Discrete Mathematics 2010-2022

Abstract:

In this talk, I will present a new flexible distribution for data on the three-dimensional torus which we call a trivariate wrapped Cauchy copula. Our trivariate copula has several attractive properties. It has a simple form of density and is unimodal. its parameters are interpretable and allow adjustable degree of dependence between every pair of variables and these can be easily estimated. The conditional distributions of the model are well studied bivariate wrapped Cauchy distributions. Furthermore, the distribution can be easily simulated. Parameter estimation via maximum likelihood for the distribution is given and we highlight the simple implementation procedure to obtain these estimates. Another interesting feature of this model is that it can be extended to cylindrical data, that is, combining angles with classical data on the real line. We illustrate our trivariate wrapped Cauchy copula on data from protein bioinformatics of conformational angles, and our cylindrical copula using climate data related to buoy in the Adriatic Sea.

Abstract:

In his study of graph codes, Alon introduced the concept of the odd-Ramsey number of a family of graphs $\mathcal{H}$ in $K_n$, defined as the minimum number of colours needed to colour the edges of $K_n$ so that every copy of a graph $H$ in $\mathcal{H}$ intersects some colour class by an odd number of edges. In recent joint work with Simona Boyadzhiyska, Shagnik Das, and Thomas Lesgourgues, we focus on the odd-Ramsey numbers of complete bipartite graphs. First, we completely resolve the problem when $\mathcal{H}$ is the family of all spanning complete bipartite graphs on $n$ vertices. We then focus on its subfamilies, that is, $\{K_{t,n-t}: t \in T \}$ for a fixed set of integers $T \subseteq [n-1]$. In this case, we establish an equivalence between the odd-Ramsey problem and a problem from coding theory, asking for the maximum dimension of a linear binary code avoiding codewords of given weights. We then use known results from coding theory to deduce asymptotically tight bounds in our setting. We conclude with bounds for the odd-Ramsey numbers of fixed (that is, non-spanning) complete bipartite subgraphs.

Abstract:

In this talk, we first introduce the Bayesian statistical paradigm for performing inference in inverse problems that are ill-conditioned or ill-posed. This is then followed by an introduction to modern high-dimensional Bayesian computation approaches based on numerical approximations of the overdamped Langevin stochastic differential equation. We pay special attention to inference strategies that intimately combine ideas from Bayesian decision theory, stochastic sampling, optimisation, and machine learning. Key concepts and techniques are illustrated through a series of numerical experiments related to computational imaging.

Abstract:

In this talk, we address the problem of finding edge-disjoint paths between adversarially selected pairs of vertices in expander graphs. We present an online algorithm that collects the key features of all previous algorithms for this classical problem, providing them in essentially their strongest form. Joint work with R. Nenadov.

Meeting link:
\[
\text{https://tugraz.webex.com/tugraz/j.php?MTID=m01b0553e547155cca576e9d6e12f2c55}
\]

Abstract:

In this talk, we examine the non-relativistic limit of the MIT boundary condition for the Dirac operator on the half-plane. Our approach differs from the standard treatment by introducing a $c$-dependent boundary condition, allowing us to recover the broader family of Schrödinger operators in the limit. Specifically, we will demonstrate that the rescaled MIT model converges in the norm resolvent sense, after subtracting the rest energy, to the Schrödinger operator with oblique transmission conditions as $c\to \infty$.

Abstract:

Place: All talks are in SR Analysis-Zahlentheorie (Kopernikusgasse 24, 2nd floor). The only exception is the talk on Friday at 11:00, which is in HS F (Kopernikusgasse 24, 3rd floor).



{\bf Thursday 5.12.2024:}



{\bf 11:00 - 11:45: Manuel Hauke (NTNU Trondheim)}:
Littlewood meets Duffin and Schaeffer

{\bf 14:00 - 14:45: Nikita Shulga (La Trobe University Bendigo)}: Diophantine approximation of a real number by its integer base expansion

{\bf 15:00 - 15:45: Dmitriy Gayfulin (Technion Haifa)}: Extreme values of Sudler products



{\bf Friday 6.12.2024:}



{\bf 11:00 - 11:45: Volker Ziegler (University Salzburg)}: On a conjecture of Levesque and Waldschmidt

{\bf 14:00 - 14:45: Nikolay Moshchevitin (TU Wien)}: Distribution of Kronecker sequences revisited

{\bf 15:00 - 15:45: Jörg Thuswaldner (MU Leoben)}: Sequences of Matrices and Substitutions



On Friday at 12:00 there will be a common lunch in the social room of the Institute of Analysis and Number Theory (Kopernikusgasse 24, 2nd floor).

Abstract:

A decision maker typically (i) incorporates training data to learn about the relative effectiveness of the treatments, and (ii) chooses an implementation mechanism that implies an "optimal" predicted outcome distribution according to some target functional. Nevertheless, a discrimination-aware decision maker may not be satisfied achieving said optimality at the cost of heavily discriminating against subgroups of the population, in the sense that the outcome distribution in a subgroup deviates strongly from the overall optimal outcome distribution. We study a framework that allows the decision maker to penalize for such deviations, while allowing for a wide range of target functionals and discrimination measures to be employed. We establish regret and consistency guarantees for empirical success policies with data-driven
tuning parameters, and provide numerical results. Furthermore, we briefly illustrate the methods in two empirical settings.

The paper is available under
https://arxiv.org/abs/2401.17909

Abstract:

A graph $G$ with an even number of edges is called even-decomposable if there is a sequence $V(G)=V_0\supset V_1\supset \dots \supset V_k=\emptyset$ such that for each $i$, $G[V_i]$ has an even number of edges and $V_i\setminus V_{i+1}$ is an independent set in $G$. The study of this property was initiated recently by Versteegen, motivated by connections to a Ramsey-type problem and questions about graph codes posed by Alon. Resolving a conjecture of Versteegen, we prove that all but an $e^{-\Omega(n^2)}$ proportion of the $n$-vertex graphs with an even number of edges are even-decomposable. Moreover, answering one of his questions, we determine the order of magnitude of the smallest $p=p(n)$ for which the probability that the random graph $G(n,1-p)$ is even-decomposable (conditional on it having an even number of edges) is at least $1/2$.

We also study the following closely related property. A graph is called even-degenerate if there is an ordering $v_1,v_2,\dots,v_n$ of its vertices such that each $v_i$ has an even number of neighbours in the set $\{v_{i+1},\dots,v_n\}$. We prove that all but an $e^{-\Omega(n)}$ proportion of the $n$-vertex graphs with an even number of edges are even-degenerate, which is tight up to the implied constant.

Joint work with Fredy Yip.

Meeting link:
\[
\text{https://tugraz.webex.com/tugraz/j.php?MTID=m01b0553e547155cca576e9d6e12f2c55}
\]

Abstract:

The theory of U-statistics was introduced by Hoeffding in 1948. Although they were introduced as a class of unbiased estimators in statistics, it turned out that they also occur naturally in counting problems. In particular, the method of U-statistics is a useful tool for proving central limit theorems for the number of occurrences of a given substructure within a large, random combinatorial structure. This famously works for pattern counts in Erdős–Rényi random graphs, and in uniform permutations.

We will see that this method can also be applied to the Mallows model, which is the permutation analog of the Erdős–Rényi model. This requires the use of an elegant, regenerative property of Mallows permutations when the bias parameter $q$ is fixed. When $q<1$, the order of magnitude of a pattern count depends on the number of blocks inside the pattern, whereas when $q=1$ (uniform distribution) it depends on the size of the pattern. When $q$ goes to 1, this order interpolates in a natural way.

Abstract:

A meandric system of size $n$ is a configuration of noncrossing loops intersecting the horizontal axis at exactly $2n$ points.
These objects were introduced in an attempt to answer a long-lasting question of Poincaré about topological configurations of two curves on the sphere.
We prove that the number of connected components in a uniformly chosen random meandric system $M_n$ behaves linearly in $n$.
The main ingredient of the proof is the convergence of $M_n$ towards an infinite discrete object called the infinite noodle.
If time allows, I will also highlight surprising connections between this model, noncrossing partitions and hypergeometric functions.

Joint work with V. Féray (Nancy)

Abstract:

Given an increasing graph property $P$, a graph $G$ is $x$-resilient with respect to $P$ if, for every spanning subgraph $H$ of $G$ where each vertex keeps more than a $(1-x)$-proportion of its neighbours, $H$ has property $P$. Since its introduction by Sudakov and Vu, the analysis of local resilience for the Erdős-Rényi random graph has seen important developments in the last 15 years, culminating in recent results of Montgomery and Nenadov, Steger and Trujić confirming that Hamiltonicity is born resilient.

This talk concerns results around the local resilience of random geometric graphs with respect to connectivity and containment of long cycles. In particular, we are going to see why the 2-dimensional random geometric graph is very far from being born resilient. This talk is based on a joint paper with Alberto Espuny Díaz (University of Heidelberg) and Alexandra Wesolek (TU Berlin).

Abstract:

In this talk, we will investigate the largest possible size of a subset of {1,2,... ,n} that does not contain k distinct elements (for a fixed k) whose product is a perfect cube. The analogous problem about avoiding products equaling a perfect square was studied by Erdos, Sarkozy and T. Sos, and very recently, also by Tao. In the case of perfect cubes we provide bounds for k=2,3,4,6 and 9, furthermore, in the general case, as well. In particular, we refute an 18-year-old conjecture of Verstraete.
Joint work with Zsigmond Fleiner, Blanka Köver, Mark Juhasz and Csaba Sandor.

Abstract:

The theory of abstract Friedrichs operators, introduced by Ern,
Guermond and Caplain (2007), proved to be a successful setting for
studying positive symmetric systems of first order partial differential
equations (Friedrichs, 1958), nowadays better known as Friedrichs systems. The theory encompasses various types of PDEs, but the main
motivation was to study the mixed-type equations.

The aim of this talk is to make the audience familiar with the theory of Friedrichs systems and briefly discuss the recent developments.
At the end, possible research directions will be discussed.

Abstract:

For the concentration vector $c(t, x) \in [0, \infty[^I$, we consider reaction-diffusion systems
on all of $\mathbb{R}^d$ with non-trivial boundary conditions at infinity. The hope is to show
that all solutions behave asymptotically self-similar. Using the parabolic scaling
variables ̃$\widetilde{c}(\tau, y) = c(t, x)$ with $t = e^\tau x = e^\tau /2}y$, we obtain the scaled system
\[
c_\tau = \mathbb{D} \Delta_y c − \frac{1}{2} y\cdot \nabla c + e^\tau R(\widetilde{c}), \quad c(\tau, y) − C^\infty(y) \rightarrow 0 \text{ for } |y|\rightarrow 0.
\]
Important is that $c \mapsto R(c)$ admits a smooth manifold of reaction equilibria. The
asymptotic self-similar profile $C^\infty$ has to satisfy $0 = \mathbb{D} \Delta C^\infty + \frac{1}{2} y \cdot \nabla C^\infty + \Lambda$ and
$R(C^\infty(y)) = 0$ in $\mathbb{R}^d$.
We show existence of nontrivial self-similar profiles in the one-dimensional case
via the theory of monotone operators. In suitable cases, the convergence to such
a profile is established to showing decay of the relative Boltzmann entropy $E(c) =
H(c|C^\infty)$ via estimates of the type $\frac{d}{d \tau} E(c) \leq -\frac{d}{2} E(c) + C_1 e^{-\sigma \tau} E(c) + C_2 e^{-\kappa \tau}$.

Abstract:

The formal sharp-interface asymptotics in a degenerate Cahn–Hilliard model for
viscoelastic phase separation with cross-diffusive coupling to a bulk stress variable
are shown to lead to non-local lower-order counterparts of the classical surface diffusion flow. The diffuse-interface model is thermodynamically consistent, and has an
Onsager gradient-flow structure with a rank-deficient mobility matrix reflecting the
ODE character of stress relaxation. We present an example where, to leading order,
the evolution of the zero level set of the order parameter approximates the so-called
intermediate surface diffusion flow. For more intricate coupling, our asymptotic
analysis leads to a variant of third order whose propagation operator is essentially
given by the square root of the minus Laplace-Beltrami operator.

Abstract:

10:00 Begrüßung}

10:10 Historischer Rückblick auf die Geschichte des Instituts} \newline
\phantom{10:00} Ernst Stadlober (TU Graz)

10:40 Random set models for hierachically structured battery cathodes} \newline \phantom{10:00} Matthias Neumann (TU Graz)

12:00 The Fundamental Theorem of Weak Optimal Transport} \newline
\phantom{10:00} Gudmund Pammer (ETH Zürich)

14:00 Dependence Modeling and Feature Selection} \newline
\phantom{10:00} Wolfgang Trutschnig (Universität Salzburg)

15:20 Von der reinen Zahlentheorie zur angewandten Finanzmathematik} \newline \phantom{10:00} Gerhard Larcher (JKU Linz)

Abstract:

A score sequence is a non-increasing sequence of natural numbers that can occur as an out-degree sequence of an orientation of the complete graph (i.e. as the scores in a round-robin tournament). The first estimates on the number of score sequences of length $n$ are due to Erdös and Moser. With Brett Kolesnik, we resolve their conjecture and show that this number grows like $cn^{-5/2}4^n$. The foundation of our proof consists of some reformulations that turn our problem into a question about the persistence probability of the area process of the simple symmetric random walk bridge. We also obtain a Brownian scaling limit of a uniformly random score sequence.

In the second part of the talk, I will also touch upon an earlier, related work with Paul Balister, Carla Groenland, Tom Johnston and Alex Scott, where we use random walk techniques to asymptotically enumerate the number of non-increasing sequences that can occur as the degree sequence of a graph, answering a question by Royle. The techniques in this work inspired the work with Kolesnik.

Meeting link:
\[
\text{https://tugraz.webex.com/tugraz/j.php?MTID=m01b0553e547155cca576e9d6e12f2c55}
\]

Abstract:

The Skolem Problem asks to determine whether a given integer linear recurrence sequence has a zero term. This problem, whose decidability has been open for 90 years , arises across a wide range of topics in computer science and dynamical system. In 1977, A generalization of this problem was made by Loxton and Van der Poorten who conjectured that for any $\epsilon >0$ and $\{u_n\}$ a linear recurrence sequence with dominant (s) roots $>1$ in absolute value, there is a effectively computable constant $C(\epsilon),$ such that if $ \vert u_n \vert < (\max_i\{ \vert \alpha_i \vert \})^{n(1-\epsilon)}$, then $n<C(\epsilon)$. Using results of Schmidt and Evertse, a complete non-effective (qualitative) proof of this conjecture was given by Fuchs and Heintze (2021) and, independently, by Karimov and al.~(2023). In this talk, we provide a survey on the study of Universal Skolem set and sketch the proof of the weak version of the conjecture by giving a effective upper bound of the number of solutions of that inequality extending Schmidt and Schlickewei previous works. The higher dimension of this conjecture will also be discussed in this presentation.


Joint work with Florian Luca, James Maynard, Joel Ouaknine and James Worrell.

Abstract:

Seminar Numerical Simulations in Technical Sciences

30.10.2024 Dr. Richard Löscher (Institut für Angewandte Mathematik, TU Graz)
17.15–18.15 Space-time finite element methods: Foundations and applications

6.11.2024 Dr. Michael Gfrerer (Institut für Baumechanik, TU Graz)
17.15–18.15 A topological-shape derivative and automated code generation
for topology optimization

27.11.2024 Dr. Dominik Mayrhofer (Institut für Grundlagen und Theorie der
17.15–18.15 Elektrotechnik, TU Graz)
Viscous acoustics on moving domains for MEMS applications

11.12.2024 Dipl.–Ing. Thomas Thaller (Institut für Mechanik, TU Graz)
17.15–18.15 Transient simulation of beam structures with clearances

Ort: Hörsaal BE01, Steyrergasse 30

Katrin Ellermann
Manfred Kaltenbacher
Martin Schanz
Olaf Steinbach

Abstract:

A classical result by Erdős and Rényi from 1960 shows that the binomial random graph $G(n,p)$ undergoes a fundamental phase transition in its component structure when the expected average degree is around $1$ (i.e., around $p=1/n$). Specifically, for $p = (1-\varepsilon)/n$, where $\varepsilon > 0$ is a constant, all connected components are typically logarithmic in $n$, whereas for $p = (1+\varepsilon)/n$ a unique giant component of linear order emerges, and all other components are of order at most logarithmic in $n$.

A similar phenomenon — the typical emergence of a unique giant component surrounded by components of at most logarithmic order — has been observed in random subgraphs $G_p$ of specific host graphs $G$, such as the $d$-dimensional binary hypercube, random $d$-regular graphs, and pseudo-random $(n,d,\lambda)$-graphs, though with quite different proofs.

This naturally leads to the question: What assumptions on a $d$-regular $n$-vertex graph $G$ suffice for its random subgraph to typically exhibit this phase transition around the critical probability $p=1/(d-1)$? Furthermore, is there a unified approach that encompasses these classical cases? In this talk, we demonstrate that it suffices for $G$ to have mild global edge expansion and (almost-optimal) expansion of sets of (poly-)logarithmic order in $n$. This result covers many previously considered concrete setups.
We also discuss the tightness of our sufficient conditions.

Based on joint works with Joshua Erde, Mihyun Kang, and Michael Krivelevich.

Abstract:

Dear colleagues,

this is a kind reminder of the invitation to the

skip

Seminar on Analysis and Algebra Alpe-Adria (SAAAAJ)} on 19th October in Graz.



The SAAAAJ is a newly established, joint seminar between Zagreb, Ljubljana and Graz in both algebra and analysis.

skip


Date and location:} 19th October (Saturday), 10 am, at the TU Graz.\newline
(Lecture Room A (HS A) at Kopernikusgasse 24)

Schedule:}\newline
10:00–10:25: Amr Ali Al-Maktry (TU Graz) -- Polynomial functions on a class of finite non-commutative rings \newline
10:30–10:55: Andoni Zozaya (University of Ljubljana) -- Verbal problems in profinite groups \newline
11:00–11:25: Filip Najman (University of Zagreb) -- Modular curves \newline
----- Coffee break ----- \newline
12:00–12:25: Magdaléna Tinková (TU Graz / Czech Technical University) -- Artin twin primes \newline
12:30–12:55: Roman Drnovšek (University of Ljubljana) -- Positive Commutators on Banach lattices \newline
13:00–13:25: Kristina Ana Škreb (University of Zagreb) -- Dimensionless $L^p$ estimates for the Riesz vector \newline
14:00: joint lunch (restaurant Dionysos, Färbergasse 6)



Registration for lunch:} \newline
If you want to join the lunch at the restaurant, please register until 15th October.

skip

If you have any further questions, please do not hesitate to ask (frisch@math.tugraz.at and nicolussi@math.tugraz.at).

skip

We are looking forward to seeing you at the seminar!

\newpage

\section*{Location}
{ All talks will be in HS A in Kopernikusgasse 24 (first floor). The precise address is:
}
Kopernikusgasse 24
Graz University of Technology (TU Graz)
8010 Graz, Austria

skip

{ The address of the restaurant is:
}
Restaurant Dionysos
Färbergasse 6
8010 Graz, Austria


\section*{Abstracts}

{\bf Polynomial functions on a class of finite non-commutative rings}[2mm]
{\bf Amr Ali Al-Maktry (TU Graz)}


Let $R$ be a finite non-commutative ring with $1\ne 0$. By a polynomial function on $R$, we mean a function $F\colon R\longrightarrow R$ induced by a polynomial $f=\sum\limits_{i=0}^{n}a_ix^i\in R[x]$ via right substitution of the variable $x$, i.e.
$F(a)=f(a)= \sum\limits_{i=0}^{n}a_ia^i$ for every $a\in R$. In this paper, we study the polynomial functions of the free $R$-algebra
with a central basis $\{1,\alfa_1,\ldots,\alfa_k\}$ ($k\ge 1$) such that $\alfa_i\alfa_j=0$ for every $1\le i,j\le k$, $R[\alfa_1,\ldots,\alfa_k]$.
Our investigation revolves around assigning a polynomial $\lambda_f(y,z)$ over $R$ in non-commutative variables $y$ and $z$ to each polynomial $f$ in $R[x]$; and describing the polynomial functions on $R[\alfa_1,\ldots,\alfa_k]$ through the polynomial functions induced on $R$ by polynomials in $R[x]$ and by their assigned polynomials in the in non-commutative variables $y$ and $z$.
By extending results from the commutative case to the non-commutative scenario, we demonstrate that several properties and theorems in the commutative case can be generalized to the non-commutative setting with appropriate adjustments.[5mm]



{\bf Positive Commutators on Banach lattices}[2mm]
{\bf Roman Drnovšek (University of Ljubljana)}

We will present several results on positive commutators of positive operators on Banach lattices. We will start with the following finite-dimensional theorem. Let $A$ and $B$ be non-negative matrices such that the commutator $C = A B - B A$ is non-negative as well. Then, up to similarity with a permutation matrix, $C$ is a strictly upper triangular matrix, and so it is nilpotent. [5mm]






{\bf Modular curves}[2mm]
{\bf Filip Najman (University of Zagreb)}




Modular curves are moduli spaces of elliptic curves with prescribed images of their Galois representations. They are a key tool in studying torsion groups, isogenies and, more generally, Galois representations of elliptic curves. In recent years great progress, in many directions, has been made in our understanding of points on modular curves of low degree. In this talk I will describe some of these recent results, their consequences, as well as open problems in the field.[5mm]





{\bf Dimensionless $L^p$ estimates for the Riesz vector}[2mm]
{\bf Kristina Ana Škreb (University of Zagreb)}



We present a fundamentally new proof of the dimensionless $L^p$ boundedness of the Bakry Riesz vector on manifolds with bounded geometry. The key ingredients of the proof are sparse domination and probabilistic representation of the Riesz vector. This type of proof has the significant advantage that it allows for a much stronger conclusion, giving us a new range of weighted estimates. The talk is based on joint work with K. Domelevo and S. Petermichl. [5mm]




{\bf Artin twin primes}[2mm]
{\bf Magdaléna Tinková (TU Graz / Czech Technical University)}




In this talk, we will join two famous conjectures for primes, namely Artins conjecture on primitive roots and the Hardy--Littlewood two-tuple conjecture. In particular, we will study the existence and the asymptotic behavior of the number of prime pairs $p$ and $p+d$ with the same prescribed primitive root. This is joint work with Ezra Waxman and Mikuláš Zindulka.[5mm]




{\bf Verbal problems in profinite groups}[2mm]
{\bf Andoni Zozaya (University of Ljubljana)}


Given a group $G$ and a group word $w$ on $k$ variables, we can define a map $w \colon G^k \rightarrow G$ though substitution. Verbal problems study the relation between the image of this map and the subgroup it generates within $G$. In particular, we will study verbal conciseness – whether the subgroup is finite when the image of the map is finite – and strong verbal conciseness – whether the subgroup is finite when the image is countable – in the context of profinite groups.
We will introduce a new family of verbally concise groups, the so-called analytic groups; and, based on recent work with de las Heras, we will show that several classes of profinite verbally concise groups are, in fact, strongly verbally concise.[5mm]

Abstract:

We will discuss the typical behavior of $M$-Lipschitz functions on $d$-regular expander graphs, where an $M$-Lipschitz function means any two adjacent vertices admit integer values differ by at most $M$. While it is easy to see that the maximum possible height of an $M$-Lipschitz function on an $n$-vertex expander graph is about $C\cdot M \cdot \log n$, where $C$ depends (only) on $d$ and the expansion of the given graph, it was shown by Peled, Samotij, and Yehudayoff (2012) that a uniformly chosen random $M$-Lipschitz function has height at most $C’\cdot M \cdot \log \log n$ with high probability, showing that the typical height of an $M$-Lipschitz function is much smaller than the extreme case. Peled-Samotij-Yehudayoff’s result holds under the condition that, roughly, subsets of the expander graph expand by the rate of about $M \cdot \log (dM)$. We will show that the same result holds under a much weaker condition assuming that $d$ is large enough. This is joint work with Robert Krueger and Lina Li.

Abstract:

The mathematical scattering theory for short-range potential is a well studied subject which
was developed by two essentially different approaches: the trace-class method and the smooth
method. The scattering problem for singular perturbations of self-adjoint operators, which is
outside the original scope of these methods, is connected with scattering from obstacles with
impenetrable or semi-transparent boundary conditions. On this side, a general scheme has
been developed by combining the construction of singular perturbations, following Posilicano’s
approach, with an abstract version of the Limiting Absorption Principle (LAP in the following)
due to W. Renger and a variant of the smooth method due to M. Schechter. Let us recall
that boundary triple theory and properties of the associated operator-valued Weyl functions
were also used to obtain similar representation of the scattering matrix for singularly coupled
self-adjoint extensions.


The target of this talk is to present a general framework for the multiple scattering with
both potential type and singular perturbations. Our concern is the scattering theory with respect to the free Laplacian and the regular and the singular parts of the perturbation are dealt
as a single object: this constitutes the main novelty of our approach. At first we provide an
abstract resolvent formula for a perturbations $A_B$ of the self-adjoint $A$ by a linear combination
of the adjoint of two bounded trace-like maps. The LAP for $A_B$ and then an aymptotic completeness criterion for the scattering couple $(A_B ; A)$ are provided, under suitable hypothesis, as
a generalisation of the scheme adopted for purely singular perturbations. Then, by a combination of LAP with stationary scattering theory in the Birman-Yafaev scheme and the invariance
principle, we obtain a representation formula for the scattering matrix of the couple $(A_B ; A)$.
Whenever $A$ is the free Laplacian in $L^2 (\mathbb{R}^3)$, such a formula contains, as subcases, both the
usual formula for the perturbation given by a short-range potential and the formula for the
case of a singular perturbation describing self-adjoint boundary conditions on a hypersurface.


After introducing the main features of this theory, we present applications where our construction is used to describe the stationary-scattering in composite acoustic or electromagnetic
dispersive media. In particular, we show how the stationary resolvent and the scattering solutions, for scalar-wave equations with the divergence-form Laplacian having discontinuous
density, can be represented in terms of a class of perturbations involving both potential terms
and specific frequency-dependent interface conditions.

Abstract:

We investigate the spectrum of Schrödinger operators with complex potentials in $L^2(R)$, perturbed with $\delta$, or $\delta^\prime$ interaction, centered at the origin
\begin{equation*}
-\partial_x^2 + ix^{2k-1} + \alpha\delta, \quad
-\partial_x^2 + ix^{2k-1} + \beta\delta^\prime,
\end{equation*}
where $\alpha \in R$, $\beta \in R$, $k\in N$.

It is well established that the spectrum of the unperturbed operators consists of countable many real, isolated and simple eigenvalues for $k\geq2$, and it is empty for $k=1$.


When $\alpha\neq0$ or $\beta\neq0$, for $k\geq1$, we observe countable many non-real eigenvalues appearing in complex conjugate pairs, and at maximum finitely many real eigenvalues. The non-real eigenvalues asymptotically converge to the eigenvalues of the unperturbed problems defined on $L^2(R_+)$ and $L^2(R_-)$ with Dirichlet, resp. with Neumann boundary conditions for $\delta$, resp. $\delta^\prime$ interaction.


Moreover, for $\alpha\leq C_1< 0$, we show the existence of a negative real eigenvalue, diverging to $-\infty$ as $\alpha\to-\infty$. Similarly for $C_2<\beta<0$, we show the existence of a negative real eigenvalue diverging to $-\infty$ as $\beta\to0_-$.

{\bf References}
{[1]} J. Behrndt, I. Semoradova, P. Siegl, The imaginary Airy operator with one-center $\delta$ interaction, {\em to appear in Pure and Applied Functional Analysis}
{[2]} M. Marletta, I. Semoradova, $\mathcal{PT}$-symmetric oscillators with one-center point interactions {\em manuscript in preparation}

Abstract:

For a class of weighted infinite metric trees we propose a definition of the boundary trace which maps $H^1$-functions on the tree to $L^2$-functions on a compact Riemannian manifold. For a range of parameters, the precise Sobolev regularity of the traces is determined. This allows one to show the well-posedness for a Laplace-type equation on infinite trees interacting with Euclidean domains through the boundary. Based on joint works with Valentina Franceschi (Padova), Maryna Kachanovska (Paris) and Konstantin Pankrashkin (Oldenburg).

Abstract:

Two nonzero $ a, b $ are said to be multiplicatively independent if the only solution in integers $ x , y $ of the Diophantine equation $ a^xb^y=1 $, is $ x=y=0 $, and multiplicately dependent otherwise.
Let $ k\ge 2 $ be a fixed integer and $ (L_n^{(k)})_{n\ge 2-k} $ be the sequence of $ k $--generalized Lucas numbers whose first $ k $ terms are $ 0,0,\ldots, 0, 2, 1 $ and each term afterwards is the sum of the preceding $ k $ terms. In this talk, we find all pairs of $ k $--generalized Lucas numbers that are multiplicatively dependent. The proof of the main result heavily employees: Baker's theory of non-zero lower bounds for linear forms in logarithms of algebraic numbers, Carmichael's Primitive Theorem, and reduction techniques involving the theory of continued fractions, in particular, the LLL algorithm. This is joint work with H. Batte, J. Kasozi and F. Luca$^{1}$.

Abstract:

Polynomials over rings exhibit various algebraic structures, such as groups, (graded) rings, modules, coalgebras, and comodules. In this work, we revisit several generalizations of polynomials and examine their properties related to grading, derivation, Noetherian conditions, cleanness, duality, and more. Additionally, we explore applications of these structures in cryptography.

Abstract:

Ausgehend von empirischen Ergebnissen und Erfahrungsberichten beginnen wir mit einer Bestandsaufnahme des prozeduralen und konzeptuellen Wissensstands von Studienanfänger:innen in Graz. Darauf aufbauend stelle ich (Forschungs-)Projekte an der Universität Graz zur Förderung dieses Wissens sowie zur Stärkung des Begründens und Argumentierens in Schule und Lehramtsstudium vor.

Abstract:

Abstract:

The subject of our analysis are discrete time processes that take their values in a (separable) Hilbert space, so called functional time series. A variety of statistics for functional time series allows for a representation as weighted average of corresponding periodogram operators over the frequency domain.
We study consistency and asymptotic normality of such spectral mean estimators under mild assumptions. As a by-product, we derive some basic first and second order properties of periodogram operators as well. We show that weak convergence of spectral mean estimators can be deduced from the (joint) weak convergence of the sample autocovariance operators. The latter is established for a large
class of weakly dependent functional time series, which admit expansions as Bernoulli shifts and the weak dependence is quantified by the condition of $L^4\!-\!m-$approximability. In order to facilitate the applicability of our results, we also propose a hybrid-bootstrap scheme, which combines a bootstrap-approach with a subsampling method to approximate the gaussian limit of spectral mean estimators.

Abstract:

This talk introduces a computational method for generating digital twins of the 3D morphology of (functional) materials through stochastic geometry models, calibrated by means of 2D image data. By means of systematic variations of model parameters a wide spectrum of structural scenarios can be investigated, such that the corresponding digital twins can be exploited as geometry input for numerical simulations of macroscopic effective properties [1,2,3]. For calibrating models that can generate virtual 3D microstructures by stochastic simulation, generative adversarial networks (GANs) have gained an increased popularity [4]. While GANs offer a data-driven approach for modeling complex 3D morphologies, the systematic variation of model parameters for generating diverse structural scenarios can be difficult. In contrast to this, relatively simple models of stochastic geometry (e.g., based on Gaussian random fields) allow parameter-driven structure variations, but can fall short in mimicking complex morphologies. Still, these “simple models” can be used to construct more advanced parametric stochastic geometry models (like excursion sets of more general random fields, or random tessellations induced by marked point processes). However, with increasing model complexity the calibration of model parameters to image data can become increasingly difficult. Combining GANs with advanced stochastic geometry models can overcome these limitations and, in addition, allows for the calibration of model parameters solely based on 2D image data of planar sections through the 3D structure [5]. These parametric hybrid models are flexible enough to stochastically model complex 3D morphologies, enabling the systematic exploration of different structures. Moreover, by combining stochastic and numerical simulations, the impact of morphological descriptors on macroscopic effective properties can be investigated and quantitative structure-property relationships can be established. Thus, the presented method, allows for the generation of a wide spectrum of virtual 3D morphologies, that can be used for identifying structures with optimized functional properties.


{References}

[1] B. Prifling et al. (2021). Front. in Mater. 8 (2021), 786502.

[2] O. Furat et al. (2021). npj Comput. Mater. 7, 105.

[3] E. Schlautmann et al. (2023). Adv. Energy Mater. 13, 2302309.

[4] S. Kench and S.J. Cooper (2021). Nat. Mach. Intell. 3, 299-305.

[5] L. Fuchs et al. (2024). Preprint. https://doi.org/10.48550/arXiv.2407.05333

Abstract:

Approximating a function from its discrete set of samples is
crucial in numerical analysis and is a fundamental step in numerous
applications. With advancements in data-collecting devices, such as modern
medical imaging, there is a growing demand for more sophisticated data
models and corresponding approximation techniques. As a result, this
classical mathematical problem becomes even more challenging with modern
applications involving massive databases, scattered data sites,
high-dimensionality, and nonlinear spaces, such as manifolds. In this
presentation, we will explore various aspects of manifold-valued
approximation, with a particular emphasis on the issue of Hermite-manifold
data, which we will define and discuss in detail.

Abstract:

There is an equivalence of categories between algebraic tori over a field $F$ and topological tori with an action of the absolute Galois group of $F$. We prove that the algebraic K-theory of an algebraic torus coincides with the equivariant homology of the associated topological torus with respect to the homology theory represented by the K-theory of the base field $F$ and its field extensions. This is joint work in progress with Shachar Carmeli, Qingyuan Bai and Florian Riedel.

(a video on that topic is available from https://youtu.be/F3w7tfBF6HA?si=BA6KyXkdtgSY1BvT})

Abstract:

How many copies of a graph $H$ can appear in a planar graph? We develop a technique which, for many graphs $H$, reduces this question to one of determining a "maximum likelihood estimator" of a graph related to $H$. For example, in the case of $H=C_{2m}$, we are led to the following question: which probability mass (on the edges of some big clique) maximizes the probability that $m$ independent samples from the mass form a copy of an $m$-cycle?

Abstract:

The self-avoiding walk is a model from statistical physics which has been
studied extensively on integer lattices. Over the last few decades, the
study of self-avoiding walks on more general graphs, in particular graphs
with a high degree of symmetry such as Cayley graphs of finitely generated
groups, has received increasing attention.

In this talk, we focus on graphs with more than one end; intuitively these
can be thought of as having some large-scale tree structure. This tree
structure allows us to decompose self-avoiding walks into smaller, more
manageable pieces, and answer questions for graphs with more than one end
whose answers for lattices currently seem out of reach.

(joint work with Christian Lindorfer and Christoforos Panagiotis)

Abstract:

Let $R_n=\mathbb{Q}[x_1,x_2,\ldots,x_n]$ be the ring of polynomials in $n$ variables and consider the ideal $\langle \mathrm{QSym}_{n}^{+}\rangle\subseteq R_n$
generated by quasisymmetric polynomials without constant term.
It was shown by J.~C.~Aval, F.~Bergeron and N.~Bergeron that $\dim\big(R_n\big/\langle \mathrm{QSym}_{n}^{+} \rangle\big)=C_n$ the $n$th Catalan number.
In the present work, we explain this phenomenon by defining a set of permutations $\mathrm{QSV}_{n}$ with the following properties:
first, $\mathrm{QSV}_{n}$ is a basis of the Temperley--Lieb algebra $\mathsf{TL}_{n}(2)$, and second,
when considering $\mathrm{QSV}_{n}$ as a collection of points in $\mathbb{Q}^{n}$,
the top-degree homogeneous component of the vanishing ideal $\mathbf{I}(\mathrm{QSV}_{n})$ is $\langle \mathrm{QSym}_{n}^{+}\rangle$.

Our construction has a few byproducts which are independently noteworthy.
We define an equivalence relation $\sim$ on the symmetric group $S_{n}$ using weak excedances
and show that its equivalence classes are naturally indexed by noncrossing partitions.
Each equivalence class is an interval in the Bruhat order between an element of $\mathrm{QSV}_{n}$ and a $321$-avoiding permutation.
Furthermore, the Bruhat order induces a well-defined order on $S_{n}\big/\!\!\sim$.
Finally, we show that any section of the quotient $S_{n}\big/\!\!\sim$ gives an (often novel) basis for $\mathsf{TL}_{n}(2)$.

This talk is based on joint work with Lucas Gagnon.

Abstract:

For two (hyper)graphs $G$ and $H$, the extremal number $ex(G,H)$ is the largest number of edges in an $H$-free subgraph of the ground graph $G$. Determining $ex(G,H)$ remains a challenge in general, even when $G$ is a complete graph $K_n$. However, in this case we know exactly what (hyper)graphs $H$ have a positive or zero Tur\'an density $\pi(H)$, where $\pi(H) = \lim_{n\rightarrow \infty} ex(K_n, H)/ ||K_n||$. When the ground graph $G$ is the hypercube $Q_n$ of dimension $n$, we don't even have such a characterisation. In this talk, I will present what we know about $ex(Q_n, H)$ and how this extremal number relates to the classical extremal numbers of hypergraphs.

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

Abstract:

We extend classical results in algebraic topology for higher homotopy groups
and singular homology groups of pseudotopological spaces. Pseudotopological
spaces are a generalization of topological spaces that also include graphs
and directed graphs. More specifically, we show the existence of a long
exact sequence for homotopy groups of pairs of closure spaces and that a
weak homotopy equivalence induces isomorphisms for homology groups.

    Our main result is the construction of a weak homotopy equivalence
between the geometric realizations of (directed) clique complexes and their
underlying (directed) graphs. This implies that singular homology groups of
finite graphs can be efficiently calculated from finite combinatorial
structures, despite their associated chain groups being infinite
dimensional. This work is similar to the work McCord did for finite
topological spaces, but in the context of pseudotopological spaces. Our
results also give a novel approach for studying (higher) homotopy groups of
discrete mathematical structures such as digital images.

Abstract:

Persistence modules in one parameter decompose into intervals, simple building blocks that capture topological features of an underlying dataset. While the notion of intervals extends to two and more parameters, not all persistence modules can be decomposed any longer into intervals. Instead, the atoms of a decomposition can become arbitrarily complicated, and such complications also arise in typical geometric scenarios.

I will present two limit theorems that support these empirical observations. These results show that in some situations, the expected frequency of intervals in a decomposition is at least $1/4$ and that the probability of obtaining only intervals in a decomposition approaches $0$ when the sample size goes to infinity. I will focus on the proof of the latter result which uses elementary algebraic and geometric arguments and combines them with basic properties of Poisson processes.

This is joint work with Angel Alonso (TU Graz) and Primoz Skraba (Queen
Mary Univ London)

Abstract:

In this talk we present basic notions and properties of finite projective planes, finite and classical inversive planes (Möbius planes), and t-designs. Using these structures we show how to construct existing and new card games (Dobble, Møbee and other possible games) and we investigate the geometry of their underlying finite geometries. After studying the mathematics behind the card games, we also play them. No previous knowledge is required to understand the talk.

Abstract:

skip

Abstract:
Many approaches that analyze and predict the results of international matches in football/soccer are based on
statistical models incorporating several potentially influential features with respect to a national team's sportive success, such
as the bookmakers' ratings or the FIFA ranking. Based on all matches from the five previous UEFA EUROs 2004-2020, we
combine a LASSO-penalized Poisson regression model with two alternative modeling classes, so-called random forests and
extreme gradient boosting, which can be seen as mixture between machine learning and statistical modeling and are known for
their high predictive power.

Moreover, we incorporate so-called hybrid predictors, i.e. features which were obtained by a separate statistical model.
For different (weighted) combinations of the three modeling techniques from above, the predictive performance with regard to
several goodness-of-fit measures is compared. Based on the estimates of the best performing method all match outcomes of
the UEFA EUROs 2024 in Germany are repeatedly simulated (1,000,000 times), resulting in winning probabilities for all
participating national teams.

Keywords: Football, UEFA EUROs, LASSO regression, Random forests, XGBoost, hybrid modeling.

Abstract:

A Latin square of order $n$ is an $n$ by $n$ grid filled with $n$ symbols so that every symbol appears exactly once in each row and each column. A partial transversal of a Latin square of order $n$ is a collection of cells in the grid which share no row, column or symbol, while a full transversal is a partial transversal with $n$ cells.

The natural extremal question here (and the subject of the Ryser-Brualdi-Stein conjecture) is: how large a partial transversal can we guarantee in any Latin square of order $n$? For a Latin square chosen uniformly at random it is known due to Kwan that we can expect to find a full transversal, so a natural probabilistic question is: can we expect to be able to decompose a random Latin square into disjoint full transversals?

I will discuss recent work on both of these questions, the latter of which is joint work with Candida Bowtell.

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

Abstract:

In this talk I'll show the existence of a packing of identical spheres in $\mathbb{R}^d$ with density $$(1-o(1))\frac{d\log d}{2^{d+1}}\, ,$$ as $d\to \infty$. This improves the best known asymptotic lower bounds for sphere packing density. The proof uses a connection with graph theory and a new result about independent sets in graphs which is proved probabilistically.
This is joint work with Matthew Jenssen, Marcus Michelen and Julian Sahasrabudhe.

Abstract:

Simplicial complexes are a natural tool for modeling structures in which
there exist interactions between objects, such as political structures and
voting games. The geometrical realizations of these simplicial complexes are
topological spaces whose properties, for example homology, give us something
about the modeled structures. We will talk about power indices, as well as
concepts such as: stability of political structure, merging, mediator,
delegation, compromise...

Abstract:

We address two current research topics, the modeling of groups of
adjacent deformable shapes and the concept of slopes that
resulted from the interpretation of LBP pyramids. In both topics
we will focus on geometrical (and topological) aspects in order
to set the basis for a fruitful scientific discussion.
Further topics that will be addressed: motivations from projects
in the past: the process of straightening/flattening enables the
separation of shape models in a normalized shape and the geometric
back-projection of its axis; super-ellipses and Bézier approximations;
the construction of topology preserving graph-pyramids and
the reconstruction with only the few colors of the top,
local binary patterns replace convolution and differentiation
to detect critical points, slope regions relate the top level
with its receptive fields in the reconstruction.

Abstract:

Abstract:


Estimating the size of a hard-to-count population is a challenging matter. We consider uni-list approaches in which the count of identifications per unit is the basis of analysis. Unseen units have a zero count and do not occur in the sample leading to a zero-truncated setting. Because of various mechanisms, one-inflation is often an occurring phenomena that can lead to seriously biased estimates of population size. The talk will review some recent advances on one-inflation and zero-truncation modelling, and furthermore focuses here on the impact it has on population size estimation. The zero-truncated one-inflated and the one-inflated zero-truncated model is compared (also with the model ignoring one-inflation) in terms of Horvitz–Thompson estimation of population size. Simulation work shows clearly the biasing effect of ignoring one-inflation. Both models, the zero-truncated one-inflated and the one-inflated zero-truncated one, are suitable to model ongoing one-inflation. It is also important to choose an appropriate base-line distributional model. Considerable emphasis is allocated to a number of case studies which illustrate the issues and the impact of the work.

Abstract:

Abstract:

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:

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.

Abstract:

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

Abstract:

{\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:

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.

Abstract:

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.

Abstract:

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 $d$.

Abstract:

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.

Abstract:

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.

Abstract:

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:

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.

Abstract:

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.

Abstract:

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.

Abstract:

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

Abstract:

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.

Abstract:

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}

Abstract:

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

Abstract:

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:

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.

Abstract:

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).

Abstract:

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:

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:

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.

Abstract:

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 $c = 3\sqrt{3}/(2\sqrt{\pi}})$.

Abstract:

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

Abstract:

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:

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.

Abstract:

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

Abstract:

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.