Discrete Mathematics 2010-2022

A vertex colouring of a hypergraph is $c$-strong if every edge $e$ sees at least $\min\{c, |e|\}$ distinct colours. Let $\chi(t,c)$ denote the least number of colours needed so that every $t$-intersecting hypergraph has a $c$-strong colouring. In 2012, Blais, Weinstein and Yoshida introduced this parameter and initiated study on when $\chi(t,c)$ is finite: they showed that $\chi(t,c)$ is finite whenever $t \geq c$ and unbounded when $t\leq c-2$. The boundary case $\chi(c-1, c)$ has remained elusive for some time: $\chi(1,2)$ is known to be finite by an easy classical result, and $\chi(2,3)$ was shown to be finite by Chung and independently by Colucci and Gy\'{a}rf\'{a}s in 2013. In this talk, we present some recent work with Kevin Hendrey, Freddie Illingworth and Nina Kam\v{c}ev in which we fill in this gap by showing that $\chi(c-1, c)$ is finite in general.

Meeting link:
\[
\text{https://tugraz.webex.com/tugraz/j.php?MTID=m8500c46344212abf0fa37925da5ef9bf}
\]

Abstract: A Calogero--Moser space describes the completed phase space of a system of finitely many indistinguishable particles with a certain Hamiltonian with quadratic inverse potential in classical physics. Since the past two decades, these spaces are also an object of ongoing study in pure mathematics. In particular, a Calogero--Moser space of $n$ particles is known to be a smooth complex-affine variety equipped with a symplectic holomorphic form, and to be diffeomorphic to the Hilbert scheme of $n$ points in the affine plane.

We establish the symplectic holomorphic density property for the Calogero--Moser spaces and describe their group of symplectic holomorphic automorphisms.

Joint work with Gaofeng Huang.

For integers $k\geq 1$ and $n\geq 2k+1$, the Kneser graph $K(n,k)$ has as vertices all $k$-element subsets of an $n$-element ground set, and an edge between any two disjoint sets. It has been conjectured since the 1970s that all Kneser graphs admit a Hamilton cycle, with one notable exception, namely the Petersen graph $K(5,2)$. This problem received considerable attention in the literature, including a recent solution for the sparsest case $n=2k+1$. The main contribution of our work is to prove the conjecture in full generality. We also extend this Hamiltonicity result to all connected generalized Johnson graphs (except the Petersen graph). The generalized Johnson graph $J(n,k,s)$ has as vertices all k-element subsets of an n-element ground set, and an edge between any two sets whose intersection has size exactly $s$. Clearly, we have $K(n,k)=J(n,k,0)$, i.e., generalized Johnson graphs include Kneser graphs as a special case. Our results imply that all known families of vertex-transitive graphs defined by intersecting set systems have a Hamilton cycle, which settles an interesting special case of Lovász’ conjecture on Hamilton cycles in vertex-transitive graphs from 1970. Our main technical innovation is to study cycles in Kneser graphs by a kinetic system of multiple gliders that move at different speeds and that interact over time, reminiscent of the gliders in Conway’s Game of Life, and to analyze this system combinatorially and via linear algebra.

This is joint work with Arturo Merino (TU Berlin) and Namrata (Warwick).

Meeting link:
\[
\text{https://tugraz.webex.com/tugraz/j.php?MTID=m8500c46344212abf0fa37925da5ef9bf}
\]

{\bf{Montag, 06.05.2024}}

09:00 - 10:00 Uhr: Dennis Schroers
10:30 - 11:30 Uhr: Verena Schwarz
13:30 - 14:30 Uhr: Martin Friesen

{\bf{Dienstag, 07.05.2024}}

09:00 - 10:00 Uhr: Stefan Tappe
10:30 - 11:30 Uhr: Larisa Yaroslavtseva
13:30 - 14:30 Uhr: Florian Bechtold

{\bf{Mittwoch, 08.05.2024}}

09:00 - 10:00 Uhr: Gudmund Pammer
10:30 - 11:30 Uhr: David Criens
13:30 - 14:30 Uhr: Daniel Bartl




{\bf{Dennis Schroers (Universität Bonn)}}

Titel: {\bf Copula Theory for Functional Data Analysis}

Abstract: The presentation will concern obstacles encountered when employing copulas in function spaces. We introduce a comprehensive approach to the notion of marginals in measurable vector spaces, along with techniques for constructing measures from copulas in the context of various function spaces such as continuous functions, Hölder spaces, Lebesgue spaces, and sequence spaces. Additionally, the talk will explore the significance of moment criteria and how the concept relates to Wasserstein spaces in determining the adequacy of these constructions.

The proposed framework aims to enhance the utility of copulas as a tool in functional data analysis, enabling their application in a range of fields. In particular, we discuss advantages for modeling returns in the context of term structures in mathematical finance. This talk is based on $[1]$, which is joint work with Fred Espen Benth and Giulia Di Nunno (University of Oslo).


Reference:

$[1]$ Benth, F.E., Di Nunno, G. and Schroers, D.: Copula measures and Sklar’s theorem in arbitrary dimensions, Scandinavian Journal of Statistics, 49 (3), 1144– 1183 (2022).



{\bf Verena Schwarz (Universität Klagenfurt)}

Titel: {\bf Numerical approximation of stochastic differential equations with jump noise}

Abstract: Stochastic differential equations (SDEs) with jump noise have applications in various areas of finance, insurance, and economics. In these contexts, however, the regularity assumptions of the standard literature are often not met, such as in control problems where discontinuous coefficients occur. Since explicit solutions are hardly available, numerical approximations are crucial for the utilisation of these models. In this talk we focus on jump-diffusion SDEs with discontinuous drift and present results on their numerical approximation. We introduce the so-called transformation-based jump-adapted quasi-Milstein scheme and provide a complete error analysis: We prove convergence of order 3/4 in $L^p$ for $p \geq 1$. Furthermore, we show lower error bounds for non-adaptive and jump-adapted approximation schemes of order 3/4 in $L^1$. We conclude the optimality of the transformation-based jump-adapted quasi-Milstein scheme.



{\bf Martin Friesen (Dublin City University)}

Titel: {\bf Affine Volterra processes: From stochastic stability to statistical inference}

Recent empirical studies of intraday stock market data suggest that the volatility, seen as a stochastic process, exhibits sample paths of very low regularity, which are not adequately captured by existing Markovian models, such as the Heston model. Additionally, classical affine processes fail to capture the observed term structure of at-the-money volatility skew. Both drawbacks can be addressed by rough analogues of stochastic volatility models described in terms of affine Volterra processes.

While the newly emerged rough volatility models have proven themselves to fit the empirical data remarkably, their mathematical properties have not been thoroughly investigated. The absence of the Markov property combined with the fact that these processes are not semimartingales constitute the main obstacles that need to be addressed.

In the first part of this presentation, we address the mean-reversion property for continuous affine Volterra processes. Based on a generalized affine transformation formula for finite-dimensional distributions, we prove the existence and uniqueness of stationary processes, characterize their dependence on the initial condition, and subsequently prove the law of large numbers. As an application, in the second part of this talk, we study the maximum-likelihood estimation for the drift parameters and outline future research directions towards a general class of stochastic Volterra equations.



{\bf Stefan Tappe (Universität Wuppertal)}

Titel: {\bf Invariance of closed convex cones for stochastic partial differential equations}

Abstract: The goal of this presentation is to clarify when a closed convex cone is invariant for a stochastic partial differential equation (SPDE) driven by a Wiener process and a Poisson random measure, and to provide conditions on the parameters of the SPDE, which are necessary and sufficient. An application from mathematical finance is presented as well.




{\bf Larisa Yaroslavtseva (Universität Graz)}

Titel: {\bf On strong approximation of SDEs with a discontinuous drift coefficient}


Abstract: Consider a scalar autonomous stochastic differential equation (SDE)
$$ dX_t = \mu(X_t) dt + \sigma(X_t) dW_t,\quad t \in [0, 1], X_0 = x_0,$$ with initial value $x_0 \in\mathbb{R}$, drift coefficient $\mu: \mathbb{R} \to \mathbb{R}$, diffusion coefficient
$\sigma: \mathbb{R} \to \mathbb{R}$ and Brownian motion $W$ . In this talk we study strong approximation of the solution $X_1$ by means of numerical methods that use finitely many evaluations of the driving Brownian motion $W$.
The classical assumption in the literature on numerical approximation of SDEs is global Lipschitz continuity of the coefficients $\mu$ and $\sigma$ of the equation. However, many SDEs arising in applications fail to have globally Lipschitz continuous coefficients.

In the last decade an intensive study of numerical approximation of SDEs with non-globally Lipschitz continuous coefficients has begun. In particular, strong approximation of SDEs with a discontinuous drift coefficient has recently gained a lot of interest. Such SDEs arise e.g. in mathematical finance, insurance and stochastic control problems. Classical techniques of error analysis are not applicable to such SDEs and well known convergence results for standard methods do not carry over in general.

In this talk I will present recent upper and lower error bounds for strong $L_p$-approximation
of such SDEs.




{\bf Florian Bechtold (Universität Bielefeld)}

Titel: {\bf Pathwise methods in stochastic analysis and applications to S(P)DEs}

Abstract: This talk provides a concise introduction to rough path theory and pathwise regularization by noise techniques as well as applications thereof in my own research. The latter include:

1) A sharp Young regime for locally monotone SPDEs

We extend the well-known Young regime for SDEs to the setting of fully non-linear locally monotone SPDEs. This shows that any driving noise with slightly more time regularity than a Hilbert-space valued Brownian motion admits a completely pathwise treatment, based on a careful combination of Besov rough analysis and monotone operator theory.

2) A law of large numbers for interacting diffusions via a mild formulation

The study of laws of large numbers for interacting particle systems is a classical topic using typically either a propagation of chaos result or analysis on Wasserstein spaces. We show how such a result can be obtained without independence or moment assumptions on the initial condition required in these approaches. Our argument is based on a complementary use of rough path theory and martingale arguments, illustrating the respective advantages and drawbacks of these theories.

3) Regularization by noise for SDEs with singular diffusion

Consider a standard SDE driven by Brownian motion with diffusion coefficient $\sigma$. We show that if $\sigma$ is singular, e.g. only in $L^p_x$, even existence of weak solutions might fail by providing an explicit counter-example. In the presence of an additional independent source of additive noise, we show that existence of weak solutions can be restored, provided said noise admits a sufficiently regular local time.

4) Pathwise regularization by noise in the multi-parameter setting

We provide a complete generalization of pathwise regularization by noise to the two parameter setting, covering in a systematic fashion regularization by space-time dependent noises for Goursat problems. We do so by extending non-linear Young theory to the two parameter setting and deriving novel regularity estimates for the local time of Gaussian fields, which are of independent interest. The latter is done by means of a newly established multi-parameter stochastic sewing lemma.



{\bf Gudmund Pammer (ETH Zürich)}

Titel: {\bf Adapted optimal transport and applications in mathematical finance}

Abstract: Wasserstein distance induces a natural Riemannian structure for the probabilities on $\mathbb R^n$. This insight of classical transport theory is fundamental for tremendous applications in various fields of pure and applied mathematics. We introduce a stochastic counterpart, the adapted Wasserstein distance AW, which can play a similar role for the class of stochastic processes. In contrast to other topologies for stochastic processes, probabilistic operations such as the Doob decomposition, optimal stopping and stochastic control are continuous w.r.t. AW. Moreover AW is a geodesic distance, and the class of martingales form a closed geodesically convex subspace. Besides these theoretical results, we also discuss applications to some of the core problems in mathematical finance.




{\bf David Criens (Univsersität Freiburg)}

Titel: {\bf Nonlinear continuous semimartingales and their applications}

Abstract: Nonlinear continuous semimartingales are families of sublinear conditional expectations that can be understood as continuous semimartingales with uncertain dynamics. The interest in them stems from robust finance, where nonlinear semimartingales can be used as market models with path-dependent drift and volatility uncertainty. In this talk, I provide a systematic discussion of nonlinear continuous semimartingales with focus on their link to the viscosity theory for path-dependent PDEs. Finally, I will comment on applications to mathematical finance and related fields.




{\bf Daniel Bartl (Universität Wien)}

Titel: {\bf On high-dimensional data and uncertainty in mathematical finance}

Abstract: In the realm of mathematical finance, the accurate choice of the probabilistic model that describes the random behavior of financial environments is not inherently known; rather, it is often derived from data or determined through various means, such as accounting for market restrictions. The first part of this presentation is dedicated to the statistical estimation of the model from empirical data. In particular, we construct procedures that exhibit the best possible statistical performance for the estimation of risk measures and convex stochastic optimization problems. The second part focuses on a non-parametric investigation (using Wasserstein-type distances) to assess how sensitive a given decision-making problem is to small model misspecifications.

Abstract:
We construct operator models for meromorphic functions of bounded type on Krein spaces. This construction is based on certain reproducing kernel Hilbert spaces which are closely related to model spaces. Specifically, we show that each function of bounded type corresponds naturally to a pair of such spaces, extending Helson’s representation theorem. This correspondence enables an explicit construction of our model, where the Krein space is a suitable sum of these identified spaces.

Matroids are a combinatorial abstraction of both graphs and linear spaces who play a central role in tropical geometry. In my talk I will take a polyhedral view on matroids and demonstrate how this perspective can be used to study classical matroid and graph invariants, tropical moduli spaces as well as problems arising in real life applications, e.g., shortest path problems in optimization, scattering processes in physics or decompositions in phylogenetics.

It is well-known that the cohomology ring has a richer structure than homology groups. However, until recently, the use of cohomology in the persistence setting has been limited to speeding up barcode computations. Some of the recently introduced invariants, namely, persistent cup-length, persistent cup modules and persistent Steenrod modules, to some extent, fill this gap. When added to the standard persistence barcode, they lead to invariants that are more discriminative than the standard persistence barcode. In this work, we devise an $O(dn^4)$ algorithm for computing the persistent $k$-cup modules for all $k\in\{2,\dots,d\}$, where $d$ denotes the dimension of the filtered complex, and $n$ denotes its size. Moreover, we note that since the persistent cup length can be obtained as a byproduct of our computations, this leads to a faster algorithm for computing it for $d \ge 3$. Finally, we introduce a new stable invariant called partition modules of cup product that is more discriminative than persistent $k$-cup modules and devise an $O(c(d)n^4)$ algorithm for computing it, where $c(d)$ is subexponential in $d$.

The Heawood graph is a remarkable graph that played a fundamental role in the development of the theory of graph colorings on surfaces in the 19th and 20th centuries. The purpose of this talk is to introduce a generalization of the classical Heawood graph indexed by a sequence of positive integers. The resulting generalized Heawood graphs are toroidal graphs with a rich combinatorial and geometric structure. In particular, they are dual to higher dimensional triangulated tori and we present explicit combinatorial formulas for their f-vectors.

This talk is based on joint work with Joseph Doolittle.

Tropical geometry by itself is a young and fast-growing area of mathematics.
Identifying where similar techniques can be used leads to an even more exciting range of applications. I will focus on two directions. First, we will see how a tropical point of view on the geometric structure of linear programming and certain two-player games helps to derive new algorithms. Second, I will present how fundamental structures of tropical geometry can be generalized to discrete convex analysis, a framework of well-behaved polyhedral functions. The latter helps to address questions from economics on one hand and the study of flag varieties on the other hand.

The free semigroup} $\mathcal{F}$ over a finite alphabet $\mathcal{A}$ is the set of all finite words with letters in $\mathcal{A}$ equipped with the operation of concatenation. A subset $S$ of $\mathcal{F}$ is $k$-product-free} if no $k$ words in $S$ concatenate to another word in $S$. How dense can a $k$-product-free subset of $\mathcal{F}$ be? What is the structure of the densest $k$-product-free subsets?

Leader, Letzter, Narayanan, and Walters proved that $2$-product-fee subsets of the free semigroup have density at most $1/2$ and asked for the structure of the densest sets. In this talk I will discuss the answer to their question as well as the answer (both density and structure) for general $k$. This generalises results of {\L}uczak and Schoen for sum-free sets in the integers although the methods used are quite different.

This is joint work with Lukas Michel and Alex Scott.

Meeting link:
\[
\text{https://tugraz.webex.com/tugraz/j.php?MTID=m8500c46344212abf0fa37925da5ef9bf}
\]

Abstract: The topic of my talk are functional inequalities for systems of orthonormal functions. One wants to have an optimal dependence of the constant on the number of functions involved. In this talk we focus on so-called spectral cluster bounds, which are concerned with $L^p$ norms of (linear combinations of) eigenfunctions of the Laplace-Beltrami operator on a compact manifold without boundary. I will review what is known in the case of a single function and for systems of orthonormal functions, then I will present some new results for operators with non-smooth coefficients or on manifolds with boundary.

The talk is partly based on joint ongoing works with Ngoc Nhi Nguyen and Xiaoyan Su.

Wachspress Geometry is a young field concerned with a family of closely related mathematical objects defined on polytopes -- the so-called ``Wachspress object''.
At its center, the ``Wachspress coordinates'' have their origin as generalized barycentric coordinates in geometric modeling and finite element analysis, but have since re-emerged in a wide range of seemingly unrelated contexts in algebraic geometry, statistics, spectral graph theory, rigidity theory and convex geometry. Explaining and exploiting this surprising ubiquity is a central motivation of Wachspress Geometry.

My first goal for this talk is to give an introduction to this fascinating subject and to elaborate on the potential of Wachspress Geometry to bridge between the algebra, geometry and combinatorics of polytopes. Subsequently I will focus on two particular applications that emerged in my own research: first, the rigidity and reconstruction of polytopes from partial combinatorial and metric data; second, a novel approach to algorithmically decide the polytopality of simplicial spheres.

Outline: The topic of the lecture is eigenfunctions of the Laplace-Beltrami operator on Riemannian manifolds. The first goal is to prove the sharp Weyl formula, which describes the asymptotic distribution of the eigenvalues. The second goal is to prove sharp bounds on Lp norms of eigenfunctions. Lp norms provide a convenient measure for the concentration of eigenfunctions and shed some light on the question `How do eigenfunctions look like’? To tackle this question we will develop tools from harmonic and microlocal analysis. If time permits, we will discuss improvements to these estimates under additional dynamical conditions on the geodesic flow.

Let $r_k(N)$ denote the size of the largest subset of $[N] = \{1,\ldots,N\}$ with no $k$-term arithmetic progression. We show that for $k\ge 5$, there exists $c_k>0$ such that

\[r_k(N)\ll N\exp(-(\log\log N)^{c_k}).\]

Our proof is a consequence of recent quasipolynomial bounds on the inverse theorem for the Gowers $U^k$-norm as well as the density increment strategy of Heath-Brown and Szemerédi as reformulated by Green and Tao.

Meeting link:
\[
\text{https://tugraz.webex.com/tugraz/j.php?MTID=m8500c46344212abf0fa37925da5ef9bf}
\]

We consider the following infection spreading process on graphs: At the beginning, each vertex is infected independently with a given probability $p$ and after that, new vertices get infected if at least $2$ of their neighbours have been infected. We say a graph percolates if eventually every vertex of the graph is infected. An interesting question in this context is above which threshold value of $p$ a graph percolates with high probability. In particular, this problem was studied on the $n$-dimensional hypercube by Balogh, Bollob\'{a}s and Morris, who established a sharp threshold when $n$ tends to infinity. We consider the process on a generalised version of the hypercube, the
Hamming graph}, which arises as the $n$-fold cartesian product of the complete graph $K_k$. We establish a threshold for percolation when $n$ tends to infinity and $k =k(n)$ is an arbitrary, positive function of $n$. The presented techniques are in parts applicable to more general classes of product graphs and might be of interest for studying the process in these cases.

This is joint work with Mihyun Kang and Michael Missethan.

Homepage: https://www.math.tugraz.at/\string~mtechnau/2024-aaa7-graz.html


\noindentThursday}

\begin{description}

\item{14:10--14:40:} Lukas Spiegelhofer:
Thue--Morse along the sequence of cubes}

\item{14:40--15:10:} Pascal Jelinek:
Square-free values of polynomials on average}

\item{15:10--15:40:} Helmut Maier:
The problem of propinquity of divisors for Gaussian integers}

\item{16:00--16:30:} Martin Widmer:
A quantitative version of the primitive element theorem for number fields}

\item{16:30--17:00:} Sumaia Saad Eddin:
Asymptotic results for a class of arithmetic functions involving the greatest common divisor}

\item{17:00--17:30:} Christian Weiß:
Uniform distribution --- the $p$-adic viewpoint}

\end{description}


\noindentFriday}

\begin{description}

\item{09:20--09:50:} Ram\={u}nas Garunk\v{s}tis:
Discrete mean value theorems for the Riemann zeta-function over its shifted nontrivial zeros}

\item{09:50--10:20:} Rainer Dietmann:
Sieving with square conditions and applications to Hilbert cubes in arithmetic sets}

\item{10:20--10:50:} Roland Miyamoto:
Fixed-points of stribolic operators and their combinatorial aspects}

\item{10:50--11:20:} Robert F.\ Tichy:
Pseudorandom measures}

\item{11:40--12:10:} Christian Bernert:
Cubic forms over imaginary quadratic number fields (and why we should care)}

\item{12:10--12:40:} Shuntaro Yamagishi:
Rational curves on complete intersections}

\end{description}

Scott and Seymour conjectured the existence of a function $f$ such that, for every graph $G$ and tournament $T$ on the same vertex set, $\chi(G)\geq f(k)$ implies that $\chi(G[N_T^+(v)])\geq k$ for some vertex $v$. We will disprove this conjecture even if $v$ is replaced by a vertex set of size $\mathcal{O}(\log{|V(G)|})$. As a consequence, we obtain a negative answer to a question of Harutyunyan, Le, Thomassé, and Wu concerning the analogous statement where the graph $G$ is replaced by another tournament. Time permitting, we will also discuss the setting in which chromatic number is replaced by degeneracy, where a quite different behaviour is exhibited.

This is joint work with António Girão, Kevin Hendrey, Freddie Illingworth, Florian Lehner, Lukas Michel, and Raphael Steiner.

Meeting link:
\[
\text{https://tugraz.webex.com/tugraz/j.php?MTID=m8500c46344212abf0fa37925da5ef9bf}
\]

Although prime numbers have been studied
since Greek antiquity, the first systematic study of the
number of prime factors is due to Hardy and Ramanujan
in 1917. Subsequently, with the work of Paul Turan, Paul
Erdos, and Marc Kac, the subject of Probabilistic Number
Theory was born around 1940 and has blossomed
considerably since then. We shall discuss classical
work on the number of prime factors of integers and more
recent results of ours with restrictions on the size of the
prime factors, that lead to very interesting variations of
the classical theme. My most recent results are jointly
with my former PhD student Todd Molnar. A variety of
techniques come into play - the classical Perron integral
approach, an analytic method of Atle Selberg, the behavior
of solutions to certain difference - differential equations,
and the Buchstab-de Bruijn iteration. The talk will be
accessible to non-experts.

Abstract:

Extensive literature has been devoted to study the operators for which the (anti-)maximum principle holds. Inspired by ideas from the recent theory of eventually positive $C_0$-semigroups, we characterise when the Dirichlet-to-Neumann operator satisfies an anti-maximum principle.

To be precise, let $\Omega \subset \mathbb{R}^d$ be a bounded domain with $C^\infty$-boundary and let $A$ be the Dirichlet-to-Neumann operator on $L^2(\partial \Omega)$. We consider the equation

\begin{equation*}
(\lambda − A)u = f

\end{equation*}
for real numbers $\lambda$ in the resolvent set of $A$. We find those $d$ for which $f \geq 0$ implies $u \leq 0$ for $\lambda$ in a ($f$-dependent) left neighbourhood of the spectral bound.

This is joint work with Jochen Gl\"uck.

In the last decades, there has been an immense interest in the behavior of
analytic objects, (e.g., invariants, measures, Green functions or Laplacian operators) on degenerating Riemann surfaces, that is, Riemann surfaces undergoing a degeneration to a singular Riemann surface. Due to their connection to geometry, the canonical measure and Arakelov Green function have been intensively studied in this context. Recently, it has become clear that their limits under degeneration are closely related to analogous analytic objects on graphs.


In this talk, we discuss recent results on degenerations of the canonical measure and related objects. A new geometric object - called hybrid curve - which mixes graphs and Riemann surfaces, plays an important role in our approach.


Based on joint work with O. Amini (Ecole Polytechnique).

The study of ordered Ramsey numbers of monotone paths for graphs and hypergraphs has a long history, going back to the celebrated work by Erdos and Szekeres in the early days of Ramsey theory. We obtain several results in this area, establishing two conjectures of Mubayi and Suk. This talk is based on a joint work with Zhihan Jin and Benny Sudakov.

Meeting link:
\[
\text{https://tugraz.webex.com/tugraz/j.php?MTID=m8500c46344212abf0fa37925da5ef9bf}
\]

Abstract:

We are interested in the local energy decay for the Schrödinger equation in an asymptotically Euclidean setting. For this, we study in particular the behavior of the corresponding resolvent for low frequencies. We will see how to use some ideas coming from the analysis of the damped wave equation to get the asymptotic profile for the resolvent, and then for the large time behavior of the solution.

Abstract:

This talk is centered around a symplectic approach to eigenvalue problems for systems of ordinary differential operators (e.g., Sturm-Liouville operators,
canonical systems, and quantum graphs), multidimensional elliptic operators on bounded domains, and abstract self-adjoint extensions of symmetric operators in Hilbert spaces. The symplectic view naturally relates spectral counts for self-adjoint problems to the topological invariant called the Maslov index. In this talk, the notion of the Malsov index will be introduced in analytic terms and an overview of recent results on its role in spectral theory will be given.

A semirestricted variant of the well-known game Rock, Paper, Scissors was recently studied by Spiro et al (Electronic J. Comb. 30 (2023), #P4.32). They assume that two players $R$ (restricted) and $N$ (normal) agree to play $3n$ rounds, where $R$ is restricted to use each of the three choices exactly $n$ times each, while $N$ can choose freely. Obviously, this gives an advantage to $N$. How large is the advantage?

The main result of Spiro et al is that the optimal strategy for $R$ is the greedy strategy, playing each round as if it were the last. (I will not give the proof, and I cannot improve on this.) They also show that with optimal play, the expected net score of $N$ is $\Theta(\sqrt{n})$. In the talk, I will show that with optimal play, the game can be regarded as a twice stopped random walk, and I will show that the expected score is asymptotic to $c \sqrt{n}$, where $c = 3\sqrt{3}/(2\sqrt{\pi}})$.

13:30 - 14:30: Ingrid Vukusic (Universität Salzburg)}
Consecutive triples of multiplicatively dependent integers

15:00 - 16:00: Victor Wang (ISTA)}
Sums of three cubes over a function field

16:15 - 17:15: Pierre-Yves Bienvenu (TU Wien)}
Intersectivity with respect to sparse sets of integers


Seminar webpage for further information:

https://sites.google.com/view/gntd/start

A $k$-uniform tight cycle} is a $k$-graph ($k$-uniform hypergraph) with a cyclic order of its vertices such that every~$k$ consecutive vertices from an edge. We show that for $k\geq 3$, every red-blue edge-coloured complete $k$-graph on~$n$ vertices contains~$k$ vertex-disjoint monochromatic tight cycles that together cover $n - o(n)$ vertices.

This talk is based on joint work with Allan Lo.

Abstract:
Landscape functions are a popular tool used to provide upper bounds for eigenvectors of Schrödinger operators on domains. After reviewing some results obtained in the last 10 years, I will show how several approaches used to achieve such bounds can be unified and extended to a large class of linear and even nonlinear operators. We also use landscape functions to derive lower estimates on the principal eigenvalue - much in the spirit of earlier results by Donsker–Varadhan and Banuelos–Carrol. Our methods solely rely on order properties of operators, which I will briefly remind. As an application, I will show how to derive hitherto unknown lower bounds on the principal eigvenalue of nonlinear Laplacian-type operators on graphs.

Given an integer $n$, let $G(n)$ be the number of integer sequences $n-1\ge d_1\ge d_2\ge \dots \ge d_n \ge 0$ that are the degree sequence of some graph. We show that $G(n)=(c+o(1))4^n/n^{3/4}$ for some constant $c>0$, improving both the previously best upper and lower bounds by a factor of $n^{1/4+o(1)}$. The proof relies on a translation of the problem into one concerning integrated random walks.

Joint work with Serte Donderwinkel, Carla Groenland, Tom Johnston and Alex Scott.


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

The weak order on permutations and permutahedron are classical
objects at the intersection of combinatorics and discrete geometry. Their
deep and interesting connections with the associahedron and the Tamari
lattice on binary trees have been an active subject of research. We present
a generalization of the weak order, the s-weak order on so-called
s-decreasing trees and their geometrical counter part the s-Permutahedron
with a possible realization as polytopal complex.