Discrete Mathematics 2010-2022

10:00 Uhr: Petr Siegl: Energy decay for the wave equation with a damping coefficient unbounded at infinity

10:45 Uhr: Johannes Ladreiter: Schrödinger operators with oblique transmission conditions and complex interaction strenghts in $\mathbb{R}^2$

13:30 Uhr: Nicolas Weber: Eigenvalues of the Schrödinger operator in 2D with a constant complex-valued singular potential supported on the unit circle

14:15 Uhr: Peter Schlosser: Quaternionic Hilbert spaces and the S-spectrum

We show that, for every $n$ and every surface $\Sigma$ (orientable or not), there is a graph $U$
embeddable on $\Sigma$ with at most $c n^2$ vertices that contains as minor every graph
embeddable on $\Sigma$ with $n$ vertices. The constant $c$ depends polynomially on the Euler
genus of $\Sigma$. This generalizes a well-known result for planar graphs due to Robertson, Seymour, and Thomas (``Quickly excluding a planar
graph'', J. Comb. Theory B, 1994) which states that the square grid on $4n^2$ vertices contains as minor every planar graph with $n$ vertices.

This is a joint work with Cyril Gavoille.

We prove that the multivariate independence polynomial of any hypergraph of maximum degree $\Delta$ has no zeroes on the complex polydisc of radius $\sim\frac{1}{e\Delta}$, centered at the origin. Up to logarithmic factors in $\Delta$, the result is optimal, even for graphs with all edge sizes greater than $2$. As a corollary, we get an FPTAS for approximating the independence polynomial in this region of the complex plane.
We furthermore prove the corresponding radius for the $k$-uniform linear hypertrees is $\Omega(\Delta^{-1/(k-1)})$, a significant discrepancy from the graph case.
Joint work with David Galvin, Gwen McKinley, Will Perkins and Prasad Tetali.


Meeting link:
\[
\text{https://tugraz.webex.com/tugraz/j.php?MTID=m313bdafcc0bb1e027dd43c65b06c1634}
\]

Abstract: Instead of solving the eigenvalue problem Tf=zf for a linear operator
T and eigenvalue z, we can use a multiplier B and instead solve the linear pencil problem BTf=zBf. This leads us to study the numerical range and essential numerical range of linear pencils. The essential numerical range is used to describe the set of spectral pollution when approximating the eigenvalue problem by projection and truncation methods. By taking intersection over various multipliers, we get sharp enclosures. We apply the results to various differential operators. This talk is based on joint work with Marco Marletta (Cardiff).

I will present results for the exact and asymptotic enumeration of generic rectangulations (i.e., tilings of a rectangle by rectangles, with no point where $4$ rectangles meet, considered under equivalence relation in a weak or strong form). These can be set in correspondence to models of decorated planar maps (bipolar orientations in the weak form, transversal structures in the strong form) that themselves can be encoded by certain quadrant walks via a bijection due to Kenyon, Miller, Sheffield and Wilson. I will also mention an extension of the results to non-generic rectangulations, which yields a continuum of models of random planar lattices that get closer to a regular lattice.

Joint work with Erkan Narmanli and Gilles Schaeffer

Meeting link:
\[
\text{https://tugraz.webex.com/tugraz/j.php?MTID=mab523a645de428d5301998280dc510ed}
\]

To analyse discrete time series, Bandt introduced a statistic called ``permutation entropy'', which is based on permutation patterns that arise consecutively in time. Inspired by this work, we introduce a new Hopf algebra on vincular permutation patterns, i.e. permutation patterns where it is further specified whether values need to appear consecutively or can be arbitrarily far apart. We encode vincular permutation patterns as pairs of finite interval partitions and permutations of equal length. The underlying blocks of this Hopf algebra are two Hopf algebras: one on interval partitions and another one on permutations. The Hopf algebra on permutations is the superinfiltration Hopf algebra introduced by Vargas. We show that the product and coproduct of this new Hopf algebra behave well with respect to linear functionals which encode occurrences of vincular patterns. This is based on the preprint https://arxiv.org/abs/2306.17800 [arxiv.org], written together with Joscha Diehl.

The application of shape analytics - topology, geometry, and graph theory- has proven its capacity for attaining new understanding of the brain or for designing new algorithms for artificial intelligence. With it, researchers have uncovered internal representations called neural manifolds in brain regions like the hippocampus, or have described the axonal morphologies.
Other efforts revolve around using shape analytics to probe how deep neural networks learn the weight distributions, or inversely, to use deep neural networks called geometric deep learning to encode the geometric structure of data. Yet, the connection between biological neural computing
and today’s artificial neural networks is quite loose, in that the models and algorithms implemented lack biological plausibility, thereby hindering explanations for the mechanisms underlying neural computation. My proposal is to study spiking neural networks, a more biologically plausible model which opens up the possibility to explain how the brain hierarchically computes information and advances artificial intelligence via neuromorphic computing. Additionally, I propose formulating questions from a systems engineering perspective, considering why a certain shape arises from the system’s constraints in terms of
trade-offs and design principles optimized for. In an aspiration for synthesis, preliminary works are presented showing how to leverage the shapes of (spiking) neural networks and their emergent dynamics to hypothesize mechanistic explanations of how neural computation works.

A long-standing conjecture of Thomassen from the 80's states that every graph with sufficiently high average degree contains a subgraph with high girth and still preserving large enough average degree. This conjecture has only been resolved in the early 2000's by Kühn and Osthus in the first non-trivial case i.e. they showed that for every $k$, there is $f(k)$ such the every graph with average degree at least $f(k)$ contains a subgraph which is $C_4$-free with average degree $k$.

We will talk about a recent result which strengthens this result of Kühn and Osthus in two ways. First, we prove an analogous induced version and secondly we give much better bounds for the function $f$ allowing us obtain few non-trivial results as simple corollaries. Finally, we use these methods to confirm a conjecture Bonamy et al.

The talk is based on two joint works; One with Du, Scott, Hunter and McCarty and the other with Hunter.

Meeting link:
\[
\text{https://tugraz.webex.com/tugraz/j.php?MTID=mab523a645de428d5301998280dc510ed}
\]

Like algebraic geometry is the geometry defined by algebraic equations, difference algebraic geometry is the geometry defined by algebraic difference equations. In this talk, we will discuss some recent interactions between difference algebraic geometry and certain problems in number theory.

I will address the following questions:[5mm]
\textbf {a.} Why stochastic analysis, isn't classical analysis enough?[2mm]
\noindent\textbf {b.} What is a rough path and how does the rough approach to stochastic analysis improve upon the Ito approach?[2mm]
\noindent\textbf {c.} How does harmonic analysis help rough analysis and how am I involved?

First, I give a brief overview about planar maps of the
disk, and what was previously known on those decorated with a
statistical physics model. The main focus of the talk will be on the
Ising model. As in my joint works with Linxiao Chen, we start from a
purely combinatorial problem of random planar triangulations of the disk
coupled with the Ising model with Dobrushin boundary conditions and at a
fixed temperature (and without external magnetic field). We identify
rigorously a phase transition by analysing the critical behaviour of the
partition functions of a large disk at and around the critical point.
Moreover, we study the random geometric implications of this, in
particular by constructing a local limit when the disk perimeter tends
to infinity. At the critical temperature, we also find some explicit
scaling limits of observables related to the interface lengths. The two
key techniques in use are singularity analysis of rational
parametrizations of generating functions, as well as a peeling process
following the Ising interface. At the end of the talk and time
permitting, I will explain how some of the ideas can be adapted to study
random maps of the disk decorated with $O(n)$ loop models (where rational
parametrizations do not necessarily exist)

Meeting link:
\[
\text{https://tugraz.webex.com/tugraz/j.php?MTID=mab523a645de428d5301998280dc510ed}
\]

Understanding solutions of Diophantine equations over a number field is one of the main problems of number theory. By the help of the modular techniques used in the proof of Fermat’s last theorem by Wiles and its generalizations, it is possible to solve other Diophantine equations too.
Understanding quadratic points on the classical modular curve also plays a role in this approach. In this talk, I will mention some recent results about these notions for the classical Fermat equation as well as some other Diophantine equations.

In this talk, we will consider the Dirac operator with infinite mass boundary conditions on a tubular neighbourhood of a smooth compact hypersurface in $\mathbb{R}^n$
without boundary. We will discuss the asymptotic behaviour of the eigenvalues of this Dirac operator when the tubular neighbourhood shrinks to the hypersurface. It turns out that this asymptotic behaviour is driven by a Schrödinger operator on the hypersurface involving electric and Yang-Mills potentials of geometric nature. The eigenvalues of the effective Schrödinger operator appear in the third term of the asymptotic expansion with respect to the thickness of the tubular neighbourhood. The analysis relies on the min-max principle applied to the square of the Dirac operator. The main difficulty in this analysis is that the boundary condition for the transverse operator depends on the direction of the normal to the hypersurface. These results are obtained in collaboration with Thomas Ourmières-Bonafos.

Diophantine geometry is a modern-day incarnation of mathematicians' perennial interest in solving algebraic equations in integers. Its fundamental idea is to study the geometric objects defined by algebraic equations in order to understand their integral solutions ("geometry determines arithmetic"). One of its major achievements, Faltings' theorem, states that a large class of algebraic equations has only finitely many primitive integral solutions, namely those associated with smooth, proper curves of genus $> 1$. In the last few years, work by Dimitrov, Gao, Habegger, and myself has led to rather "uniform" bounds on the number of these solutions. Even more, our techniques yield purely geometric statements (uniform Manin-Mumford conjecture in characteristic 0). I will conclude with an optimistic outlook towards more general unlikely intersection problems in the framework of the conjectures of Zilber and Pink.

Let K be a number field. We consider the multiplicative group S of elements of K that are mapped to the unit circle in the complex plane under each embedding of K. The group S characterizes CM-fields, and further, if K is a CM-field then S is the kernel of a canonical norm map. Petersen and Sinclair studied the distribution of S on the unit circle when K is imaginary quadratic. We present a result that generalizes their result to arbitrary CM-fields. As an outlook we consider a new type of unit equation. All this is part of ongoing joint work with Shabnam Akhtari and Jeffrey Vaaler.

Der Kategoriensatz von Baire ist ein fundamentales Resultat der Topologie und bildet die Grundlage für zahlreiche Resultate der Funktionalanalysis. Er besagt, dass (a) jeder vollständige pseudometrische Raum und (b) jeder lokal kompakte reguläre Raum ein Baire-Raum ist. Hierbei heißt ein Raum Baire-Raum, falls jeder abzählbare Durchschnitt offener und dichter Mengen ebenfalls dicht ist. Nach einem Beweis dieses Satzes werden wir einige seiner Folgerungen skizzieren.

Given an inhomogeneous linear differential equation on a domain, it is a standard method to solve it using Green's function. This also works for solving Poisson's equation on a Riemannian manifold. When the domain is furthermore a moduli space, then special values of Green's functions could have arithmetic applications. In this talk, we will consider the case where the domain is the moduli space of isomorphism classes of elliptic curves over the complex numbers, and give some recent results about special values of Green's functions.

I will start with an introduction to moduli spaces of Higgs bundles and then explain some results on these spaces which are motivated by mirror symmetry.

In this lecture, I will present some of my main results on moments of L-functions, an important subject in number theory.
If time permits, I intend to discuss some aspects of my recent collaborative work with Bergström, Petersen, and Westerland, aiming to understand the moments of quadratic L-functions in the rational function field setting.

Energy and discrepancy are two different, yet closely related, ways to quantify the quality of a discrete point distribution. In the former, points are treated as particles which interact according to a certain potential, while the discrepancy compares the actual and expected number of points within a certain class of test sets (e.g., spherical caps). We shall talk about some recent results on these topics. In particular, we shall be interested in the clustering effect (minimization of certain energy integrals yields discrete rather than uniform measures), the associated phase transitions, and relations to some problems and conjectures of discrete geometry, such as the Fejes Toth problem on the sum of angles between N lines.

\section*{Programm:}
\begin{description}

\item{10:00-11:00} Peter Kritzer (Österreichische Akademie der Wissenschaften, Linz)

Quasi-Monte Carlo using Points with Non-Negative Local Discrepancy}

\item{11:00-12:00} Christoph Aistleitner (TU Graz)

Point Configurations in the Cube and on the Sphere}

\item{12:00-14:00} Gemeinsames Mittagessen

\item{14:00-15:00} Martin Ehler (Universität Wien)

t-Design Curves}

\item{15:00-16:00} Josef Dick (University of New South Wales, Sydney)

Low-Discrepancy Points on the Sphere}

\end{description}

The graph reconstruction conjecture states that each graph $G$ on at least $3$ vertices can be reconstructed from the multiset of all induced subgraphs on $n-1$ vertices. Although this conjecture is notoriously difficult, it is easy to reconstruct some information about the graph, e.g. the degree sequence of $G$ and whether $G$ is connected. We study a set-up with smaller induced subgraphs ('small cards'). Using a theorem about the number of positive real roots of a complex polynomial, we show the induced subgraphs on $0.9n$ vertices determine whether an $n$-vertex graph is connected and we also develop counting techniques to reconstruct trees from subgraphs of size about $8n/9$.

This is based on joint work with Tom Johnston, Alex Scott and Jane Tan.

Meeting link:
\[
\text{https://tugraz.webex.com/tugraz/j.php?MTID=mab523a645de428d5301998280dc510ed}
\]

Bootstrap percolation} is a spreading process on graphs where starting with an initial set $A$ of infected vertices, other vertices become infected once a certain threshold $r$ of their neighbors are infected. We say that the set $A$ percolates if eventually all vertices of the graph are infected. When $r = d(v)/2$, where $d(v)$ is the degree of a vertex, that is, when a vertex becomes infected once half of its neighbors are already infected, we call the process majority bootstrap percolation}.

We analyse this process when the initial set $A$ is chosen randomly, with each vertex being assigned to $A_p$ with a fixed probability $p$ independently, which in some way represents the ‘typical’ behaviour for sets of density $p$. Balogh, Bollob\'{a}s and Morris considered this process on the hypercube, a well-studied geometric graph, and demonstrated a sharp threshold for the property that the set $A_p$ percolates. We determine a similar sharp threshold for a broader class of product graphs}, those arising as the cartesian product of many graphs of fixed order.

Joint work with Mauricio Collares, Joshua Erde and Mihyun Kang.

In a series of papers with J\'ulia Komj\'athy and Dieter Mitsche, we uncover the deep relation between three types of connected components in a large class of supercritical spatial random graphs that includes long-range percolation, geometric inhomogeneous random graphs, and the age-dependent random connection model. Let $\mathcal{C}_n^\sss{(1)}$ and $\mathcal{C}_n^\sss{(2)}$ denote the largest and second-largest component in the graph restricted to a $d$-dimensional box/torus of volume $n$, and let $\mathcal{C}(0)$ be the component in the infinite graph that contains a vertex at the origin. We identify $\zeta\in(0,1)$ such that
\begin{align}
\Prob\big(|\mathcal{C}_n^\sss{(1)}|\le (1-\varepsilon)\E\big[|\mathcal{C}_n^\sss{(1)}|\big]\big)=\exp\big(-\Theta\big(n^{\zeta}\big)\big),\label{eq:ltld}
|\mathcal{C}_n^\sss{(2)}|\, \big/\, (\log n)^{1/\zeta}=\Theta_{\zeta}(1),\nonumber
\Prob\big(n\le |\mathcal{C}(0)|<\infty\big)=\exp\big(-\Theta\big(n^{\zeta}\big)\big),\nonumber
\end{align}
as $n$ tends to infinity.
In words, there is a single exponent $\zeta$ (depending only on a few parameters of the model) that is guiding the speed of the lower tail of large deviations for the giant, the size of the second largest component, and the decay of the finite cluster size distribution of the origin.
During the talk, I will explain intuition about the relation between the three quantities through the lens of an example. Time permitting, I discuss also the upper tail of large deviations of the giant component, i.e., $\Prob\big(|\mathcal{C}_n^\sss{(1)}|\ge (1+\varepsilon)\E\big[|\mathcal{C}_n^\sss{(1)}|\big]\big)$, and show that it behaves drastically different compared to the lower tail in~\eqref{eq:ltld}.

Set-valued tableaux are a generalization of young tableaux in which the cells of the tableaux are filled with nonempty sets of integers. These tableaux originated in the study of the $K$-theory of the Grassmannian, and they also have connections to Brill-Noether theory and to dynamical algebraic combinatorics. In this talk I will present some recent work with Sam Hopkins (Howard University) and Svante Linusson (KTH) which is motivated by the connection to combinatorial dynamical systems. Using probabilistic reasoning, we derive elegant product formulas for the $q$-enumeration of "barely set-valued" tableaux (that is, tableaux with exactly one extra element). Time permitting, I will also discuss some ongoing work with Linusson in which we extend our results to arbitrary set-valued tableaux of certain special shapes.

Meeting link:
\[
\text{https://tugraz.webex.com/tugraz/j.php?MTID=m4b9e22ad5fa0dc60554bf1575845ee90}
\]

In 1995, Kuperberg introduced a remarkable collection of trivalent web bases which encode tensor invariants of $SL_3$. Extending these bases to general $SL_r$ has been an open problem ever since. We present a solution to the $r=4$ case by introducing a new generalization of Postnikov's plabic graphs. In this talk I will mostly focus on the beautiful combinatorics of these graphs and their relation to rectangular standard Young tableaux.

This is joint work with Christian Gaetz, Oliver Pechenik, Jessica Striker and Joshua Swanson.

We show that, for $k \geq 3$ and $n$ divisible by $k$, if a $k$-uniform hypergraph $H$ on $n$ vertices has large enough minimum $(k-1)$-degree to guarantee a perfect matching, then asymptotically almost surely a $p$-random subhypergraph of $H$ also contains a perfect matching, provided that $p > \frac{C \log n}{ n^{k-1}}$. Our result strengthens Johansson, Kahn, and Vu's seminal solution to Shamir's problem and can be viewed as a 'robust' version of a hypergraph Dirac-type result by Rödl, Ruciński, and Szemerédi.

This is joint work with Dong Yeap Kang, Tom Kelly, Daniela Kühn, and Deryk Osthus.

Meeting link:
\[
\text{https://tugraz.webex.com/tugraz/j.php?MTID=mb8d3461791088d2393546b3e2a9d4d8d}
\]

Abstract:
I will offer an invitation to spectral geometry of metric graphs. Several bounds on the spectral gap of the Laplacian on metric graphs with standard or Dirichlet vertex conditions will be derived. I will especially present estimates based on recently introdcuced metric quantities, such as the length of a shortest cycle (girth), the avoidance diameter, the torsional rigidity and the mean distance. Using known results about Ramanujan graphs, a class of expander graphs, we also prove that some of these metric quantities, or combinations thereof, do not deliver any spectral bounds with the correct scaling.
If time allows, I will briefly discuss the case of graphs with infintely many edges.

In this talk, we have structured our presentation into two distinct parts.
First, we provide a concise review of the literature on Dirac's type problems for uniform hypergraphs,
summarizing results and methodologies.
In the second part, we delve into our ongoing research,
which focuses on the emergence of loose Hamilton cycles within subgraphs of random uniform hypergraphs.
Here, we outline our strategy and discuss the limitations of our proof.
Our main result states that:
the minimum $d$-degree threshold for loose Hamiltonicity relative to the
random $k$-uniform hypergraph $\mathbf{G}^{(k)}(n,p)$ coincides with
its dense analogue whenever $p$ exceeds $n^{- (k-1)/2+o(1)}$ (and the minimum $d$-degree threshold $\Omega(n^{-(k-d)}\log{n}))$.

Meeting link:
\[
\text{https://tugraz.webex.com/tugraz/j.php?MTID=m000462befb8d64231d5ee72ca1ff5e9e}
\]

The Bombieri-Vinogradov theorem asserts that for $q< x^{1/2} \log(x)^{-A}$, the number of primes $p < x$ with $p \equiv a$ mod $q$ is on average very close to $\pi(x)/\phi(q)$. An extension of this result to $q > x^{1/2}$ is a longstanding open problem. Following Zhang's breakthrough result on bounded gaps between primes~[1], attention turned towards moduli $q$ which are $x^\delta$-smooth. Polymath~[2] showed that the primes have exponent of distribution $0.52\dot{3}$ to smooth moduli.

Following the arguments of Polymath~[3], in this talk we sketch how equidistribution estimates for primes in arithmetic progressions to smooth moduli can be used to give bounds on $H_m = \liminf_{n \rightarrow \infty} (p_{n+m}-p_n)$. We also show how an alternative version of the $q$-van der Corput method can be used to improve the exponent of distribution to smooth moduli to $0.525$.

tba

Die Künstliche Intelligenz und Maschinelles Lernen haben in den letzten
Jahrzehnten unglaubliche Fortschritte erfahren, sei es in der Bilderkennung,
der Sprachverarbeitung oder der personalisierten Werbung. Auch in der
Wissenschaft konnten dank dieser Fortschritte in den letzten Jahren
bedeutende Durchbrüche erzielt werden, zum Beispiel in der Simulation der
Proteinfaltung. In diesem Vortrag werden wir die Rolle der Mathematik in der
Künstlichen Intelligenz erörtern, sowie einige grundlegenden mathematischen
Resultate besprechen.

We report on recent progress on the problem of building a stochastic process
that admits the hypothetical Yang-Mills measure as its invariant measure.
One interesting feature of our construction is that it preserves
gauge-covariance in the limit even though it is broken by our UV
regularisation. This is based on joint work with Ajay Chandra, Ilya
Chevyrev, and Hao Shen.

Given $n$ points in the plane, how many pairs among these points can have distance exactly $1$? More formally, what is the maximum possible number of unit distances among a set of $n$ points in the plane? This problem is a very famous and still largely open problem, called the Erd\H{o}s unit distance problem. One can also study this problem for other norms on $\mathbb{R}^2$ (or more generally on $\mathbb{R}^d$ for any dimension $d$) that are different from the Euclidean norm. This direction has been suggested in the 1980s by Ulam and Erd\H{o}s and attracted a lot of attention over the years. We give an almost tight answer to this question for almost all norms on $\mathbb{R}^d$ (for any given $d$). Furthermore, for almost all norms on $\mathbb{R}^d$, we prove an asymptotically tight bound for a related problem, the so-called Erd\H{o}s distinct distances problem. Our proofs combine combinatorial and geometric ideas with algebraic and topological tools.

Joint work with Noga Alon and Matija Buci\'{c}.

tba

n this talk, some recent developments on matrix distributions and their connection
to absorption times of inhomogeneous Markov processes are discussed. We show how and why
such constructions are natural tools for the modelling of insurance risks, for both non-life and
life insurance applications. A particular emphasis will be given on the higher-dimensional
case. We also illustrate how certain extensions to the non-Markovian case involve fractional
calculus and lead to matrix Mittag-Leffler distributions, which turn out to be a flexible and
parsimonious class for the modelling of large but rare insurance loss events.

Im Rahmen eines Kolloquiums im Fachgebiet ``Statistical Modeling and Data Science'' finden folgende Vorträge statt: \newline

21.09., 09:00 - 10:00 Uhr, Michael Scholz: ``Interpretable Local Machine Learning for Huge and Distributed Data''}\newline \newline
27.09., 09:00 - 10:00 Uhr, Jana Lasser: ``From Alternative conceptions of honesty to alternative facts in communications by U.S. politicians''} \newline \newline
27.09., 11:00 - 12:00 Uhr, Alejandra Avalos-Pacheco: ``Integrative Large-Scale Bayesian Learning: From Factor Analysis to Graphical Models''} \newline \newline
29.09., 09:00 - 10:00 Uhr, Dennis Schroers: ``Functional Data Analysis for Term Structure Models''} \newline \newline
29.09., 11:00 - 12:00 Uhr, Matthias Neumann: ``Parametric random set models for virtual testing of electrode materials''}
\newline

{\bf Hauptvortragende:} [2mm]
Hansjörg Albrecher (Univ. Lausanne)
Mathias Beiglböck (Univ. Wien)
Martin Hairer (EPFL)
Robert Scheichl (Univ. Heidelberg)
Oliver Roche-Newton (Univ. Linz)
Sandra Müller (TU Wien)
Angkana Rüland (Univ. Heidelberg)
Lisa Sauermann (MIT)
Damaris Schindler (Univ. Göttingen)
Karin Schnass (Univ. Innsbruck)
Elisabeth Werner (CWRU, Cleveland)
Vortrag zum Förderungspreis 2023[2mm]

{\bf Öffentlicher Vortrag:} Philipp Grohs[2mm]

{\bf Minisymposien:}
{\it Additive Combinatorics} (A. Geroldinger, W. A. Schmid)
{\it Computational Topology and Geometry} (M. Kerber, O. Aichholzer, B. Vogtenhuber)
{\it Stochastic Mass Transport} (G. Pammer, M. Beiglböck)
{\it Inverse Problems in Imaging} (M. Holler, K. Bredies)
{\it Versicherungs- und Finanzmathematik} (S. Thonhauser)
{\it Lehrerinnen- und Lehrertag} (C. Dorner, C. Krause)
{\it High-Dimensional Approximation} (M. Ullrich, D. Krieg)
{\it Probabilistic Methods in Convexity} (J. Prochno, E. Werner)
{\it Set Theory} (V. Fischer, S. Müller)
{\it Continuous Optimization Methods} (R. Bot, C. Clason)
{\it Diophantine analysis and equidistribution} (C. Aistleitner, J. Thuswaldner)[2mm]

{\bf Registrierung:} {\tt https://oemg-tagung-2023.at/}

In analogy to  the classical surface area,  a notion of affine surface area -- invariant under affine transformations -- has been defined.
The isoperimetric inequality states that the usual surface area is minimized for a Euclidean ball. Affine isoperimetric inequality states that
affine surface area is maximized for ellipsoids.

Due to this inequality and its many other remarkable properties, the affine surface area
finds applications in many areas of mathematics and applied mathematics. This has led to intense research in recent years and numerous
new directions have been developed.

We will discuss some of them and we will show how affine surface area is related to a geometric object, that is interesting in its own right,
the floating body.

Abstract:
The aim of this talk is twofold.
Firstly, it focuses on the analysis of distributed optimal control problems constrained by partial differential equations (PDEs).
Under some natural assumptions on the involved differential operators, it is shown that this class of problems
admits a certain structure and can be analyzed in an abstract framework. This structure carries over to every conforming discrete setting, where
optimal approximation results are gained, depending on the regularity of the desired target and the cost parameter. Various
model problems, including a space-time optimal control problem for the wave equation, are shown to fit into this framework and numerical examples
will be given that support the theory.
Secondly, the stable solution of (ill-posed) problems of PDEs is discussed when using a least squares approach. Again, this will be
done in an abstract framework. Under merely the same assumptions as in the case of optimal control problems, a full analysis of the
continuous and the discrete setting are carried out, revealing that the approach provides a reliable error estimator under a
standard saturation assumption. Again, various applications of the framework are discussed and numerical examples are given.

Regular Approximation and T-Compatibility

In the 1970s Stummel initiated the study of ``discrete approximation schemes'', which is a very generic framework to conduct the convergence analysis of numerical methods for e.g. source problems, holomorphic eigenvalue problems, nonlinear problems, etc.. In this context a most important property is the regularity of approximations. However, until recently this framework was only applied to weakly coercive problems for which the regularity of Galerkin approximations is easy to obtain.
In the last two decades the notion of weak T-coercivity became popular to study the well-posedness (Fredholmness) of all kinds of partial differential equations, e.g. Maxwells equations, interior transmission eigenvalue problems and dispersive transmission problems.
In this talk I present a technique to mimic the weak T-coercivity analysis on the discrete level to obtain the regularity and hence convergence of approximations. I present the application to Maxwells equations, mixed systems and the equations of stellar oscillations.

{
We consider a game-theoretical scenario in quantum information theory involving two parties, say Alice and Bob. At each round, Alice chooses a
quantum state from a given ensemble, known to both parties, and sends it to Bob. Bob is allowed to perform any quantum operation on the state and
to query Alice multiple times, one state at a time, until he correctly guesses the state. The game is repeated many times, and Bob's aim is to
minimize the average number of queries needed. This problem, known as quantum guesswork, can be reframed as an instance of quantum hypothesis
testing, and has therefore long been conjectured not to admit analytical solutions except for the cases in which the hypothesis testing problem
is solvable, that is, for binary and symmetric ensembles.
Here, we disprove such a belief by deriving conditions under which the guesswork problem (by definition, a continuous optimization) can be recast
as a combinatorial problem, and therefore can be solved analytically by exhaustive search. We further show that such conditions are verified by
any qubit ensemble, thus conclusively settling the problem in dimension two, and we show that in that case the guesswork is recast as an instance
of the quadratic assignment problem (QAP). In particular, the problem consists of assigning a given set of weights to a given set of
three-dimensional real vectors (the Bloch vectors of the states of the ensemble) in order to maximize the norm of the baricenter, thus
generalizing the (maximization version of the) turbine balancing problem to the three-dimensional, not-necessarily-symmetric case.
By showing that the corresponding Gram matrix is benevolent, we explicitly solve the QAP for a class of qubit ensembles whose image in the Bloch
sphere is an anti-prism, including uniform anti-prisms such as symmetric, informationally complete (SIC) and mutually unbiased basis (MUBs)
ensembles. For qubit ensembles whose image is centrally symmetric, we show that the size of the exhaustive search, as well as its complexity, can
be quadratically reduced. We exploit this fact to devise and implement a parallel branch and bound algorithm in the C programming language that
computes the guesswork of qubit ensembles of up to 30 states within hours.}

Vorgestellt werden ausgewählte geometrische Eigenschaften von Banachräumen, im Besonderen die klassischen Konvexitätseigenschaften strikte, lokal gleichmäßige und gleichmäßige Konvexität mit ausgewählten Beispielen und Anwendungen. Ferner wird ein kurzer Einblick in das junge Begriffsfeld der sogenannten Durchmesser zwei Eigenschaften gegeben, die als "maximale" Kontrastbegriffe zu Regularitätseigenschaften wie der Radon-Nikodym-Eigenschaft geschaffen wurden.

Many conjectures -- but few results -- exist for the statistical properties of large "random" matroids. For example, the question of which matroids appear as a minor of almost every matroid has been settled for only a few matroids. I will present recent progress in this direction: almost every matroid has at least one of two particular matroids, the rank-3 wheel $M(K_4)$ or the rank-3 whirl $W^3$, as a minor.

At the heart of the argument lies a counting version of the Ruzsa--Szemerédi (6,3)-theorem on 3-uniform hypergraphs, which is then generalised in several ways to obtain the main result.

For this talk, no knowledge about matroids is assumed.



--------------------

Meeting link:
\[
\text{https://tugraz.webex.com/tugraz/j.php?MTID=m49b4af86f873e7d36187a851c7fd52e4}
\]

Meeting number:
2734 674 7217

--------------------

When there are several endpoints (response variables) and different predictors, one typically wants to find out which predictors are relevant, and for which endpoints. We present two rather general approaches towards inference for multivariate data, one accommodating binary, ordinal, and metric endpoints, and the other allowing for a factorial design structure. The first approach uses rank-based statistics and an F-approximation of the sampling distribution, the second employs asymptotically valid resampling techniques (bootstrap). We also try to address the question of how well the proposed methods actually accomplish their goals, and how to use the respective toolboxes that have been developed.

The inverse problem in differential Galois theory consists of asking
which algebraic groups over a field k can be realized as Galois groups
of differential equations over a differential field K with constants k.
If k is algebraically closed of characteristic zero and K = k(z), the
answer has been known since the work of J.Hartmann in 2005, and the
answer is that every algebraic group can be realized.

Over \mathbb{C}(z), a stronger result has been known for a longer time: every
algebraic group can be realized as the differential Galois group of a
regular singular differential equation.

We intend to give an overview of the theory, and explain how a recent
result of R.Feng and M.Wibmer (2022) lets us extend the result known
over \mathbb{C} to arbitrary algebraically closed fields.

A graph $\Gamma$ is universal for a family $\mathcal{F}$ of graphs if for each $F \in \mathcal{F}$ there is a copy of $F \in \Gamma$ (not necessarily induced, and the copies are not necessarily disjoint).

Alon and Capalbo considered the case that $\mathcal{F}$ is the family of $n$-vertex graphs with maximum degree $k$, and showed that there is a universal graph for this family with $O\left(n^{2-2/k}\right)$ edges, which is sharp. Alon asked what the answer is if one replaces 'maximum degree' with 'degeneracy'.

We give a probabilistic construction of a universal graph for the family of $n$-vertex $D$-degenerate graphs with $\tilde{O}\left(n^{2-1/D}\right)$ edges, which is optimal up to the polylog. In this talk, I will describe the construction and give most of the details of the proof of its universality.

This is joint work with Julia Boettcher and Anita Liebenau.

Meeting link:
\[
\text{https://tugraz.webex.com/tugraz/j.php?MTID=m3162bb7e6bef850e659f657a18095a1c}
\]

Meeting number: 2733 453 3442

Password: bSDVGJDp976

Additively indecomposable integers are a useful tool in the study of universal quadratic forms or the Pythagoras number in totally real number fields. In the case of real quadratic fields, they can be derived from the continued fraction of concrete quadratic irrationalities. Thus, it is natural to ask whether there exists analogous relation between indecomposable integers and multidimensional continued fractions. In this talk, we will focus on the Jacobi–Perron algorithm and discuss elements originating from expansions of concrete vectors for several families of cubic fields. This is joint work with Vítězslav Kala and Ester Sgallová.

A tournament is an orientation of the complete graph. The score sequence lists the out-degrees in non-decreasing order. Works by Erdős and Moser, Winston and Kleitman, and Kim and Pittel bounded the number of score sequences. In this talk, we locate the precise asymptotics. These agree numerically with those conjectured by Takács. We will discuss connections to geometry, renewal theory, infinitely divisible probability distributions, the Erdős–Ginzburg–Ziv numbers, and random lattice walks.

Meeting link:
\[
\text{https://tugraz.webex.com/tugraz/j.php?MTID=m3162bb7e6bef850e659f657a18095a1c}
\]

Meeting number: 2733 453 3442

Password: bSDVGJDp976

We study normalized volumes of a family of polytopes associated with column-convex {0,1}-matrices. Such a family is a generalization of the family of consecutive coordinate polytopes, studied by Ayyer, Josuat-Vergès, and Ramassamy, which in turn generalizes a family of polytopes originally proposed by Stanley in EC1. We prove that a polytope associated with a column-convex {0,1}-matrix is integrally equivalent to a certain flow polytope. We use the recently developed machinery in the theory of volumes and lattice point enumeration of flow polytopes to find various expressions for the volume of these polytopes, providing new proofs and extending results of Ayyer, Josuat-Vergès, and Ramassamy. This is joint work with Chris Hanusa, Alejandro Morales, and Martha Yip.

10 Uhr: Prof. Dr. Carmelo Antonio FINOCCHIARO (University of Catania)
Titel: A Topological Insight on the Ideal Theory of Rings of Integer-Valued Polynomials

11 Uhr: Prof. Dr. Giulio PERUGINELLI (University of Padova)
Titel: The Ubiquity of Integer-Valued Polynomials: Polynomial Dedekind Domains

Gemeinsames Mittagessen

14 Uhr: Univ.-Prof. Dr. Daniel SMERTNIG (Universität Graz)
Titel: From Bass Rings to Monoids of Graph Agglomerations

15 Uhr: Dr. Roswitha RISSNER (Universität Klagenfurt)
Titel: Powers of Irreducibles in Rings of Integer-Valued Polynomials

The recent development of boundary triples theory allowed us to study self-adjointness of the Dirac operator perturbed by the singular potential $D_l^A=D+A\delta_\Sigma$, where $A$ is arbitrary hermitian matrix and $\delta_\Sigma$ stands for the single-layer distribution supported on closed non-self-intersecting Lipschitz-smooth curve or surface $\Sigma$ in $\mathbb{R}^n$, $n\in\{2,3\}$. It was proved that the operator $D_l^A$ corresponds to a self-adjoint extension of a symmetric operator $D_0\varphi = D\varphi$, $\varphi \in H^1_0(\mathbb{R}^n\setminus \Sigma)$. Contrary to the general belief, by studying seemingly similar formal operator $D^A_{nl} = D+A|\delta_\Sigma\rangle \langle \delta_\Sigma|$ we will be able to define completely new self-adjoint extensions of the operator $D_0$. Let us mention that this is not the case in one dimension, where operators $D^A_l$ and $D^A_{nl}$ are one and the same. Finally, in case of $C^2$ curve resp. surface we will be able to construct regular approximations and prove the norm-resolvent convergence to the operator $D_{nl}^A$ without renormalization.

Abstract:
When the one-dimensional relativistic point interactions are approximated by scaled regular potentials, the coupling constants have to be renormalized in the limit. This effect was observed by P. Šeba in 1989 for the first time in the paper called ”Klein’s Paradox and the Relativistic Point Interaction” (Lett. Math.Phys. 18). However, the paper does not contain any explicit explanation of a relation between the Klein paradox and the renormalization of the coupling constants. The aim of this short talk is to trace such a relation.

Greedy algorithms are surprisingly effective in a wide range of optimization problems, though have only recently been considered as a possible way to find point configurations with low discrepancy or energy.
In this talk, we will discuss the performance of a greedy algorithm on the sphere to minimize the Riesz energies
$$ E_s(\{ z_1, ..., z_N\}) = \sum_{i \neq j} \frac{1}{s} \|x-y\|^{-s}$$
and the quadratic spherical cap discrepancy
$$ D_2(\{ z_1, ..., z_N\}) = \int_{-1}^{1} \int_{\mathbb{S}^d} \Big| \frac{\# ( C(x,t) \cap \{ z_1, ..., z_N\} )}{N} - \sigma(C(x,t)) \Big|^2 d\sigma(x) dt.$$
As an intermediate step, we study the maximal Riesz polarization on the sphere,
$$ \mathcal{P}_s( \mathbb{S}^d , N) = \sup_{\omega_N \subset \mathbb{S}^{d}, |\omega_N| = N} \; \inf_{x \in \mathbb{S}^d} \sum_{y \in \omega_N} \frac{1}{s} \|x - y \|^{-s}.$$
In an analogue of the connection between energy and best packings, as $s \rightarrow \infty$, optimal polarization configurations become best coverings. We show that that the first and second order asymptotics of optimal Riesz polarization nicely parallel those of energy minimization, and greedily generated point sets are nearly optimal for polarization.

The research in this presentation is joint work with Dmitriy Bilyk (University of Minnesota), Michelle Mastrianni (University of Minnesota), and Stefan Steinerberger (University of Washington).

The last decade has seen an ever growing research interest in the study of generalizations of spaces of “diagonal harmonics” for the symmetric group. Indeed, this bring into play natural interactions between Algebraic Combinatorics, Symmetric Function Theory, Representation Theory, Algebraic Geometry, Knot Theory, and Theoretical Physics. On the combinatorial side, the subject has evolved from the special case of “Classical Catalan Combinatorics” to the current all inclusive notion of “Triangular Catalan Combinatorics”, successively going through the contexts of “Rational Catalan Combinatorics” and then “Rectangular Catalan Combinatorics”, in increasing level of generality. Each of these has raised new questions in other fields. Just to illustrate, the rational case relates to calculation of the Khovanov-Rozansky homology of torus knots, whilst the rectangular case extends this to torus links.

We will describe interesting aspects of the combinatorics of the new triangular context, showing that it makes natural many relevant notions that previously appeared to be rather mysterious. We will also see that this context has nice closure properties that where lacking in previous ones. We will conclude with many open natural questions that remain to be explored.

The linear arboricity of a graph $G$, denoted by $\textrm{la}(G)$, is the minimum number of edge-disjoint linear forests (i.e. collections of disjoint paths) in $G$ whose union is all the edges of $G$. It is known that the linear arboricity of every cubic graph is $2$. In 1987 Wormald conjectured that every cubic graph with even number of edges, can be partitioned such that the two parts are isomorphic linear forests.

This is known to hold for Jeager graphs and for some further classes of cubic graphs (see, Bermond-Fouquet-Habib-Peroche, Wormald, Jackson-Wormald, Fouquet-Thuillier-Vanherpe-Wojda). In this talk, we will present a proof of Wormald's conjecture for all large connected cubic graphs.

This is joint work with Shoham Letzter, Alexey Pokrovskiy, and Liana Yepremyan.

Meeting link:
\[
\text{https://tugraz.webex.com/tugraz/j.php?MTID=m3162bb7e6bef850e659f657a18095a1c}
\]

Meeting number: 2733 453 3442

Password: bSDVGJDp976

A hypergraph is linear} if every pair of the vertices is contained in at most one edge. In 1972, Erd\H{o}s, Faber, and Lov\'{a}sz conjectured that every linear hypergraph on $n$ vertices with maximum degree at most $n$ has chromatic index at most $n$.

In this talk, we discuss a proof of the conjecture for all large $n$ and some generalisations. This is joint work with Deryk Osthus, Daniela Kühn, Abhishek Methuku, and Tom Kelly.


Meeting link:
\[
\text{https://tugraz.webex.com/tugraz/j.php?MTID=m3162bb7e6bef850e659f657a18095a1c}
\]

Meeting number: 2733 453 3442

Password: bSDVGJDp976

Given an irrational number $\alpha$ consider its irrationality measure function
$$
\psi_{\alpha}(t)=\min\limits_{1\le q\le t, q\in\mathbb{Z}}\|q\alpha\|.
$$
The set of all values of
$$
\lambda(\alpha)=\bigl(\limsup\limits_{t\to\infty} t\psi_{\alpha}(t)\bigr)^{-1},
$$
where $\alpha $ runs through the set $\mathbb{R}\setminus\mathbb{Q}$ is called the Lagrange spectrum $\mathbb{L}$. Denote by $\mathcal{Q}=\{q_1,q_2,\ldots,q_n,\ldots\}$ the set of denominators of the convergents to $\alpha$. One can consider another irrationality measure function
$$
\psi^{[2]}_{\alpha}(t)=\min\limits_{1\le q\le t, q\in\mathbb{Z},q\not\in\mathcal{Q}}\|q\alpha\|
$$ connected with the properties of so-called second best approximations. Or, in other words, approximations by rational numbers, whose denominators are not the denominators of the convergents to $\alpha$. Replacing the function $\psi_{\alpha}$ in the definition of $\mathbb{L}$ by $\psi^{[2]}_{\alpha}$, one can get a set $\mathbb{L}_2$ which is called the ''second'' Lagrange spectrum. In my talk I give the complete structure of the discrete part of $\mathbb{L}_2$.

How dense are the rational points close to a compact and smooth hyper-surface $M$ in $R^n$?
In fact, consider all rationals whose (common) denominator is at most $Q$ and which are $\delta$-close to $M$.
We discuss this quantitative question without requiring any knowledge of number theory at all, using only basic Fourier analysis instead.

One knows that as long as the Gaussian curvature of $M$ is uniformly bounded from below, the number of such rational points
is of size $\delta Q^n$ when $\delta > Q^{\varepsilon -1}$ --- as one expects from a simple volume-informed random model.

In this talk, we consider a rich class of hyper-surfaces M where the Gaussian curvature
vanishes at one point to a very high order, and we establish a sharp counting theorem.
Indeed, the aforementioned random model is now inadequate.Hence, we propose a refined random model.
This is based on joint work with Rajula Srivastava (U. Bonn).

Configurations of axis-parallel boxes in $\mathbb{R}^d$ are extensively studied in combinatorial geometry. Despite their innocent appearance, there are many old problems involving their structure that are still not well understood. I will talk about a construction, which shows that starting from dimension $d\geq 3$, configurations of boxes may be more complex than people conjectured.

Meeting link:
\[
\text{https://tugraz.webex.com/tugraz/j.php?MTID=m3162bb7e6bef850e659f657a18095a1c}
\]

Meeting number: 2733 453 3442

Password: bSDVGJDp976

The ``no solution could be found dilemma'' is omnipresent in various application contexts. If a user specifies a set of requirements which
cannot be completely supported, the question arises, which of those requirements should be adapted. In this presentation, I will summarize some
approaches to deal with such situations in a personalized fashion.



Meeting link:
\[
\text{https://tugraz.webex.com/tugraz/j.php?MTID=m08eab8cd8af8502205a8cf177eebc3c7}
\]

Meeting number: 2730 761 9040

Password: Bp2tsHfiC24

We will look at the following problem on time—frequency localization: What is the norm of the point evaluation functional in the classical Paley—Wiener $L^p$ spaces for $0<p<\infty$? The central challenge in this problem is how to go beyond the “power trick” which allows you to relate an estimate for a given $p$ to that for $kp$ for a positive integer $k$. I will discuss some results and multiple conjectures around this problem. The talk is based on recent joint work with Ole Fredrik Brevig, Andrés Chirre, and Joaquim Ortega-Cerdà (to appear in J. Anal. Math., see
https://arxiv.org/abs/2210.13922).

We consider multi-stage robust optimization problems like recoverable robust optimization and two-stage robust optimization. Recoverable robust
optimization is a multi-stage approach, in which it is possible to adjust a first-stage solution after the uncertain cost scenario is revealed.
Two-stage robust optimization can be seen as games between a decision maker and an adversary. After the decision maker fixes part of the solution,
the adversary chooses a scenario from a specified uncertainty set. Afterwards, the decision maker can react to this scenario by completing the
partial first-stage solution to a full solution.

In this talk we focus on various hardness results for multi-stage robust optimization problems and efficient algorithms that can be obtained for
specific variants and different types of uncertainty sets. We will discuss under which conditions these types of problems become hard for higher
levels of the polynomial hierarchy. We will also discuss the complexity of multi-stage robust optimization for (representative) selection in more
detail.

Abstract:

We intend to give an overview of some recent results on (optimal) Birman-Hardy-Rellich-type inequalities. In particular, we will discuss power-weighted inequalities and some of their logarithmic refinements in all space dimensions $n \geq 1$. In dimension one we will present a new perturbative Hardy-type inequality which sheds new light on the origin of Hardy inequalities.

Hardy-type inequalities are a basic tool in the spectral theory of differential operators and we intend to briefly illustrate their relevance.

This is based on various joint work with L. Littlejohn, I. Michael, R. Nichols, and M. M. H. Pang.

09:30 - 10:15 - {\bf Mauricio Collares} (TU Graz) ``Set-colouring Ramsey numbers''}

(SR 1 Geometry Institute, Kopernikusgasse 24)
\vspace{.5cm}

10:30 - 11:00 Coffee Break

(Geometry Institute, Kopernikusgasse 24)
\vspace{.5cm}

11:00 - 11:45 - {\bf Sahar Diskin} (Tel Aviv University) ``The Erdős-Rényi component phenomenon: component sizes in percolation on high-dimensional product graphs''} (Part of the Advanced Topics seminar)

(SR 2 Geometry Institute, Kopernikusgasse 24)
\vspace{.5cm}

12:00 - 14:00 - Lunch Break
\vspace{.5cm}

14:00 - 15:00 - Rigorosum of Tuan Anh Do

(AE06, Steyrergasse 30)
\vspace{.5cm}

16:00 - 16:30 - Coffee Break

(2nd Floor, Steyrergasse 30)
\vspace{.5cm}

16:30 - 17:30 - {\bf Michael Krivelevich} (Tel Aviv University) ``Fast construction on a restricted budget''}

(BE01, Steyrergasse 30)
\vspace{.5cm}

Abstracts can be found at

https://www.math.tugraz.at/comb/workshops/Workshop2023Abstracts.pdf

The ratio-limit boundary $\partial_R \Gamma$ for a random walk on a group  $\Gamma$ is defined as the remainder obtained after compactifying $\Gamma$ with respect to ratio-limit kernels $H(x,y) = \lim_n P^n(x,y)/P^n(e,y)$, where $P^n(x,y)$ is the probability to pass from $x$ to $y$ in $n$ steps. These limits were shown to exist for various classes of groups, normally by establishing a local limit theorem which measures the asymptotic behavior of $P^n(x,y)$ as $n \rightarrow \infty$. In increasing level of generality, works of Woess, Lalley, Gouezel and Dussaule established such local limit theorems for certain symmetric random walks on relatively hyperbolic groups. Techniques developed in these works then allow us to study $\partial_R \Gamma$. For instance, when $\Gamma$ is hyperbolic, Woess was
able to show that $\partial_R \Gamma$ is the Gromov boundary of $\Gamma$ . 

In this talk we will explain how for a large class of random walks on relatively hyperbolic groups, the ratio-limit boundary is essentially minimal. That is, there is a unique minimal closed $\Gamma$ -invariant subspace of $\partial_R \Gamma$ . This result is motivated by applications in operator algebras, and indeed, by using it we are able to show the existence of a  co-universal equivariant quotient of Toeplitz C*-algebras for a large
class of random walks on relatively hyperbolic groups. The talk will focus
mostly on geometry, topology, and dynamics, and if time permits I will
explain some operator algebraic aspects and motivation.

This talk is based on joint work with Matthieu Dussaule and Ilya Gekhtman.

Erdos, Kleitman and Rothschild proved that the number of triangle-free graphs on $n$ vertices is asymptotic to the number of bipartite graphs; or in other words, a typical triangle-free graph is bipartite. Osthus, Promel and Taraz proved a sparse analogue of this result: if $m > (\sqrt{3}/4 +\epsilon) n^{3/2} \sqrt{\log n}$, a typical triangle-free graph with $m$ edges is bipartite (and this no longer holds below this threshold).

What do typical triangle-free graphs at sparser densities look like and how many of them are there? We consider what we call the ordered regime, where typical triangle-free graphs are not bipartite but have a dense max-cut. In this regime we prove asymptotic formulas for the number of triangle-free graphs and give a precise probabilistic description of their structure. This leads to further results such as determining the threshold at which typical triangle-free graphs are $q$-colourable for $q \geq 3$, determining the threshold for the emergence of a giant component in the complement of a max-cut, and many others.

This is joint work with Will Perkins and Aditya Potukuchi.

Meeting link:
\[
\text{https://tugraz.webex.com/tugraz/j.php?MTID=m3162bb7e6bef850e659f657a18095a1c}
\]

Meeting number: 2733 453 3442

Password: bSDVGJDp976

Let $R(k)$ be the $k$th diagonal Ramsey number: the smallest $n$ for which every red/blue edge-colouring of the complete graph on $n$ vertices contains a monochromatic complete graph of size $k$. I will discuss the proof that $R(k) < (4-c)^k$, for some absolute constant $c>0$. This is the first exponential improvement over the bound of Erdős and Szekeres, proved in 1935.

This is based on joint work with Marcelo Campos, Simon Griffiths and Rob Morris.

Meeting link:
\[
\text{https://tugraz.webex.com/tugraz/j.php?MTID=m3162bb7e6bef850e659f657a18095a1c}
\]

Meeting number: 2733 453 3442

Password: bSDVGJDp976

The Ramsey number $r(G)$ of a graph $G$ is the minimum $N$ so that every red/blue edge-coloring of $K_N$ contains a monochromatic copy of $G$. Conlon, Fox, and Sudakov conjectured that if we delete a single vertex from $G$, then the Ramsey number can decrease by at most a constant factor. Though very natural (and true in a variety of special cases), this conjecture turns out to be false in general. In this talk, I'll explain how one disproves this conjecture, as well as the connections this problem has to a number of other questions in Ramsey theory.

Meeting link:
\[
\text{https://tugraz.webex.com/tugraz/j.php?MTID=m3162bb7e6bef850e659f657a18095a1c}
\]

Meeting number: 2733 453 3442

Password: bSDVGJDp976

The vertex-persuit game Cops and Robbers} is usually played on a graph, in which a group of cops attempt to catch a robber moving along the edges of the graph. The cop number} of a graph is the minimum number of cops required to win the game.
An important conjecture in this area due to Meyniel states that the cop number of a connected graph is $O(\sqrt{n})$. In 2016, Pra\l at and Wormald showed that this conjecture holds with high probability for connected random graphs. Moreoever, \L uczak and Pra\l at found that on a log-scale the cop number shows a surprising zigzag} behaviour in dense random graphs. In this paper, we consider the game of cops and robbers on a hypergraph, where the players move along hyperedges instead of edges. We conjecture that the cop number of a connected $k$-uniform hypergraph is $O\left(\sqrt{\frac{n}{k}}\right)$ and show that this conjecture holds with high probability up to $\log$-factors for the random binomial $k$-uniform hypergraph $G^k(n,p)$ for a broad range of parameters $p$ and $k$. As opposed to the case of $G(n,p)$, on a log-scale our upper bound on the cop number arises as the minimum of two complementary zigzag curves.

We consider inference on principal directions in non-standard asymptotic scenarios where principal directions are unidentifiable in the limit. To fix ideas, we first tackle the problem of testing the null hypothesis $H_0 : \theta_1=\theta^0_1$ against the alternative $H_1 : \theta_1 \neq \theta_1^0$, where $\theta_1$ is the ”first” eigenvector of the underlying covariance matrix and $\theta_1^0$ is a fixed unit $p$-vector. In the classical setup where eigenvalues $\lambda_1>\lambda_2 \geq \ldots \geq \lambda_p$ are fixed, the likelihood ratio test (LRT) and the Le Cam optimal test for this problem are asymptotically equivalent under the null hypothesis, hence also under sequences of contiguous alternatives. We show that this equivalence does not survive asymptotic scenarios where $\lambda_{n1}/\lambda_{n2} = 1 + O(r_n)$ with $r_n = O(1/\sqrt{n})$. For such scenarios, the Le Cam optimal test still asymptotically meets the nominal level constraint, whereas the LRT becomes extremely liberal. Consequently, the former test should be favored over the latter one whenever the two largest sample eigenvalues are close to each other. By relying on the Le Cam theory of asymptotic experiments, we study in the aforementioned asymptotic scenarios the non-null and optimality properties of the Le Cam optimal test and show that the null robustness of this test is not obtained at the expense of efficiency. Our asymptotic investigation is extensive in the sense that it allows $r_n$ to converge to zero at an arbitrary rate. While these results relate to robustness to weak identifiability, we also introduce sign tests that combine such non-standard robustness with more classical robustness to outliers and/or heavy tails. Finally, we tackle the corresponding point estimation problem by considering in this framework the asymptotic behaviour of the celebrated scatter estimator from Tyler (1987).

In this talk I will present a simple method, inspired by the works of Ratner and Burger, to study ergodic integrals for the classical horocycle flow. The asymptotic expansion we can prove following this approach is a strengthening of the result by Flaminio and Forni in two ways: the coefficients in the expansion are shown to be Hölder continuous with respect to the base point and the term corresponding to the functions in the kernel of the Casimir operator is explicitly described.
Furthermore, we recover the spatial limit theorems by Bufetov and Forni and the temporal limit theorem by Dolgopyat and Sarig (this latter proof is by E. Corso).

I will present two approaches to the common theme of a topological space changing over time. The first approach is a way to understand algebraic changes in the homology of a simplicial complex of a point cloud, as its points move around in Euclidean space. The second approach is about binary dynamics of a fixed point cloud, where edges are determined by a biologically-motivated reconstruction of a mammalian brain. The dynamics in both cases are arbitrary, but the second approach focuses on recovering long-term patterns with topology at the local scale. This is joint work with Barbara Giunti, Ran Levi, and others.

Recently, there was a lot of activity regarding an old conjecture of van der Waerden, culminating in its solution by Bhargava, and including joint work by Sam Chow and myself on which I want to report in this talk:
We showed that the number of irreducible monic integer polynomials of degree n, with coefficients in absolute value bounded by H, which have Galois group different from S_n and A_n, is of order of magnitude O(H^{n-1.017}), providing that n is at least 3 and is different from 7,8,10. Apart from the alternating group and excluding degrees 7,8,10, this establishes the aforementioned conjecture to the effect that irreducible non-S_n polynomials are significantly less frequent than reducible polynomials.
The method is quite flexible and can also be applied to other problems.

Let $m$ be a positive integer and $b_{m}(n)$ be the number of partitions of $n$ with parts being powers of 2, where each part can take $m$ colors. We show that if $m=2^{k}-1$, then there exists the natural density of integers $n$ such that $b_{m}(n)$ can not be represented as a sum of three squares and it is equal to $1/12$ for $k=1, 2$ and $1/6$ for $k\geq 3$. In particular, for $m=1$ the equation $b_{1}(n)=x^2+y^2+z^2$ has a solution in integers if and only if $n$ is not of the form $2^{2k+2}(8s+2t_{s}+3)+i$ for $i=0, 1$ and $k, s$ are non-negative integers, and where $t_{n}$ is the $n$th term in the Prouhet-Thue-Morse sequence. A similar characterization is obtained for the solutions in $n$ of the equation $b_{2^k-1}(n)=x^2+y^2+z^2$. This talk is based on a joint work with Bartosz Sobolewski (Jagiellonian University).

13:30 - 14:30: Efthymios Sofos (Glasgow):}
Averages of arithmetic functions over values of multivariable polynomials

15:00 - 16:00: Jakob Glas (ISTA):}
Rational points on del Pezzo surfaces over function fields

16:15 - 17:15: Shuntaro Yamagishi (ISTA):}
On Poonen's question regarding polynomial representing all natural numbers


Seminar webpage for abstracts and further information:

https://sites.google.com/view/gntd/upcoming-seminars

The talk considers the following problem which got suggested to the speaker
as last problem by the late Gerhard Woeginger who passed away on April 1, 2022.

There are $n$ people who like to take a walk with a group of some size.
Each person has its own interval $[a_i,b_i]$, which restricts its own preferred group size.
Is there an assignment for all people to groups of the preferred size?
We will show that this problem can be solved in polynomial time and
discuss some variants.

In Europe, we have embarked on the journey towards net-zero power systems and want to reach full decarbonization by
2050 (European Commission ``A clean planet for all''). Austria wants to reach carbon neutrality in the power system
already by 2030 (Erneuerbaren Ausbau Gesetz) and the decarbonization of the entire energy system by 2040, which
will require coupling the power, transport, heat and gas sectors. In this talk we discuss how mathematical
modeling, optimization and game theory are key ingredients to achieving such ambitious climate goals by tackling
relevant challenges in energy economics. In particular, complex systems such as the electric power system are
governed by physical laws and technical limitations that often lead to non-convex, mixed-integer, large-scale
optimization models. Moreover, electric energy is traded through liberalized electricity markets -- competitive
environments that can be assessed via game theory and that can lead to interesting hierarchical equilibrium
problems that call for bilevel programming. This talk emphasizes the significant contributions that mathematicians
can make to tackle the climate crisis.


Bio:
Since Summer 2021 Sonja Wogrin has been the head of the Institute of Electricity Economics and Energy Innovation. She studied mathematics at TU Graz and obtained her diploma degree in 2008 and then moved to Spain where she obtained her Ph.D. at the Instituto de Investigación Tecnológica (IIT) of the Universidad Pontificia Comillas in Madrid in 2013. She worked at the IIT from 2009 to 2021. At TU Graz Sonja Wogrin is involved in many interdisciplinary research projects and is the initiator of the planned TU Graz research center ``Energy Economics & Energy Analytics'' in which contributions from all areas of mathematics are very welcome.

We consider the enumeration problem of bipartie and quasi-bipartite planar maps with internal faces, counted with Boltzmann weights, and external faces with prescribed degrees. the generating series for these objects solves a simple functional equation discovered by Eynard and generalized by Collet and Fusy, who also proposed a bijective derivation based on the Bouttier-Di Francesco-Guitter bijection. One observation is that the functional equation becomes even simpler if one asks that the external face boundaries be tight, in the sense that their lengths are minimal in their respective free homotopy class, in the surface obtained by removing one point from every external face. We give bijective interpretations, based on geometric decompositions, for this simpler formula in two particular cases: when there are three external faces (pairs of pants), and when there are no internal faces (tight maps). I will discuss implications of these for the statistics of large planar maps, as well as for the quasi-polynomiality for the number of plane tight maps observed by Norbury. This is based on joint work with Jérémie Bouttier and Emmanuel Guitter.

Meeting link:
\[
\text{https://tugraz.webex.com/tugraz/j.php?MTID=m44797227fd680cc7956ebb840b6f033a}
\]

Meeting number: 2730 500 3129

Password: vQydpg372D4

A rainbow subgraph in an edge-coloured graph is one in which all edges have different colours. This talk will be about finding rainbow subgraphs in colourings of graphs that come from groups. An old question of this type was asked by Hall and Paige. Their question was equivalent to the following ``Let $G$ be a group of order $n$ and consider an edge-coloured $K_{n,n}$, whose parts are each a copy of $G$ and with the edge $\{x,y\}$ coloured by the group element $xy$. For which groups $G$, does this coloured $K_{n,n}$ contain a perfect rainbow matching?'' This question is equivalent to asking ``which groups G contain a complete mapping'' and also ``which multiplication tables of groups contain transversals''. Hall and Paige conjectured that the answer is ``all groups in which the product of all the elements is in the commutator subgroup of $G$''. They proved that this is a necessary condition, so the main part of the conjecture is to prove that that rainbow matchings exist under their condition.
The Hall-Paige Conjecture was confirmed in 2009 by Wilcox, Evans, and Bray with a proof using the classification of finite simple groups. Recently, Eberhard, Manners, and Mrazovic found an alternative proof of the conjecture for sufficiently large groups using ideas from analytic number theory. Their proof gives a very precise estimate on the number of complete mappings that each group has.
In this talk, a third proof of the conjecture will be presented using a different set of techniques, this time coming from probabilistic combinatorics. This proof only works for sufficiently large groups, but generalizes the conjecture in a new direction. Specifically we not only characterize when the edge coloured $K_{n,n}$ contains a perfect rainbow matching, but also when random subgraphs of it contain a perfect rainbow matching. This extension has a number of applications, such as to problems of Snevily, Cichacz, Tannenbaum, Evans.
This is joint work with Alp Muyesser.



Meeting link:
\[
\text{https://tugraz.webex.com/tugraz/j.php?MTID=m44797227fd680cc7956ebb840b6f033a}
\]

Meeting number: 2730 500 3129

Password: vQydpg372D4

When can (the edge-set of) a graph $G$ be decomposed into copies of a given graph $H$? This question goes all the way back to Euler; despite this, the setting where the number of vertices in $G$ and $H$ are comparable is not yet well-understood. I will talk about the resolution of a conjecture of Ringel from 1963 where $G$ is the complete graph on $2n+1$ vertices and $H$ is any given tree with $n$ edges. This is joint work with Peter Keevash; the conjecture was independently resolved by Montgomery, Pokrovskiy and Sudakov.

Meeting link:
\[
\text{https://tugraz.webex.com/tugraz/j.php?MTID=m44797227fd680cc7956ebb840b6f033a}
\]

Meeting number: 2730 500 3129

Password: vQydpg372D4

In this talk I will describe a class of automaton groups,
generated by so-called reducible automata, which are shown to be
direct limit of finite index subgroups of the direct product of free
groups. This result partially supports a conjecture by T. Brough
regarding the general structure of groups with poly-context-free word
problem, i.e., whose word problem is the intersection of finitely
many context-free languages.
Joint work with M.Cavaleri, A.Donno, and E.Rodaro