and other combinatorial structures. Recently, a unified abstract framework, called separation systems, has been developed, within which many of these results can be expressed.

I will talk about some of my work on the theory of tree-structure in separation systems, and its application to problems in various combinatorial objects.

Meeting link:

\[

\text{https://tugraz.webex.com/tugraz/j.php?MTID=mfb7c5958c15df928aa27f1296d6264c3}

\]

Meeting number (access code): 174 286 6219

Meeting password: hfJHu2sEV83

#### Seminar für Kombinatorik und Optimierung

**Title:**Distribution of tree parameters via martingale Central Limit Theorem

**Speaker:**Mikhail Isaev (Monash University)

**Date:**Friday 13th November 14:15

**Room:**Online meeting (Webex)

**Abstract:**

Tree parameters, like pattern or symmetry counts, have been extensively studied in the literature for various random models. It is known, in particular, that many important examples exhibit normal or log-normal limit law. A typical approach relies on the recursive nature of trees and properties of its generating functions. We propose a purely probabilistic argument based on the general theory of Central Limit Theorems for martingales. For uniform random labelled tree, we find the limiting distribution of tree parameters which are stable (in some sense) with respect to local perturbations of a tree structure. Our results are general enough to get the asymptotical normality of the number of occurrences of any given small pattern and the asymptotical log-normality of the number of automorphisms. More details can be found at https://arxiv.org/abs/1912.09838.

Meeting link:

\[

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

\]

Meeting number:

137 149 1265

Password:

JYc3B3dunG2

#### Strukturtheorie-Seminar

**Title:**Star automaton groups and graphs

**Speaker:**Doz. Dr. Daniele D'Angeli (Univ. Cusano, Rome)

**Date:**Thursday, 12.Nov.2020, 11:00 on time

**Room:**Webex virtual meeting

**Abstract:**

Automaton groups act by automorphisms on rooted regular trees and give rise to examples of groups with interesting properties: groups with intermediate growth, non elementary amenable groups, Burnside groups etc.

In this talk I will explain a new construction to obtain automaton groups from finite graphs. In case the initial graph is a star we are able to classify the corresponding Schreier graph and describe explicitely the spectrum. We get an uncountable family of infinite graphs whose spectrum is a Cantor set.

webex meeting

Thursday, 12 Nov, 2020 10:30 (Rome, Stockholm, Vienna)

Meeting number: 174 528 8380

Password: gnXUfXfC698

https://tugraz.webex.com/tugraz-de/j.php?MTID=m0ef97d04285f42a6e8b6d91bc5cd7be8

[tugraz.webex.com]

meeting to be opened at 10:30, the talk will start at 11:00

#### Seminar für Kombinatorik und Optimierung

**Title:**Flip processes on finite graph and dynamical systems they induce on graphons

**Speaker:**Jan Hladky (Institute of Mathematics of the Czech Academy of Sciences)

**Date:**Friday 6th November 14:15

**Room:**Online meeting (Webex)

**Abstract:**

We introduce a class of random graph processes, which we call flip processes}. Each such process is given by a rule} which is just a function $\mathcal{R}:\mathcal{H}_k\rightarrow \mathcal{H}_k$ from all labelled $k$-vertex graphs into itself ($k$ is fixed). Now, the process starts with a given $n$-vertex graph $G_0$. In each step, the graph $G_i$ is obtained by sampling $k$ random vertices $v_1,\ldots,v_k$ of $G_{i-1}$ and replacing the induced graph $G_{i-1}[v_1,\ldots,v_k]$ by $\mathcal{R}(G_{i-1}[v_1,\ldots,v_k])$. This class contains several previously studied processes including the Erd\H{o}s--R\'enyi random graph process and the random triangle removal.

Given a flip processes with a rule $\mathcal{R}$ we construct time-indexed trajectories $\Phi:\mathcal{W}\times [0,\infty)\rightarrow\mathcal{W}$ in the space of graphons. We prove that with high probability, starting with a large finite graph $G_0$ which is close to a graphon $W_0$, the flip process will follow the trajectory $(\Phi(W_0,t))_{t=0}^\infty$ (with appropriate rescaling of the time).

These graphon trajectories are then studied from the perspective of dynamical systems. We prove that two trajectories cannot form a confluence, give an example of a process with an oscilatory trajectory, and study stability and instability of fixed points.

Joint work with Frederik Garbe, Matas Sileikis and Fiona Skerman.

Meeting link:

\[

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

\]

Meeting number:

137 149 1265

Password:

JYc3B3dunG2

#### Strukturtheorie-Seminar

**Title:**Limit behaviour of random walks on Schreier and Cayley graphs

**Speaker:**Dr. Bogdan Stankov (ENS Paris)

**Date:**Thursday, 5.Nov.2020, 11:00 on time

**Room:**Webex virtual meeting

**Abstract:**

The Poisson boundary is a probability space that encodes the limit behaviour of random walks. It is known that a group is amenable if and only if there exists a

non-degenerate measure such that the random walk on its Cayley graph has trivial

Poisson boundary. When a group acts on a space, the Poisson boundary of the

induced walk on the Schreier graph is a quotient of the Poisson boundary of the

random walk on the Cayley graph. We discuss results around the non-triviality of

the Poisson boundary of the induced walk on the Schreier graph under the

hypothesis of a measure with finite first moment. We apply it to Thompson's

group F, which extends a result by Kaimanovich about finitely supported

measures. We also adapt a similar approach to prove non-triviality of Poisson

boundary on subgroups that are not locally solvable of a group H(Z) of

piecewise projective homeomorphisms. Monod studied the class of groups H(A) of

piecewise projective homeomorphisms where A is a subring of the real numbers,

and proved that unless A=Z (the integers), H(A) is non-amenable without free

subgroup.

webex meeting

Thursday, 5 Nov, 2020 10:30 (Rome, Stockholm, Vienna)

Meeting number: 137 699 7178

Password: wZrkf6e7rE7

https://tugraz.webex.com/tugraz-de/j.php?MTID=mc01f97d96751dac6cea676386f5d8f07

[tugraz.webex.com]

meeting to be opened at 10:30, the talk will start at 11:00

#### 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

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#### 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.

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#### 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)

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Meeting password: dT6yWAuqu64

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#### 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.