Discrete Mathematics 2010-2022

Consider a random subgraph of the square integer lattice obtained by including each edge independently at random with probability $p$, and leaving it out otherwise. The Harris--Kesten theorem states that if $p$ is at most $1/2$, then almost surely all connected components in this random subgraph are finite, while if $p>1/2$ then almost surely there exists a unique infinite connected component.

But now what if we introduced some local dependencies between the edges? More precisely, suppose each edge still has a probability $p$ of being included in our random subgraph, but its state (present/absent) may depend on the states of edges it shares a vertex with. To what extent can we exploit such local dependencies to delay the appearance of an infinite component?

In this talk I will discuss this question, which first arose in work of Balister, Bollobás and Walters in 2005, as well as some recent progress around it.

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

Meeting number: 2730 500 3129

Password: vQydpg372D4

15.00 Uhr - Oliver Roche-Newton, Johannes Kepler Universität Linz}

Title: The sum-product problem, convexity, and elementary methods

Abstract: Sum-product theorems state that a given set must determine many distinct sums or products, and this idea is captured in a famous conjecture of Erdos and Szemeredi. In this talk, I will give an introduction to this problem, and then focus on what can be proven using elementary methods. It turns out that, for many sum-product type problems, the best known results are attained via simple arguments that do not rely on any heavy theoretical tools. In particular, I will discuss some recent advances in this area that use an elementary "squeezing" argument.


16.00 Uhr - Alan Filipin, University of Zagreb}

Title: Triangular Diophantine $m$-tuples

Abstract: We call a set of $m$ positive distinct integers a triangular Diophantine $m$-tuple, if the product of any two of them increased by
1 is a triangular number. In this talk, we prove that there does not exist a triangular Diophantine quadruple of the form $\{1, 2, c, d\}$. In the proof we use the standard methods like solving simultaneous Pell-like equations, a congruence method and linear forms in logarithms. At the end of the talk we also give some ideas for possible future research. This is joint work with Laszlo Szalay.

16.40 Uhr - Vinko Petri\v{c}evi\'{c}, University of Zagreb}

Title: Searching for elliptic curves with high rank in the PARI/GP software package

Abstract: In the new version of PARI/GP software package, they added the command ellrank, which in some situations can calculate the rank of the elliptic curve very well. Using this and connecting PARI/GP and
C++, we were able to find record-breaking curves for some torsion
groups.

This talk is based on the topics of my dissertation in which I considered two generalizations of polynomial rings with respect to algebraic and arithmetic properties.

First, I give a general overview on concepts related to the questions and results of my thesis.

Afterwards, I introduce certain irreducible elements of the ring of integer-valued polynomials.
It turns out that these so-called binomial polynomials are absolutely irreducible in this ring, which is to say, all their powers factor uniquely into irreducible elements.

A large part of the thesis is devoted to monoid algebras.
We consider class groups and prime divisors of monoid algebras, characterizations of the weakly Krull property for monoid algebras and then combine these results to obtain information about the multiplicative arithmetic of monoid algebras.

Finally, we deal with a type of uniform spaces that is important in algebraic number theory and commutative algebra (especially for rings whose elements can be considered as functions, such as polynomial rings). These non-Archimedean uniform spaces are described with respect to all three notions of uniformities.
We give a separation axiom for a topology that is equivalent to be induced by a non-Archimedean uniformity and we characterize those non-Archimedean uniform spaces which are pseudo-metric spaces.

The thesis is based on joint papers with G.W.~Chang, V.~Fadinger, and R.~Rissner.


https://tugraz.webex.com/tugraz/j.php?MTID=md4da8bda2944bd437ab9f60f5eac3b4d
Meeting number (access code): 2733 680 6636 Meeting password: ugBhSBCJqnT

We study the Set Covering (or Hitting Set) problem within the
paradigm of finding a ``diverse'' collection of $r$ solutions of size
at most $k$ each. This type of problem is related to multi-objective
optimization, and it has recently been introduced to the field of parameterized complexity. [Baste, Fellows, Jaffke, Masa{\v r}{\'i}k , Oliveira, Philip, and Rosamond, arXiv 2019].

We use two measures for the diversity of such a collection: the sum of all pairwise Hamming distances, and the
minimum pairwise Hamming distance. We show that our problem is fixed-parameter tractable (FPT) in $k+r$ for both
diversity measures. A key ingredient in our algorithms is a problem-independent network flow formulation that,
given a set of base solutions, computes a maximally diverse collection of solutions. This could be of independent
interest.

We also discuss the modelling aspects of finding an appropriate
formalization for ``diversity''.

This is joint work with J. Baste, L. Jaffke, T. Masa{\v r}{\'i}k, and G. Philip.


{ Streaming of the walk via Webex could be offered upon request in advance.}

Since the seminal work of Erdos and Renyi on random graphs in 1959, random discrete objects have been a source of fascinating results. The most exciting results concern thresholds at which major changes in behaviour of the random model occur, such as planarity, connectedness, or the size of the largest component.

In this talk, we focus on topological aspects and properties of random models. How do topological constraints, such as being embeddable on a specific surface, influence the structure of random graphs? What behaviour do higher-dimensional random models, such as random simplicial complexes, exhibit with respect to fundamental properties such as connectedness?

Consider the probability that a random graph selected uniformly from the set of $r$-uniform hypergraphs with $n$ vertices and $m$ edges, has a given degree sequence. Previously the value of this probability has been investigated by Kamčev, Liebenau and Wormald, where they examined degree sequences from very sparse to moderately dense hypergraphs when $r=o\left(n^{1/4}\right)$ and the variation of the degrees is small, but exceeds the typical degree variation in random hypergraphs.

We extend their results, by establishing this result for dense hypergraphs, which hold for any value of $r$ and allow for a greater variation on the degrees.

This is joint work with Catherine Greenhill, Mikhail Isaev and Brendan McKay.

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

Meeting number: 2730 500 3129

Password: vQydpg372D4

In this presentation, I discuss the global-scale behavior of random matrix eigenvalues and random partitions which depend on the "inverse temperature" parameter beta. I will focus on explaining the method of moments, powered by algebraic and combinatorial tools, to prove the Law of Large Numbers in the regimes of high and low temperatures, i.e. when beta converges to zero and to infinite, respectively. This talk is based on joint works with Benaych-Georges--Gorin, and with Dolega--Moll.

One of the most classical classes of combinatorial questions consists of decomposition problems, the scientific study of which dates back to the work of Euler in the 18th century. In a very broad sense, a decomposition problem asks whether we can decompose a large object into isomorphic (in an appropriate sense) copies of a smaller object. In this talk I will give a brief introduction to decomposition problems and discuss two classical conjectures in the area, namely Rota's basis conjecture and the Erdos-Gallai conjecture.

In Mathematics and Computer Science we frequently encounter different kinds of combinatorial objects, such as permutations, partitions, binary trees, spanning trees of a graph, linear extensions of a poset etc.
There are a number of closely related fundamental tasks that we want to perform with these objects, such as counting (how many objects are there?), random sampling (pick one of the objects uniformly at random), or optimization (find the best object subject to some objective function). In this talk the focus is on combinatorial generation, where the goal is to efficiently list all the objects from the class, each object exactly once.
In this talk I give an overview over some long-standing problems and recent advances in this area, highlighting newly discovered connections to related mathematical problems in graph theory, algebra, order theory, geometry, and algorithms.

My research focuses mainly on the large-scale behavior of directed acyclic graphs, Young tableaux, and lattice paths. I am primarily interested in universal phenomena such as specific terms in the asymptotic growth (e.g., critical exponents $n^{-3/2}$ or stretched exponentials $\mu^{n^{1/3}}$) and limit laws (e.g., normal or Mittag-Leffler distributions). I will present several general tools I have developed to prove such phenomena extending methods from Analytic Combinatorics and Probability Theory. They allowed me to solve open problems on, e.g., the number of minimal automata and phylogenetic networks, or the limiting behavior of the sum-of-digits function, periodic Young tableaux and Pólya urns.

Trees are one of the most fundamental classes of graphs, and many different models of random trees have been put forward. Results on the distribution of tree parameters play a key role, and it is therefore desirable to develop general methods for this purpose. I will discuss how the notion of additive tree functionals is a unifying tool to obtain limit theorems for a wide variety of tree parameters under different models of randomness. I will illustrate these results with topical examples and applications.

We study precise asymptotic estimates of running times of algorithms linked with the divide and conquer paradigm. In particular, we investigate variants of quicksort and related lattice paths, regular sequences which are a generalisation of the divide and conquer paradigm and finally sequences defined by finite transducer automata.

The study of the asymptotic behaviour of discrete structures is a growing field of research that lies at the intersection of combinatorics and probability theory. It deals with the enumeration, random sampling, and study of shape characteristics of combinatorial objects such as trees, permutations, and graphs. The methods employed range from bijective combinatorics and singularity analysis of analytic generating functions to stochastic process techniques. The talk focuses on selected archetypes of random structures and intends to highlight universal phenomena.

We will discuss connections between combinatorics and other areas that study analytic objects (such as measure theory, descriptive set theory, measure-preserving group actions, etc), focusing on how combinatorial methods and ideas were successfully applied to find constructive answers to foundational problems, such as Tarski's Circle Squaring Problem from 1925. If time permits, we will also present some applications of analytic methods to finite combinatorics.

We discuss recent results and methods to construct new examples, e.g

(1) on large gaps in the value set of sums of two integer squares

(2) large caps over small fields, i.e. point sets in $(F_p)^n$ without three points on any line, where $p$ is small, and $n$ is large.
Here the case $p=3$ has attracted a great deal of attention. We show that a novel type of construction gives for $p=5$ sets of comparable density, und announce a considerably improved record.
Large caps seem to look considerably different than was known before.

We use combinatorial, computational, number theoretic and probabilistic tools.

An expander graph is a graph which satisfies a particular type of discrete isoperimetric inequality, asserting that large sets in the graph have correspondingly large boundaries. The notion of graph expansion has turned out to be fundamental to many areas of mathematics, uncovering deep links between combinatorics, geometry, probability, algebra as well as to many topics in computer science. In this talk I will discuss some of my research related to expansion, in particular how expansion can be used as a tool to study the structure of random graph models, and the problem of determining the expansion of lattice-like families of graphs.

13:00 Marco ZANK (U Wien): Compressive aspects of space-time methods

13:30 Peter GANGL (RICAM Linz): Space-time design optimization of electric machines

14:00 Andreas SCHAFELNER (RICAM Linz): Parallel space-time solvers for parabolic evolution problems

14:30 Mario MALLY (TU Darmstadt): Gauging and domain decomposition for efficient magnetoquasistatic simulation

15:30 Mario GOBRIAL (TU Graz): Parallel space-time solvers for eddy current problems

16:00 Christian KÖTHE (TU Graz): Adaptive least-squares space-time FEM

16:30 Richard LÖSCHER (TU Graz): Adaptive space-time FEM for the wave equation

How long does it take for a random walk to cover all the vertices of a graph?
And what is the structure of the uncovered set (the set of points not yet visited by the walk) close to the cover time?
We show that on vertex-transitive graphs of bounded degree, this set is decorrelated (it is close to a product measure) if and only if a simple geometric condition on the diameter of the graph holds. In this case, the cover time has universal fluctuations: properly scaled, the cover time converges to a Gumbel distribution.
To prove this result we rely on recent breakthroughs in geometric group theory which give a quantitative form of Gromov's theorem on groups of polynomial growth.
We also prove refined quantitative estimates showing that the hitting time of any set of vertices is (irrespective of its geometry) approximately an exponential random variable.

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

Meeting number: 2730 500 3129

Password: vQydpg372D4

By a celebrated result of Ax from the 60s, everything non-trivial that can be said about 'large'
finite fields follows from the Lang-Weil estimates. This may be made precise with the help
of so-called pseudofinite fields.

A twisted version of the Lang-Weil estimates, due to Hrushovski, leads to an analogous
result on the q-Frobenius automorphism, where q is a 'large' prime power, ultimately
showing that the non-standard Frobenius automorphism satisfies a Nullstellensatz for
difference polynomials.

In the talk, we will first give an overview of these results, and we will then present a
classification of definable quotients in algebraically closed fields endowed with a
non-standard Frobenius automorphism, also in case a non-trivial valuation is added to
the structure.

In this seminar we discuss how to provably determine all rational points on curves $X_0^+(p)$ of genus $g$ up to $6$ (for prime levels $p$). Let $r$ be the rank of the Jacobian of the curve (over the rationals). As these curves satisfy $r = g$, we use Quadratic Chabauty. We also determine all rational points on hyperelliptic curves $X_0^*(N)$ where we used other methods as well: rank $0$ quotients, Mordell-Weil Sieve and variations of Chabauty’s method. Since the points on these
curves parametrize elliptic curves with additional structure, we also classify rational points on all $X_0^+(p)$ and on
hyperelliptic $X_0^*(N)$ for $N$ squarefree. We also discuss future work and applications.
This seminar presents several works written in co-authorship with V. Arul, L. Beneish, M. Chen, S. Chidambaram, T. Keller, O. Padurariu and B. Wen.

Let $F(x,y) \in \mathbb{Q}[x,y]$ be an irreducible polynomial of degree $d>1$ in $y$. Hilbert's Irreducibility Theorem
(HIT) states that for the vast majority of integers $n$ the polynomial
$F(n,y) \in \mathbb{Q}[y]$ is irreducible, i.e.
$[\mathbb{Q}(\theta_{n}):\mathbb{Q}]=d$ for any root $\theta_{n}$ of $F(n,y)$. However, HIT does not answer the following questions:
\begin{itemize}
\item Given an integer $N$, what is the degree of $\mathbb{Q}(\theta_{1},\theta_{2}, \dots, \theta_{N})$?
\item How many distinct fields are there among $\mathbb{Q}(\theta_{j})$, $1 \leq j \leq N$?
\end{itemize}
These questions were first studied by Dvornicich and Zannier, who showed that there is a positive constant $c$ such that $\mathbb{Q}(\theta_{1},\dots, \theta_{N}) \geq e^{cN/\log N}$, and consquently that there are at least $c' N/\log N$ many distinct fields among $\mathbb{Q}(\theta_{j})$ with $j \leq N$.

We will consider the larger set of fields $\mathbb{Q}(\theta_{r})$ where $r \in \mathbb{Q}$ varies over rational numbers of height $H(r) \leq N$. Under some assumptions on $F$ we will obtain a lower bound on the number of distinct fields among $\mathbb{Q}(\theta_{r})$, $H(r) \leq N$.

We study the planted-dense-subgraph model where an Erdős–Rényi base graph is augmented by adding one or more `communities' - subsets of vertices with a higher-than-average connection probability between them. The detection problem is to distinguish between the vanilla Erdős–Rényi model and that with one or many planted communities and the recovery problem is to approximately recover the position of the community structure within the graph. A detection-recovery gap is known to occur, for some parameter combinations (sizes of structure, edge probabilities), we have fast algorithms to perform detection but recovery is not possible. We investigate something in-between: we want to infer some property of the community structure without needing to recover it. We say counting is the problem of distinguishing a single community from many. Our result is to show counting is no easier than recovery in the low degree framework. The proof involves proving some properties of a recursively defined sequence indexed by graphs.

Joint work with Cynthia Rush, Alex Wein and Dana Yang.

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

Meeting number: 2730 500 3129

Password: vQydpg372D4

The cluster expansion is a classical tool from statistical physics used to understand systems of weakly interacting particles in the high temperature regime of statistical physics models. It can also be a very useful tool in probabilistic, extremal, and enumerative combinatorics and in the study of large deviations in probability theory. I will give an introduction to the cluster expansion, present some examples of combinatorial applications, and try to provide some intuition about when the cluster expansion should or should not be a useful tool for a particular problem.

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

Meeting number: 2730 500 3129

Password: vQydpg372D4

Abstract:
In this talk we discuss some bounds about sets avoiding certain arithmetic or geometric configurations in $\mathbb{F}_p^n$ (or more generally, in $\mathbb{Z}_m^n$).
In particular, sets avoiding right angles in $\mathbb{F}_p^n$ will be considered. We will also give a brief introduction of the so-called slice rank of tensors which plays an important role in our proofs. Our methods can also be used to bound the maximum possible size of a binary code where no two codewords have Hamming distance divisible by a fixed prime $p$. 

Joint work with Bursics, Matolcsi and Schrettner.

We will discuss recent results on moderate deviations (between the order of the standard deviation and the mean) for arithmetic progressions in random sets and subgraphs of random graphs.

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

Meeting number: 2730 500 3129

Password: vQydpg372D4

PROGRAM
=======

13:10–13:30 Opening words

13:30–14:20 Tullio Ceccherini-Silberstein (Benevento/Rome)
“Entropy tightness of irreducible sofic subshifts”

14:25–15:15 Tatiana Nagnibeda-Smirnova (Geneva)
“On dominant vertices and spectral measures on graphs”

15:20–16:00 Coffee break

16:00–16:50 Vadim A. Kaimanovich (Ottawa)
“Singularity of stationary measures”

16:55–17:45 Ecaterina Sava-Huss (Innsbruck)
“Aggregation models with multiple sources on fractal graphs”

17:50 Closing words


REGISTRATION
==========

If you plan to attend, please register until October 17, 2022, 3 pm by sending an e-mail to office5050@math.tugraz.at to allow us to plan the coffee break.

~

We prove that a $1 - o(1)$ fraction of all $k$-SAT functions on $n$ Boolean variables are unate (i.e., monotone after first negating some variables), for any fixed positive integer $k$. This resolves a conjecture by Bollobás, Brightwell, and Leader from 2003.

Joint work with József Balogh, Dingding Dong, Bernard Lidický, and Nitya Mani

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

Meeting number: 2730 500 3129

Password: vQydpg372D4

In bilevel optimization problems, there are two players, the leader and the follower, who make their
decisions in a hierarchy, and both decisions influence each other. Usually one assumes that both players have
full knowledge also of the other player's data and objective. Although bilevel problems are often already NP-
or even $\Sigma^p_2$-hard to solve under this assumption, a more realistic model might sometimes be needed.
Uncertainty can be quantified, for example, using the robust optimization approach: Assume that the leader
does not know the follower's objective function precisely, but only knows an uncertainty set of potential
follower's objectives, and the leader's aim is to optimize the worst case of the corresponding scenarios. Now
the question arises how the computational complexity of bilevel optimization problems changes under the
additional complications of this type of uncertainty.



We make a step towards answering this question by investigating a rather simple bilevel problem that can be
solved in polynomial time without uncertainty. In the bilevel selection problem, the leader and the follower
each select a number of items, while a common number of items to select in total is given, and each of the
two players maximizes the total value of the selected items, according to different sets of item values. We
compare several variants of this problem and show that all of them can be solved in polynomial time. We then
investigate the complexity of the robust version of the problem. If the item sets controlled by the leader
and by the follower are disjoint, it can still be solved in polynomial time in case of a finite uncertainty
set or interval uncertainty. If the two item sets are not disjoint, the robust problem version becomes
NP-hard, even for a finite uncertainty set, which shows that uncertainty can indeed add additional complexity to a bilevel optimization problem.

If the need arises, the talk can be streamed via Webex. Please contact Bettina Klinz in advance to be provided with a link.

Given an $r$-edge-colouring of the complete graph $K_n$, what is the largest
number of edges in a monochromatic connected component? This natural question has only recently received the attention it deserves, with work by two disjoint subsets of the authors resolving it for the first two special cases, when $r = 2$ or $3$. Here we introduce a general framework for studying this problem and apply it to fully resolve the $r = 4$ case, showing that such a coloring always yields a monochromatic component with at least $\frac 1 {12} \binom{n}{2}$ edges, where the constant $\frac 1 {12}$ is optimal only when the coloring matches a certain construction of Gyarfas.

Joint work with David Conlon and Sammy Luo.

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

Meeting number: 2730 500 3129

Password: vQydpg372D4

14.00–14.45 Fredi Tröltzsch (TU Berlin):
On elliptic optimal control problems with control appearing nonlinearly
in the state equation
14.45–15.30 Olaf Steinbach (TU Graz):
(Space-time) finite element methods for optimal control problems with
energy regularization
16.00–16.45 Richard Löscher (TU Graz):
Numerical illustration of an abstract setting for distributed
optimal control problems
16.45–17.30 Ulrich Langer (JKU Linz):
Efficient iterative solvers for finite element optimality systems

How many spheres are needed to construct a knotted necklace? Behind this harmless question lies a deep connection between knot theory, sphere packings, polytopes, number theory and Lorentz geometry. In this talk, we will see how these different theories are connected, and we will use them to describe a method for the construction of optimal knotted necklaces, which at the same time produces solutions of certain Diophantine equations.

This is a joint work with Jorge Ramírez Alfonsín.

We will discuss several problems at the interface of analytic number theory and Fourier analysis, involving the distribution of values and zeros of the Riemann zeta-function, and also the distribution of integers and primes represented by quadratic forms. First, we discuss Selberg's celebrated central limit theorem, which roughly states that the logarithm of the Riemann zeta-function is normally distributed on the critical line. We will discuss related joint works with Andrés Chirre and with Meghann Moriah Lugar and Micah B. Milinovich. In particular, we discuss our proof of a conjecture of Berry (1988), conditional on RH and on a strong version of the pair correlation conjecture. Finally, we also discuss a Fourier analysis approach to the study of integers and primes represented by binary quadratic forms -- a classical problem going back to Fermat (joint work with Andrés Chirre).

Abstract:}
In this talk, we provide statistical guarantees for a kernel-based estimation of time-varying parameters driving the dynamic of infinite memory processes introduced by Doukhan and Wintenberger (2008). We then extend the results of Dahlhaus, Richter, and Wu (2019) on local stationary Markov processes to other models, such as the GARCH model. The estimators are computed as localized M-estimators of any contrast satisfying appropriate regularity conditions. We prove the uniform consistency and the pointwise asymptotic normality of such kernel-based estimators. We apply our results to usual contrasts such as least-square, least absolute value, or quasi-maximum likelihood contrasts. We consider various time-varying models such as AR($\infty$), ARCH($\infty$), and LARCH($\infty$). We discuss their approximation of locally stationary ARMA and GARCH models under contraction conditions. Numerical experiments demonstrate the efficiency of the estimators on both simulated and real data sets.
This talk is based on a joint work with J.-M. Bardet and P. Doukhan

The precise reference is (if needed)
Bardet, J. M., Doukhan, P., & Wintenberger, O. (2022).
Contrast estimation of time-varying infinite memory processes. SPA, 152, 32-85.

A complex reflection is an invertible linear transformation of a finite-dimensional complex vector space that has finite order and acts trivially on a complex subspace of codimension one. A complex reflection group is a finite group generated by complex reflections. Replacing “complex” with “real”, one obtains precisely the finite Coxeter groups, which in turn comprise a generalization of Weyl groups of semisimple Lie algebras. Complex reflection groups have many applications, including to: representation theory of reductive algebraic groups, Hecke algebras, knot theory, moduli spaces, low-dimensional algebraic topology, invariant theory and algebraic geometry, differential equations, and mathematical physics.

In joint work with Nathan Williams, we study normal reflection subgroups of complex reflections groups (that is, normal subgroups that are also generated by complex reflections). Our approach leads to a refinement of the celebrated theorem of Orlik and Solomon that the generating function for fixed-space dimension over a complex reflection group is a product of linear factors involving generalized exponents. Our refinement gives a uniform proof and generalization of a recent theorem of Nathan Williams.

Poisson summation, in particular in combination with the theory of oscillatory integrals, is a very versatile tool in analytic number theory.
The talk illustrates this statement in the context of pseudo-randomness of sequences. We report on joint work with
1) Athanasios Sourmelidis (TU Graz) and Christopher Lutsko (Rutgers) on fine-scale statistics of slowly growing sequences, and
2) with Rajula Srivastava (U. Bonn) on the density of rational points close to rough hyper-surfaces.

The complexity classes Unique End of Potential Line (UEOPL) and its
promise  version PUEOPL were introduced in 2018  by Fearnly et al. PUEOPL
captures search problems where the instances  are promised to have a unique
solution. UEOPL captures total search  versions of these promise problems.
The promise problems can be made  total by defining violations that are
returned as  a short certificate of an unfulfilled promise.
GridUSO  is the problem of finding the sink in a grid with a unique sink
orientation. It was introduced by Gärtner  et al. in 2008. We describe a
promise preserving reduction from GridUSO to UniqueForwardEOPL, a
UEOPL-complete problem. Thus, we show that  GridUSO is in UEOPL and its
promise version is in PUEOPL.

Abstract:

Groups acting on trees play a particularly important role in the broader theory of locally compact groups. Generalising a construction of Burger-Mozes, I define a class of groups acting on regular trees by imposing restrictions on how elements are allowed to act on balls of a given radius around the vertices of the tree. Following basic properties and examples, I use these groups to classify all locally transitive groups that contain an edge inversion of order 2 and satisfy one of several independence properties.



Webex Meeting-Link:

https://tugraz.webex.com/tugraz/j.php?MTID=me1e8ee701b05885ed2a24adfa7b644fd

Meeting-Kennnummer:
2733 296 8400

Meeting-Passwort:
awSFXzcj352

On 1.+2. July 2022 we hold a small workshop with talks by Rainer Dietmann, Florian Luca, Igor Shparlinski, Cathy Swaenepoel, Arne Winterhof, Michael Drmota, Lukas Spiegelhofer, Kübra Benli, Pierre Bienvenu, Andrei Shubin, Paul Peringuey.

The programme can be found on the workshop webpage:
https://www.math.tugraz.at/arithrand22}

The Tur\'an density} of an $r$-graph $H$, denoted $\pi(H)$, is the limit of the maximum density of an $n$-vertex $r$-graph not containing a copy of $H$, as $n \to \infty$. Denote by $C_{\ell}$ the $3$-uniform tight cycle on $\ell$ vertices.

A conjecture due to Mubayi and R\"odl (2002) asserts that $\pi(C_5) = 2\sqrt{3} - 3 \approx 0.464$, with the conjectured extremal example being an iterated construction which does not contain tight cycles whose length is not divisible by $3$.

We prove that any sufficiently long tight cycle whose length is not divisible by $3$ has the above conjectured Tur\'an density. Namely, if $\ell$ is sufficienly large and not divisible by $3$ then $\pi(C_{\ell}) = 2\sqrt{3} - 3$.
This is joint work with Nina Kam\v{c}ev and Alexey Pokrovskiy.

Meeting link:
\[
\text{https://tugraz.webex.com/tugraz/j.php?MTID=m40f85343e56ff5051d731ace1bea82e4}
\]

Meeting number: 2731 089 0467

Password: btHRJxCa252

The study of the partition function $p(n)$, counting the number of solutions of
the equation $n=a_{1} + \dots + a_{\ell}$ over integers
$1 \leq a_{1} \leq \dots \leq a_{\ell}$, has a long history in combinatorics. In
the present talk we consider the following variant of this question: partitions
in integers of the form
\begin{equation*}
n=\lfloor a_1^\alpha\rfloor+\cdots+\lfloor a_\ell^\alpha\rfloor
\end{equation*}
with $1\leq a_1< \cdots < a_\ell$ and $0 < \alpha < 1$ a fixed real. Using the
saddle point method we show a central limit theorem for the number of summands
in a random partition of that kind.

This is joint work with Gabriel Lipnik and Robert Tichy.

Erdős and Pósa proved in 1965 that there is a duality between the maximum size of a packing of cycles and the minimum size of a vertex set hitting all cycles.
We therefore say that cycles satisfy the Erdős-Pósa property.
However, while odd cycles do not satisfy the Erdős-Pósa property, Reed proved in 1999 an analogue by relaxing packing to half-integral packing, where each vertex is allowed to be contained in at most two such cycles.
Moreover, he gave a structural characterisation for when the Erdős-Pósa property for odd cycles fails.
We prove a far-reaching generalisation of the theorem of Reed; if the edges of a graph are labelled by finitely many abelian groups, then the cycles whose values avoid a fixed finite set for each abelian group satisfy the half-integral Erdős-Pósa property, and we similarly give a structural characterisation for the failure of the Erdős-Pósa property.
A multitude of natural properties of cycles can be encoded in this setting.
For example, we show that the cycles of length $\ell$ modulo $m$ satisfy the half-integral Erdős-Pósa property, and we characterise for which values of $\ell$ and $m$ these cycles satisfy the Erdős-Pósa property.

This is the work of Pascal Gollin, Kevin Hendrey, Ken-ichi Kawarabayashi, O-joung Kwon, Sang-il Oum, and Youngho Yoo.

Meeting link:
\[
\text{https://tugraz.webex.com/tugraz/j.php?MTID=m40f85343e56ff5051d731ace1bea82e4}
\]

Meeting number: 2731 089 0467

Password: btHRJxCa252

Let $L_{c,n}$ be the length of the longest cycle in a sparse binomial random graph $G_{n,p}$, $p = c/n$, $c>1$. Erd\H{o}s conjectured that if $c>1$ then w.h.p. $L_{c,n}\geq \ell(c)n$ for some strictly positive function on $(1,\infty)$ that is independent of $n$. His conjecture was proved by Ajtai,
Koml\'os and Szemer\'edi and in a slightly weaker form by Fernandez de la Vega. Henceforward there has been a line of research in trying to bound $L_{c,n}$ for $c>1$. In this talk we will discuss how one can identify a set of vertices that spans a longest cycle in $G_{n,p}$ w.h.p for sufficiently large $p$. We will then show that $\frac{L_{c,n}}{n}$ converges to some continuous function $f(c)$ almost surely which can be evaluated within arbitrary accuracy for $c>C_0$ where $C_0$ is a sufficiently large constant. This talk is based on a joint work with Alan Frieze.

Meeting link:
\[
\text{https://tugraz.webex.com/tugraz/j.php?MTID=m40f85343e56ff5051d731ace1bea82e4}
\]

Meeting number: 2731 089 0467

Password: btHRJxCa252

The study of relations between moments and cumulants play a central role in both classical and non-commutative probability theory. Using the group of
characters on a combinatorial Hopf algebra H of ``words on words'', and its corresponding Lie algebra of infinitesimal characters, it is possible to
study distinct families of cumulants corresponding to different types of independences: free, boolean and monotone. We discuss several formulas for
the (known) free-to-moment and boolean-to-moment relations, obtained from the antipode of H. Also, using a weighted Möbius inversion, we deduce a new relation of monotone cumulants in terms of moments.

How and when to pay out dividends is a crucial question for many companies.
In stochastic control theory, this question can be modelled as the problem of choosing a càdlàg, adapted process D (representing the accumulated dividend payments) which maximises the expected discounted value of dividends up to the time when the post-dividend surplus becomes negative. Often, the discounting function is modelled in such a way that money `today' is preferred to money `tomorrow'. However, these preferences can change over time and are subject to various exogenous and endogenous factors, such as changes in management, the influence of (im-)patient investors, regulatory requirements and market crashes.
In this talk, we consider an extension of the dividend maximisation problem for an insurance company by allowing switches between a positive and a negative time preference rate. The negative preference reflects the `tendency to wait' many companies show in times of uncertainty. We model the surplus by a Brownian risk process and the preference rate by a two-state Markov chain. We solve the problem of finding the optimal dividend payout strategy for the setting with a classical ruin concept as well as for the case of Parisian ruin with an exponential delay.
The talk is based on joint work with Julia Eisenberg (TU Vienna).

We study the algorithmic complexity of computing persistent homology of
a randomly chosen filtration. Specifically, we prove upper bounds for the average fill-up (number of non-zero entries) of the boundary matrix on Erd\H{o}s--R\'enyi and Vietoris--Rips filtrations after matrix reduction.
Our bounds show that, in both cases, the reduced matrix is expected to
be significantly sparser than what the general worst-case predicts.
Our method is based on previous results on the expected first Betti
numbers of corresponding complexes. We establish a link between these results to the fill-up of the boundary matrix.
Our bound for Vietoris--Rips complexes is asymptotically tight up to
logarithmic factors. We also provide an Erd\H{o}s--R\'enyi filtration realising the worst-case.

Joint work with Barbara Giunti (TU Graz) and Guillaume Houry
(Ecole Polytechnique)

Meeting link:
\[
\text{https://tugraz.webex.com/tugraz/j.php?MTID=m40f85343e56ff5051d731ace1bea82e4}
\]

Meeting number: 2731 089 0467

Password: btHRJxCa252

In the first part of the talk I will make a brief introduction to free probability and motivate the results presented in the second part.

Next I will discuss the problem of the tail behaviour of free multiplicative convolution powers $\mu^{\boxtimes t}$, $t\geq 1$. We consider measures $\mu$ on the positive half-line with regularly varying tail, which means that \begin{align}\label{eqn:1}\mu\left((x,+\infty)\right)\sim x^{-\alpha} L(x), \end{align} where $L$ is a slowly varying function, and $f(x)\sim g(x)$ means that $g(x)/f(x)\to 1$ as $x\to +\infty$.

We will characterize the behaviour of the S-transform at $0^-$ of measures with regularly varying tails.

Next I will discuss some applications of this result. In particular we will show that there is a phase transition in the tail behaviour of $\mu^{\boxplus t}$, for measures as in \eqref{eqn:1}, between regimes $\alpha<1$ and $\alpha>1$.

Talk based on a joint work with Bartosz Kołodziejek.

Given an $r$-edge-coloured complete graph $K_n$, how many monochromatic connected components does one need in order to cover its vertex set? This natural question is a well-known essentially equivalent formulation of the classical Ryser's conjecture which, despite a lot of attention over the last 50 years, still remains open. A number of recent papers consider a sparse random analogue of this question, asking for the minimum number of monochromatic components needed to cover the vertex set of an $r$-edge-coloured random graph $G(n,p)$.

We discover a very strong connection between this problem and a certain Helly-type local to global hypergraph covering question raised about 30 years ago by Erdős, Hajnal and Tuza. This allows us to obtain a good understanding of the answer to the general problem giving in particular some very surprising answers to questions raised by Bal and DeBiasio; Kohayakawa, Mota and Schacht; Lang and Lo; Girão, Letzter and Sahasrabudhe; and Kohayakawa, Mendonça, Mota and Schülke.

Based on joint works with Korándi and Sudakov and with Bradač.

Meeting link:
\[
\text{https://tugraz.webex.com/tugraz/j.php?MTID=m40f85343e56ff5051d731ace1bea82e4}
\]

Meeting number: 2731 089 0467

Password: btHRJxCa252

We discuss the behavior of random walks driven by measures that have stable-like tails on nilpotent groups and groups with polynomial volume growth. This talk is based on joint work with Alexander Bendikov, with Tianyi Zheng and with Zhen-Qing Chen, Takashi Kumagai, Jian Wang, and Tianyi Zheng. 



Webex meeting, meeting link:

https://tugraz.webex.com/tugraz-de/j.php?MTID=me5ef9882738c84a9bc781b0c706c69c1}

Meeting number: 2734 763 6683
Password: bgZMVUVf283

The meeting opens at 14:3, the 50min talk starts at 15:00

Reed-Solomon codes are an important and intensively studied class of error-correcting codes. After giving some background, this talk will discuss the so-called list-decoding problem for Reed-Solomon codes. More specifically, we prove that for any fixed list-decoding parameters, there exist Reed-Solomon codes with a certain rate, which is optimal up to a constant factor. This in particular answers a question of Guo, Li, Shangguan, Tamo, and Wootters about list-decodability of Reed-Solomon codes with radius close to 1. Joint work with Asaf Ferber and Matthew Kwan.

Meeting link:
\[
\text{https://tugraz.webex.com/tugraz/j.php?MTID=m40f85343e56ff5051d731ace1bea82e4}
\]

Meeting number: 2731 089 0467

Password: btHRJxCa252

10:00 -- 11:00: Prof. Christopher Frei (TU Graz)}
Titel: Quadratic polynomials represented by relative norms
Abstract: We discuss work in progress, joint with Jonathan Chapman, on
the arithmetic of equations of the shape P(t) = N(x), where P is an
irreducible quadratic polynomial and N a norm form of an extension of
number fields. Our main contribution is an extension of analytic methods
due to Browning, Heath-Brown and Schindler to the relative setting.



11:00 -- 12:00: Prof. Janos Pintz (Hungarian Academy of Sciences)}
Title: Density estimates for the Riemann zeta-function
Abstract: We present new density estimates for the number N(a,T) of zeta-zeros in the domain Res$>$a, $-$T $<$Ims$<$ T for the case a $>$ 19$/$20.
These improve earlier results of Heath-Brown and Bourgain. The results use beside an idea of Halasz an optimal form of Vinogradov's mean value theorem proved by Bourgain, Demeter and Guth and ideas of Heath-Brown.




14:00 -- 15:00: Prof. P\'eter P\'al Pach (Universität Budapest)}
Title: The Alon-Jaeger-Tarsi conjecture via group ring identities
Abstract: The Alon-Jaeger-Tarsi conjecture states that for any finite field $F$ of size at least 4 and any nonsingular matrix $M$ over $F$ there exists a vector $x$ such that neither $x$ nor $Mx$ has a 0 component. In joint work with János Nagy we proved this conjecture when the size of the field is sufficiently large, namely, when $61<|F|\ne 79$. In this talk we will discuss this result. Our proof uses group ring identities and additive combinatorial lemmas.




15:00 -- 16:00: Prof. Tim Browning (IST Austria)}
Titel: Manin-Peyre for some threefolds involving quadratic forms
Abstract: I'll summarise the state of affairs for the Manin-Peyre conjecture in dimension 3, with particular emphasis on Fano threefolds of Picard rank 2, before discussing its resolution for an explicit family that have the structure of a quadric bundle. This gives a new case where a "thin set" of rational points needs to be removed in order for the predicted density of rational points to be valid. This is joint work with Dante Bonolis and Zhizhong Huang.

The Erdös-Ko-Rado theorem is a classical result in extremal set theory which has remarkable generalizations in various areas of mathematics.  In this talk, we are interested in a generalization of the Erdös-Ko-Rado theorem for permutation groups. In this talk, we review some of the early results of Erdös on intersecting sets of permutations and some recent developments.

Webex meeting, meeting link:

https://tugraz.webex.com/tugraz-de/j.php?MTID=mf9da2b68041ce22f6d11d9f22cca6e24}

Meeting number: 2733 974 6457

Password: A4Ch3mZP8Yp

The meeting opens at 9:40, the 50min talk starts at 10:00

The Gromov-Hausdorff distance between two metric spaces is an
important tool in geometry, but it is difficult to compute. For example,
the Gromov-Hausdorff distance between unit spheres of different
dimensions is unknown in nearly all cases. I will introduce recent work
by Lim, Mémoli, and Smith that lower bounds the Gromov-Hausdorff
distance between spheres using Borsuk-Ulam theorems. We improve these
lower bounds by connecting this story to Vietoris-Rips complexes,
providing new generalizations of the Borsuk-Ulam theorem. This is joint
work in a polymath-style project with many people, most of whom are
currently or formerly at Colorado State, Ohio State, Carnegie Mellon, or
Freie Universität Berlin.

In this talk we apply a greedy framework to derive exhaustive generation algorithms for two classes of combinatorial objects, as well as Hamilton paths and cycles on their corresponding polytopes:
(1) different classes of rectangulations, which are subdivisions of a rectangle into smaller rectangles (see www.combos.org/rect);
(2) elimination trees of chordal graphs, which generalize several interesting combinatorial/geometric objects such as permutations, binary trees and bitstrings (see www.combos.org/elim).
If time permits, we will also discuss a dual approach which allows for efficiently generating all of the bases/independent sets of a matroid, as well as Hamilton paths on their corresponding polytopes.

This talk is based on joint work with Jean Cardinal, Torsten Mütze and Aaron Williams.

Providing reliable and efficient supply with medical treatment and health related services is a truly challenging task. Some of these challenges including travel time, coverage and reachability are increased by the fact that health care infrastructure is usually designed to be stationary, i.e. patients need to seek treatment at a hospital or a doctor’s office. However, mobile health care services such as vaccination buses or motorcycles become more and more popular, especially in rural areas.

In this talk, I will explain how the planning of mobile medical services relates to the budgeted colored matching problem. The budgeted colored matching problem is a variant of the perfect matching problem on edge colored graphs where budgets restrain the number of matching edges of a specific color. We will discuss the complexity of this problem as well as its solvability on special graph classes.




Meeting link:
\[
\text{https://tugraz.webex.com/tugraz/j.php?MTID=mbf592cad71f71db26df1fc108970740a
}
\]

Meeting number: 2734 364 9879

Meeting password: ESpbcZY3Q32

The goal of this paper is to establish relative perturbation bounds, tailored for empirical covariance operators. Our main results are expansions for empirical eigenvalues and spectral projectors, leading to concentration inequalities and limit theorems. One of the key ingredients is a specific separation measure for population eigenvalues, which we call the relative rank, giving rise to a sharp invariance principle in terms of limit theorems, concentration inequalities and inconsistency results. Our framework is very general, requiring only $p>4$ moments and allows for a huge variety of dependence structures.

In this talk, we will discuss evolutionary games on a binomial random graph $G(n,p)$. These games are determined through $2$-player symmetric game with $2$ strategies played between the incident members of the vertex set. Players/vertices update their strategies synchronously: at each round, each player selects the strategy that is the best response to the current set of strategies its neighbours play. We show that such a system reduces to generalised majority and minority dynamics. We show rapid convergence to unanimity for $p$ in a range that depends on a certain characteristic of the payoff matrix. In the presence of a bias among the pure Nash equilibria of the game, we determine a sharp threshold on p above which the largest connected component reaches unanimity with high probability, and below which this does not happen.

This is joint work with Jordan Chellig and Calina Durbac.

Meeting link:
\[
\text{https://tugraz.webex.com/tugraz/j.php?MTID=m40f85343e56ff5051d731ace1bea82e4}
\]

Meeting number: 2731 089 0467

Password: btHRJxCa252

Erd\H{o}s proved that there are infinitely many weakly prime numbers (also called (digitally) delicate primes), i.e. prime numbers such that changing any single one of the digits, in a given base, with any other digit always results in a composite number. Tao proved that weakly prime numbers constitute a positive proportion in all prime numbers.
In this talk, we are going to discuss
further quantitative refinements on the distribution of weakly prime numbers.

Abstract: The celebrated Skolem--Mahler--Lech theorem asserts that if ${\bf u}:=(u_n)_{n\ge 0}$ is a linearly recurrent sequence of integers then the set of its zeros, that is the set of positive integers $n$ such $u_n=0$, form a union of finitely many infinite arithmetic progressions together with a (possibly empty) finite set. Except for some special cases, is not known how to bound effectively all the zeros of ${\bf u}$. This is called {\it the Skolem problem}. In this talk we present the notion of a {\it universal Skolem set}, which an infinite set of positive integers ${\mathcal S}$ such that for every linearly recurrent sequence ${\bf u}$, the solutions $u_n=0$ with $n\in {\mathcal S}$ are effectively computable. We present a couple of examples of universal Skolem sets, one of which has positive lower density as a subset of all the positive integers.

I will discuss percolation on the Voronoi tessellation generated by a
homogeneous Poisson point process on the hyperbolic plane. That is, to each point $z$ of a constant intensity Poisson point process $Z$ on the hyperbolic plane we assign its Voronoi cell -- the region consisting of all points that are closer to $z$ than to any other $z'$ in $Z$ -- and we colour each cell black with probability $p$ and white with probability $1-p$, independently of the colours of all other cells. We say that percolation occurs if there is an infinite connected cluster of black cells.

Hyperbolic Poisson-Voronoi percolation was first studied by Benjamini and Schramm about twenty years ago. Their results show that there are spectacular differences with the corresponding model in the Euclidean plane.

I will sketch joint work with my recently graduated doctoral student Ben Hansen that resolves a conjecture and an open question, posed by Benjamini and Schramm, on the behaviour of the ``critical probability for percolation'' as a function of the intensity parameter of the underlying Poisson process. (Unlike in Euclidean Poisson-Voronoi percolation, this critical value depends on the intensity of the underlying Poisson process.)

Based on joint work with Benjamin Hansen.

Meeting link:
\[
\text{https://tugraz.webex.com/tugraz/j.php?MTID=m40f85343e56ff5051d731ace1bea82e4}
\]

Meeting number: 2731 089 0467

Password: btHRJxCa252

We develop a limit theory for Latin squares, paralleling the recent limit theories of dense graphs and permutations. We introduce a notion of density, an appropriate version of the cut distance, and a space of limit objects – so-called Latinons. Key results of our theory are the compactness of the limit space and the equivalence of the topologies induced by the cut distance and the left-convergence. Last, using Keevash's recent results on combinatorial designs, we prove that each Latinon can be approximated by a finite Latin square.

This is joint work with Robert Hancock, Jan Hladký and Maryam Sharifzadeh.

Meeting link:
\[
\text{https://tugraz.webex.com/tugraz/j.php?MTID=m40f85343e56ff5051d731ace1bea82e4}
\]

Meeting number: 2731 089 0467

Password: btHRJxCa252

The Følner function of a group is defined on positive integers n as the smallest size of a Følner set, the boundary of which is at most 1/n times the size of the set. Its values are then finite if and only if the group is amenable. It can be thought of as encoding `how amenable a group is'. The functions obtained on the same group by different generating sets are distinct, but asymptotically equivalent. Numerous results on Følner functions are known, however only up to asymptotic equivalence. In this talk, we will consider fixed generating sets and obtain (to our knowledge) the first results on the exact values of exponential Følner functions. We will present possible applications, in particular to the Coulhon -- Saloff-Coste inequality -- in terms of whether its constants can be improved. The inequality in question provides, as a corollary, a lower bound for the Følner function. In joint work with Christophe Pittet, for groups of exponential growth we obtain an explicit expression of the optimal multiplicative constant in the Coulhon -- Saloff-Coste inequality. We prove that the optimal value over all groups for that constant is between 1 and 2.



Webex meeting, meeting link:

https://tugraz.webex.com/tugraz-de/j.php?MTID=mb470ebdf01d053878567d11a89947bb5}

Meeting number: 2731 075 8334

Password: Q7Wf9ZT2ijP

The meeting opens at 10:40, the 50min talk starts at 11:00

We will present two theorems concerning the global and local structure of a uniform spanning tree (UST) of a connected simple graph with degrees linear in the number of vertices. The first theorem states that the diameter of the UST on such graphs is of order square root of the number of vertices with high probability. The second theorem provides the asymptotic frequency of the appearance of any small subtree (i.e., the local limit of the UST) in terms of the associated graphon. The proof of both theorems illustrates how to combine Szemeredi-like partition theorems with Kirchhoff's electric network theory.

Based on joint works with Noga Alon and Matan Shalev and with Jan Hldaky and Tuan Tran.

Meeting link:
\[
\text{https://tugraz.webex.com/tugraz/j.php?MTID=m40f85343e56ff5051d731ace1bea82e4}
\]

Meeting number: 2731 089 0467

Password: btHRJxCa252

The Green-Tao theorem states that the set of primes contain arbitrarily long arithmetic progressions. It is reasonable to wonder whether the same holds for certain subsets of the primes, and whether arithmetic progressions may be replaced by other linear configurations, i.e. solutions to systems of linear equations. I will review known (and unknown) results in that area and talk about a recent joint work with Joni Teräväinen and Fernando Shao, where we provide a general criterion for a set of integers to contain linear configurations and show that this criterion applies to certain interesting sets of primes such as Chen primes or almost twin primes.

The size Ramsey number of a graph $H$ is defined as the minimum number of edges in a graph $G$ such that there is a monochromatic copy of H in every two-coloring of the edges of G. The size Ramsey number was introduced by Erdős, Faudree, Rousseau, and Schelp in 1978 and they ended their foundational paper by asking whether one can determine up to a constant factor the size Ramsey numbers of three families of graphs: complete bipartite graphs, book graphs, and starburst graphs. In this talk, we completely resolve the latter two questions and make substantial progress on the first by determining up to a constant factor the size Ramsey number of complete bipartite graphs with parts of size $s$ and $t$ for all $t = \Omega(s \log s)$. The proofs involve a variety of unusual random graph models.

Based on joint work with David Conlon and Yuval Wigderson.

Meeting link:
\[
\text{https://tugraz.webex.com/tugraz/j.php?MTID=m40f85343e56ff5051d731ace1bea82e4}
\]

Meeting number: 2731 089 0467

Password: btHRJxCa252

How many $d$-regular $k$-uniform hypergraphs are there on $n$ vertices? We provide an asymptotic formula covering a wide range of parameters, including regular $k$-uniform hypergraphs of density up to $\epsilon k^{-1}$ for $3 \leq k<n^{1/10}$ and $\epsilon >0$. We will discuss the perturbation method, originally applied to the graph enumeration problem by Liebenau and Wormald.

This is joint work with Anita Liebenau and Nick Wormald.

Meeting link:
\[
\text{https://tugraz.webex.com/tugraz/j.php?MTID=m40f85343e56ff5051d731ace1bea82e4}
\]

Meeting number: 2731 089 0467

Password: btHRJxCa252

I will discuss novel definitions of boundary and curvature. The boundary will include an earlier definition of Chartrand-Erwin-Johns-Zhang and satisfy an isoperimetric principle: graphs with many vertices have a large boundary (or gigantic diameter: the path graph only has 2 boundary vertices independently of
length).  The notion of boundary will be inspired by potential theory and satisfy a Bonnet-Myers inequality (graphs with curvature bounded from below have diameter bounded from above) as well as reverse Bonnet-Myers inequality and other desirable properties.



Webex meeting, meeting link:

https://tugraz.webex.com/tugraz-de/j.php?MTID=mf16036a90f945f9a410f7e76246c2ba4}

Meeting number: 2733 846 1857

Password: K4SiDmmvp73

The meeting opens at 15:30, the 50min talk starts at 16:00

Many physical systems can be described by partial differential equations which in turn generate operators between Banach spaces. A well-known illustration of such interplay is quantum mechanics together with the spectral theory of self-adjoint operators in a Hilbert space. However, in several branches of physics like hydrodynamics, damped systems, quantum resonances, superconductivity or balanced loss/gain materials, the occurring partial differential equations contain non-symmetric terms and thus lead to non-self-adjoint spectral problems.

Due to the lack of powerful self-adjoint tools like the spectral theorem or variational principles, the encountered phenomena are very different from the self-adjoint case and include spectral instabilities, divergence of expansions in eigenvectors or non-reliability of (numerical) approximations. Moreover, the analysis of non-self-adjoint problems is much more fragmented, mainly comprising a collection of diverse advanced tools such as pseudospectra, semi-classical methods or Lieb-Thirring inequalities.

In the first part of the talk, we give a general introduction to spectral theory and its motivation originating in the study of partial differential equations. Particular attention will be paid to the non-self-adjoint aspects mentioned above. In the second part, we present a new eigenvalue enclosure for a particular spectral problem related to the stability of the so-called Ekman spiral in hydrodynamics.

{\bf Abstract:} We consider the Recoverable Traveling Salesman Problem (TSP). Here the task is to find two tours simultaneously, such that the intersection between the tours is at least a given minimum size, while the sum of travel distances with respect to two different distance metrics is minimized. Building upon the classic double-tree method, we derive a 4-approximation algorithm for the Recoverable TSP.

(Joint work with Marc Goerigk and Stefan Lendl)




Meeting link:
\[
\text{https://tugraz.webex.com/tugraz/j.php?MTID=mdcccda2bb2e2830bec0a3414b1b2ffd7}
\]

Meeting number:
2732 952 1193

Password:
RdHWamcW933

$4$-regular planar maps can be counted bijectively via two constructions: one with blossoming trees due to Schaeffer, and one with well-labeled trees due to Cori-Vauquelin-Schaeffer.
As shown by Bouttier, Di Francesco, and Guitter, these bijections imply that the so-called 2-point function $R_i(g)$ of $4$-regular planar maps (generating function of $4$-regular maps with two points at dual distance smaller than $i$) satisfies the recurrence $R_i(g)=1+R_i(g)*(R_{i-1}(g)+R_i(g)+R_{i+1}(g))$ (with boundary condition $R_0(g)=0$), which remarkably can be solved explicitly.

I will show how the construction based on blossoming trees can be extended to $4$-regular maps of unfixed genus. This provides a bijective proof of the fact that the corresponding counting series is expressed as the first term $r_1(g)$ of a sequence of series $(r_i(g))_{i\geq 1}$ that are now related by the recurrence $r_i(g)=i+r_i(g)*(r_{i-1}(g)+r_i(g)+r_{i+1}(g))$, with boundary condition $r_0(g)=0$ (this is the same recurrence as for the $R_i(g)$, except for the constant term $i$ instead of $1$), an expression that had previously been obtained via the configuration model and matrix integrals. Our construction also contains the one for $4$-regular planar maps, and it explains the similarity between the series expressions for the planar case and the unfixed genus case.

This is joint work with Emmanuel Guitter.

Meeting link:
\[
\text{https://tugraz.webex.com/tugraz/j.php?MTID=ma70275cd258e7748417214793956c7bf }
\]

Meeting number: 188 980 7021

Password: ahMZ84fJYQ2

We consider a SIR model on the complete graph with n nodes. Each vertex represents an individual which may be susceptible, infected or removed. At the beginning only one node is infected and each node $x$ has a random spread capacity $L_x$. Once infected, the node $x$ becomes an emitter and has $L_x$ attempts to infect other nodes, which are randomly chosen. If the attempt is towards a susceptible node $y$, then $y$ becomes infected, if it is towards an infected or removed node, then $x$ loses an infection attempt.
When $x$ has used all its spread capacity, it becomes removed and no longer interacts with the rest of the population. This model was studied by Comets et al. in the case where the spread capacities are i.i.d. and the nodes which one attempt to infect are chosen uniformly.
We study the case where the population is divided into J types and each type has a possibly different susceptibility, that is, some individuals have a larger probability of being chosen as targets. Also the law of the spread capacity may depend on the type. We obtain a law of large numbers and a central limit theorem for the duration of the process and the proportion of infected nodes. We prove that when the spread capacity is homogeneous, then a population with inhomogeneous susceptibility is overall less affected by the epidemics than a population with homogeneous susceptibility.

This is joint work with J.Kampf, E.Sava-Huss and F.Zucca.


Webex meeting, meeting link:

https://tugraz.webex.com/tugraz-de/j.php?MTID=mfb7a477bc53fac8b0a6db53cf1f2690e}

Meeting number: 2734 226 6767

Password: Je3q3w37mJG

The meeting opens at 10:40, the 50min talk starts at 11:00

The size Ramsey number of a graph $H$ is the smallest number of edges in a graph $G$ with the property that every two-colouring of the edges of $G$ contains a monochromatic copy of $H$. In this talk, we will discuss recent progress on several classical questions about size Ramsey numbers. For instance, what is the size Ramsey number of the complete bipartite graph $K_{s,t}$? And what is the size Ramsey number of a graph with maximum degree three? Parts of this talk represent joint work with Rajko Nenadov and Milos Trujic and also with Jacob Fox and Yuval Wigderson.

\[
\text{https://tugraz.webex.com/tugraz/j.php?MTID=m6c5170bb9fefd922f6a2c6f0ae61c425}
\]

Meeting number: 2731 791 2435

Password: JPmsWgYG332

Abstract:

The presentation consists of two parts. In the first part, we study term structure movements in the spirit of the Heath-Jarrow-Morton methodology under volatility uncertainty. We model the instantaneous forward rate as a diffusion process driven by a G-Brownian motion. The G-Brownian motion represents the (Knightian) uncertainty about the volatility. Within this framework, we derive a sufficient condition for the absence of arbitrage, known as the drift condition. The drift condition allows to construct arbitrage-free term structure models that are completely robust with respect to the volatility. In particular, we obtain robust versions of classical term structure models. In the second part, we study the pricing of contracts in fixed income markets under volatility uncertainty, based on the first part. The setting leads to a sublinear pricing measure for contracts, which yields either a single price or a range of prices. We develop pricing methods for several types of contracts. With these tools, we derive robust pricing formulas for all major interest rate derivatives.

Webex Information:

https://tugraz.webex.com/tugraz/j.php?MTID=me666b8f3ab32db5b443406d6a7763b4c

Meeting number: 2734 301 6604

Password: ymYwJnsW395

Abstract:

Financial institutions have to satisfy capital adequacy tests required, e.g., by the Basel Accords for banks or Solvency II for insurers. At the same time, they would like to maximize their expected utility. Combining both aspects leads to a portfolio optimization problem under a risk constraint. In this talk, we give an example of a tight financial situation in which no reallocation of the initial endowment exists such that the capital adequacy test is passed. The classical portfolio optimization approach breaks down and a capital increase is needed. We introduce the scalarized utility-based multi-asset (SUBMA) risk measure, which optimizes the hedging costs and the expected utility of an agent simultaneously subject to the capital adequacy test. We find out that the SUBMA risk measure is coherent if the utility function has constant relative risk aversion and the capital adequacy test leads to a coherent acceptance set. In a
one-period financial market model we present a sufficient condition for the SUBMA risk measure to be finite-valued and continuous. Under further assumptions on the utility function we
obtain existence and uniqueness results for the optimal hedging strategies. Finally, we calculate the SUBMA risk measure in a continuous-time financial market model for two benchmark
capital adequacy tests.

Webex Information:

https://tugraz.webex.com/tugraz/j.php?MTID=me666b8f3ab32db5b443406d6a7763b4c

Meeting number: 2734 301 6604

Password: ymYwJnsW395

We consider random digraphs in a directed version of the classical Erdős–Rényi model: given n vertices, each possible directed edge is inserted with probability $p$, independently of the others. It turns out that these graphs undergo a phase transition when $p$ is about $1/n$, which can be seen in the answer to questions such as: what is the probability that there are no directed cycles (equivalently, that all strongly connected components are singletons)? Using methods from analytic combinatorics, we obtain very precise asymptotic answers to questions of this kind.

Meeting link:
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\text{https://tugraz.webex.com/tugraz/j.php?MTID=ma70275cd258e7748417214793956c7bf }
\]

Meeting number: 188 980 7021

Password: ahMZ84fJYQ2

We consider Galton-Watson branching processes with countable typeset $X$. We study the vectors $\mathbf{q}(A)=(q_x(A))_{x \in X}$ recording the conditional probabilities of extinction in subsets of types $A \subseteq X$, given that the type of the initial individual is $x$. We first investigate the location of the vectors $\mathbf{q}(A)$ in the set of fixed points. Next, we present equivalent conditions for $\mathbf{q}_x(A) < \mathbf{q}_x(B)$ for any initial type $x$ and $A,B \subseteq X$. Finally, we develop a general framework to characterise all distinct extinction probability vectors, and thereby to determine whether there are finitely many, countably many, or uncountably many distinct vectors. We illustrate our results with examples.

This is based on joint work with D.Bertacchi, P.Braunsteins and S.Hautphenne.


Webex meeting, meeting link:

https://tugraz.webex.com/tugraz-de/j.php?MTID=m32158f79ff92d881d5eb9b4f2502cfef}

Meeting number: 2730 713 4197

Password: kFvAawdJ332

The meeting opens at 10:40, the 50min talk starts at 11:00