Discrete Mathematics 2010-2022

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

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

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:
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\]

Meeting number:
137 149 1265

Password:
JYc3B3dunG2

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

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

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

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

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:
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\]

Meeting number:
137 149 1265

Password:
JYc3B3dunG2

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

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

Meeting-Link:
https://tugraz.webex.com/tugraz-de/j.php?MTID=mfb1bbed334c4aaef1c50084da208ddc9
Meeting-Kennnummer:
174 076 1584
Passwort:
CJgM3nZU3m6

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

Meeting number (access code): 174 286 6219

Meeting password: hfJHu2sEV83

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

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

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

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

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.

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

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

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.   

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)

\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}

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:
\[
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\]

Meeting number:
137 149 1265

Password:
JYc3B3dunG2

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

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.

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.

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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Webex meeting info

Meeting number (access code): 137 382 3722

Meeting password: 8diN3Pd9q34


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

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}

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}

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
 

Der Link fuer den Vortrag wird rechtzeitig bekanntgegeben.

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.

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.

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.

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.

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

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.

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.

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.

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.

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) $\subset$ $E$.

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

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.

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.