abstracts

Plant autotoxicity has proved to play an essential role in the behaviour of local vegetation. We analyse a reaction-diffusion-ODE model describing the interactions between vegetation, water, and autotoxicity. The presence of autotoxicity is seen to induce movement and deformation of spot patterns in some parameter regimes, a phenomenon which does not occur in classical biomass-water models. We aim to analytically quantify this novel feature, by studying travelling wave solutions in one spatial dimension. We use geometric singular perturbation theory to prove the existence of symmetric, stationary and non-symmetric, travelling pulse solutions, by constructing appropriate homoclinic orbits in the associated 5-dimensional dynamical system. In the singularly perturbed context, we perform an extensive scaling analysis of the dynamical system, identifying multiple asymptotic scaling regimes where (travelling) pulses may or may not be constructed. We discuss the agreement and discrepancy between the analytical results and numerical simulations. Our findings indicate how the inclusion of an additional ODE may significantly influence the properties of classical biomass-water models of Klausmeier/Gray–Scott type.

We study two interacting quantum particles forming a bound state in d-dimensional free space, and constrain the particles to half-space in k directions, with Neumann boundary conditions. First, we show that the particles stick to the corner where all boundary planes intersect. Second, we prove that the resulting Hamiltonian, after removing the free part of the kinetic energy, has only finitely many eigenvalues below the essential spectrum. This generalizes the work of Egger, Kerner and Pankrashkin (J. Spectr. Theory 10(4):1413--1444, 2020) to dimensions d>1 and was obtained in collaboration with Robert Seiringer.

In a 3D bounded and C^{1,α}-smooth domain Ω, 0<α<1, we distribute a cluster of nanoparticles enjoying moderately contrasting relative permittivity and permeability which can be anisotropic. We show that the effective permittivity and permeability generated by such cluster is explicitly characterized by the corresponding electric and magnetic polarization tensors of the fixed shape. The error of the approximation of the scattered fields corresponding to the cluster and the effective medium is inversely proportional to the dilution parameter.

In this talk, I will talk about a cross-diffusion PDE model describing the interaction of two competing gangs. The gang members perform a biased random walk adding graffiti markings on the walls as they walk and avoid the graffiti markings done by the rival gang members. Depending on the strength of the avoidance and other parameters, the equilibrium state consists in either segregation or a well-mixed state.

I will present some of my previous work with optical coherence tomography (OCT) data. OCT is a non-invasive and state of the art technique for imaging layers of the human retina. Some typical aims for measuring and predicting vision loss are the identification of retinal fluid, the photoreceptor layer and retinal blood vessels. These problems are usually very complicated and go beyond evaluation algorithms provided by manufacturers. On the other hand, manual evaluation of large-scale data is often not feasible in the clinical context and may suffer from inconsistencies. Therefore, well designed algorithms are needed to tackle these problems yielding feasible results for subsequent diagnosis.

On metric spaces, certain maps, the quasi-isometries, provide an equivalence relation. Since we can understand groups as geometric objects by considering their Cayley graphs, this leads to one the fundamental problems in geometric group theory, the Quasi-Isometry Problem: Given two finitely generated groups, decide whether they are equivalent to each other or not.

A class of groups for which we would like to solve the Quasi-Isometry Problem are the Right-angled Coxeter groups, which arise as a generalization of reflection groups. Conveniently, their presentations can be given in terms of a defining graph. In this talk I want to illustrate with some hands-on examples how we can solve the Quasi-Isometry Problem for a wide class of Right-angled Coxeter groups just by comparing their defining graphs.

I will give a basic introduction into totally disconnected, locally compact groups.

In this talk we shall discuss how Dupin cyclides (that is, surfaces foliated by circular curvature lines) can be generated by a pair of initial data: an initial circle and a 1-parameter family of suitable Möbius transformations.

We then generalize this idea to orthogonal coordinate systems in 3-space and demonstrate how systems with Dupin cyclides as coordinate surfaces arise in this way. Thus, this evolution approach provides a purely geometric construction for these systems that were already intensively studied by various geometers in terms of their metrics.

Dynamic programming and Bellman’s principle are versatile tools with applications not only in financial mathematics , but also in many other scientific fields. While multi-objective problems of optimal control have been solved via dynamic programming in the past, an explicit form of the Bellman equation has not been available until now. In this talk we will present an appropriate counter part of the Bellman’s principle for problem with multiple objectives. We will illustrate this on a well-known bi-objective problem from financial mathematics, the mean-risk problem.

The prophet inequality problems is an Optimal Stopping problem. In its classical version, samples from independent random variables (possibly differently distributed) arrive online in an adversarial order. A gambler that knows the distributions, but cannot see the future, must decide at each point in time whether to stop and pick the current sample or to continue and lose that sample forever. The gambler’s goal is to maximize the expected value of what she picks. The performance measure is the worst-case ratio between the expected value the gambler gets and what a prophet that sees all the realizations in advance gets. Recently, there has been a lot of work in generalizing these models to multiple items because of their applications in mechanism design. A particularly interesting variant is the so-called Prophet Secretary problem, in which the only difference is that the samples arrive in a uniformly random order. For this variant several (1−1/e)-competitive algorithms are known. But what happens in real world applications is different. There is no information about the distribution which variables are drawn from. Current algorithms don’t take this into account. We aim to analyze an extension form of Prophet Inequality which we call Repeated Prophet Inequality when the algorithm is applied many times under partial knowledge. That means gambler knows nothing about the distribution except for some samples. The main challenge is the trade-off between exploration and exploitation of information.

In this talk an existence result for the regular conditional distribution a large class of semi-martingale driven SDEs is presented. For this we show that the solution of these SDEs can be written as a measurable function of its driving processes into the space of all càdlàg functions equipped with the Borel algebra generated by all open sets with respect to the Skorohod metric. Our result is relevant, for example, in computational stochastics: as corollary it provides a Markov property which is essential for proving convergence results of numerical methods for general semi-martingale driven SDEs. This is joint work with Michaela Szölgyenyi (University of Klagenfurt) and Paweł Przybyłowicz (AGH University of Science and Technology).

We investigate the problem of learning an unknown nonlinearity in parameter-dependent PDEs. The nonlinearity is represented via a neural network of an unknown state. The learning-informed PDE model has three unknowns: physical parameter, state and nonlinearity. We propose an all-at-once approach to the minimization problem. (Joint work: Martin Holler, Christian Aarset) More generally, the representation via neural networks can be realized as a discretization scheme. We study convergence of Tikhonov and Landweber methods for the discretized inverse problems, and prove convergence when the discretization error approaches zero. (Joint work: Barbara Kaltenbacher)

The study of the combinatorics of some special subsets of the reals has been central for Set Theory in the last decades. In particular, I will refer to the concept of an independent family, which was introduced by Fichtenholz and Kantorovic. I will mention many interesting results and open problems the study of these objects has to offer, as well as some generalizations I have studied in the last years.

We call an n-tuple (z_1, ..., z_n) ∈ \C^n multiplicatively dependent (m.d.) if there exists a non-zero integer vector (k_1, ..., k_n) ∈ \Z^n such that z_1^k_1 ... z_n^k_n = 1. For instance, (2,8) is m.d. and so is (3,9). Therefore, we say that (2,8) and (3,9) are consecutive pairs of m.d. integers. But do there exist any other consecutive pairs of m.d. integers? In the talk, we answer this question and pose many more. Some of the questions can be answered easily, others are still open. In particular, we present a result about consecutive triples of m.d. integers. This is joint work with Volker Ziegler.

Generating functions are great tools for counting discrete structures. For instance, we can often represent a decompositions of an object into smaller instances by a functional equation. Unfortunately, it might be very hard or impossible to solve this equation explicitly. In this talk, we present some recent results on solutions to functional equations with one catalytic variable and explain how to use them in some easy examples.

Pseudorandom sequences, i.e. sequences which are generated with deterministic algorithms but look random, have many applications, for example in cryptography, in wireless communication or in numerical methods. In this work, we are interested in studying the properties of pseudorandomness of sequences derived from hyperelliptic curves of genus 2. In particular, we study the linear complexity of these sequences and provide a lower bound for the same. In this talk, we will introduce the N-th linear complexity of a sequence, the group structure on hyperelliptic curves of genus 2 and look at the main result. This is joint work with László Mérai (RICAM, Linz).