abstracts

The so called combinatorial sets of reals are sets of reals, which originate in analysis, topology and algebra, enjoy a specific infinitary combinatorial structure (which can hugely vary) and are maximal in a natural sense. Even though, these are easily defined objects, the combinatorics they represent is highly non-trivial and complex.  The talk will give a brief introduction to the subject and point towards some interesting open questions.

A quiver is a directed graph and a quiver representation assigns each vertex a vector space and each arrow a linear map between the corresponding vector spaces. The usual notions such as morphisms between representations, direct sums and decomposability can be defined very naturally. Every representation can be written uniquely as a direct sum of indecomposable ones. So it is important to describe the latter: Here Gabriel’s Theorem steps in and provides a classification for those quivers that admit only finitely many indecomposable representations. They are precisely given by the ADE Dynkin quivers that also appear in the theory of Lie algebras and cluster algebras. I will give an introduction to the topic and comment on areas of research where this is currently used.

Post-Lie algebra structures have been studied in various areas of mathematics, such as affine structures on Lie groups, where they appear as generalizations of pre-Lie algebra structures, homology of partition posets, Rota-Baxter operators and the classical Yang-Baxter equation, étale and prehomogenous modules or decompositions of Lie algebras. The existence and classification of these structures often involves the study of decompositions of Lie algebras. In this talk we will focus on how post-Lie algebra structures are related to decompositions of Lie algebras. Moreover, we will show how post-Lie algebra structures are also linked to étale and prehomogenous modules, presenting results on strongly disemisimple Lie algebras, i.e. Lie algebras that decompose as a direct vector space sum into the sum of two semisimple subalgebras.

This talk is based on joint work with Dietrich Burde and Karel Dekimpe. (D. Burde, K. Dekimpe and M.Monadjem: International Journal of Algebra and Computation, Vol. 32, Issue 08, 1495-1511 (2022))

If we parameterize the positions of all vertices of a given graph in the plane such that distances between adjacent vertices are fixed, we obtain a moving graph.  An L-linkage is a realization of a moving graph in 3D-space, by representing edges using horizontal bars and vertices by vertical sticks. Vertical sticks are parallel revolute joints, while horizontal bars are links connecting them.  We give a sufficient condition for a moving graph to have a collision-free L-linkage. Furthermore, we provide an algorithm guiding the construction of such a linkage when the moving graph fulfills the sufficient condition, via computing a height function for the edges (horizontal bars).

In particular, we prove that any Dixon-1 moving graph has a collision-free L-linkage and no Dixon-2 moving graphs have collision-free L-linkages, where Dixon-1 and Dixon-2 moving graphs are two classic families of moving graphs. 

I'll motivate and introduce the higher order Cech complexes in the context of data analysis. Then, I'll explain why they are challenging to compute in practice and how a sparsification would solve that issue. I'll finish with our result that a sparsification does exist - and let you know how much that improves the computational complexity.

The task of recognizing an algebraic surface from a single apparent contour can be reduced to the recovering of a homogeneous equation in four variables from its discriminant. In this paper, we use the fact that Darboux cyclides have a singularity along the absolute conic in order to recognize them up to Euclidean similarity transformations.

Let us fix two integers a and b, for example a = 2 and b = 3. Then some integers c can be represented as differences of powers of a and b. For example, 5 = 2^3 - 3^1, but it is easy to check that 6 = 2^x - 3^y is not possible with integer exponents x and y. Already in the 1930s S. S. Pillai proved an asymptotic formula for the number of c's that have such a representation for fixed a and b.  More recently, M. Bennett proved that for fixed a and b, any c has at most two distinct representations and there exist several other related results and conjectures. In this talk, we discuss whether such results can be extended to linear recurrence sequences on the one hand (e.g. replace a^x by the Fibonacci numbers F_x), and to transcendental numbers on the other hand (e.g. replace a^x by e^x). This talk is based on three papers that are joint work with Sebastian Heintze, Robert Tichy, Daodao Yang and Volker Ziegler. The goal is to give an idea of how approximation properties can be used to solve Diophantine problems. 

Many applied problems are aimed at finding and stabilizing of an optimal configuration of a system, while its parameters and the analytical description of the corresponding cost function are partially or completely unknown. One of the powerful tools to tackle such problems is extremum seeking theory, which allows to minimize (or maximize) the output of the control system using only some limited information. This talk presents a gradient-free model-free approach for generating extremum seeking controllers based on Lie bracket approximation techniques. The proposed algorithms exploit time-periodic controls whose coefficients depend only on the values of the cost function and do not exploit its derivative. Conditions for the practical asymptotic stability and asymptotic stability in the sense of Lyapunov are proposed depending on the properties of the system and control vector fields. The obtained results are illustrated via numerical simulations and experiments with a mobile robot.

The accurate finite element approximation of wave propagation problems may entail specifically tailored meshes and discrete spaces to deal with singularities and high-frequency oscillations. The implementation of these methods typically requires specialized analytical tools. Uniformly refined low-order finite element methods (FEMs) may be employed to avoid these involved numerical schemes. In the uniformly refined low-order setting, suitable compression techniques are needed to tackle the large computational complexity that results from the strong mesh resolution requirements. The Quantized Tensor Train (QTT) decomposition is a multilevel construction that successively separates the levels of the data, resulting in data compression and complexity reduction for suitable arrays. In this talk, we will devise a non-dissipative conforming piecewise-linear FEM for the acoustic wave equation and we will exploit its multilevel low-rank tensor structure to perform QTT complexity reduction.

In this talk, we introduce a fractional variant of the Cahn-Hilliard equation settled in a bounded domain and complemented with homogeneous Dirichlet boundary conditions of solid type. After briefly showing how to prove existence and uniqueness, we investigate the convergence of solutions for this class of nonlocal Cahn-Hilliard problems to their local counterparts, as the order of the fractional Laplacian appearing in the equation is let tend to 1.

This is a joint work with E. Davoli and L. Lombardini (TU Wien).

More precisely, f(t,x) only has to be integrable with respect to the first variable t and convex with respect to the second variable x, but it is not necessary to impose any specific growth condition of f.

My talk is based on joint work with Jarkko Siltakoski.

We consider a boundary value problem for a q-Dirac equation. We prove orthogonality of the eigenfunctions, realness of the eigenvalues, and we study asymptotic formulas of the eigenfunctions. We show that the eigenfunctions form a complete system, we obtain the expansion formula with respect to the eigenfunctions, and we derive Parseval's equality. We construct the Weyl solution and the Weyl function. We prove a uniqueness theorem for the solution of the inverse problem with respect to the Weyl function. 

We consider the problem of elastic diffraction tomography, which consists in reconstructing elastic properties (i.e. mass density and elastic Lamé parameters) of a weakly scattering medium from full-field data of scattered waves outside the medium. Elastic diffraction tomography refers to the elastic inverse scattering problem after linearization using a first-order Born approximation. In this paper, we prove the Fourier diffraction theorem, which relates the 2D Fourier transform of scattered waves with the Fourier transform of the scatterer in the 3D spatial Fourier domain. Elastic wave mode separation is performed, which decomposes a wave into four modes. A new two-step inversion process is developed, providing information on the modes first and secondly on the elastic parameters. Finally, we discuss reconstructions with plane wave excitation experiments for different tomographic setups and with different plane wave excitation frequencies, respectively.

In this talk, we will introduce the short-time Fourier transform (STFT) phase retrieval problem and review recent advances.

Abstract phase retrieval stands for the recovery of an element f of a given Banach space X from the amplitudes |Tf|, where T is a linear operator from X to a function space. The nature of the problem allows this only up to a global phase, i.e., up to a factor of modulus 1.

The problem is clearly ill-posed without any a priori information on X, often a Hilbert space of functions. Typically, we consider for the operator T the Fourier transform, the STFT, or the wavelet transform. Even with such well-investigated operators, it is a highly non-trivial matter whether the injectivity of the original operator is preserved when the phase is lost. A second point is the stability of the reconstruction: assuming the linear operator has a bounded inverse and phase retrieval is possible on its range, there isn’t necessarily a stable way to determine the function. It turns out that uniform stability is not possible for phase retrieval in infinite-dimensional Hilbert spaces, in stark contrast to the finite-dimensional case.

The sampling aspect further asks to recover a function, not from its magnitudes, but from the restriction of the magnitudes on a discrete (relatively) separated set. While only constraints are known for the sampled STFT phase retrieval of square-integrable functions, we obtained new results under additional support conditions.

This is joint work with Philipp Grohs and Lukas Liehr (University of Vienna).

When listening to a piece of music, the concept of variable bandwidth is very intuitive: the highest frequency in a musical composition is time-varying. With this motivation, it is reasonable to allow different local bandwidths to different intervals of a signal. However, producing a rigorous definition of variable bandwidth is a challenging task, since bandwidth is global by definition and the assignment of a local bandwidth finds an obstruction in the uncertainty principle. 

We introduce a new concept of variable bandwidth based on the truncation of Wilson expansions. Analyzing some MATLAB experiments, we motivate why these new spaces could be useful for the reconstruction of particular classes of functions.

A stochastic hierarchical model for the evolution of low grade gliomas is proposed. Starting with the description of cell motion using a piecewise diffusion Markov process (PDifMP) at the cellular level, we derive an equation for the density of the transition probability of this Markov process based on the generalised Fokker-Planck equation. Then, a macroscopic model is derived via parabolic limit and Hilbert expansions in the moment equations. After setting up the model, we perform several numerical tests to study the role of the local characteristics and the extended generator of the PDifMP in the process of tumour progression. The main aim focuses on understanding how the variations of the jump rate function of this process at the microscopic scale and the diffusion coefficient at the macroscopic scale are related to the diffusive behaviour of the glioma cells and to the onset of malignancy, i.e., the transition from low-grade to high-grade gliomas.

Nowadays, magnetic materials play an important role in modern technology.

Such materials find applications in the construction of electric and medical devices, and data storage to name a few. Very often, the efficiency of the device is based on the magnetic properties of the materials used. From a mathematical viewpoint, the study of these materials is challenging since the material-dependent parameters vary at a very small scale. 

This suggests carrying out a homogenization procedure.

In this talk, we provide an effective description of magnetic composites whose heterogeneities are randomly distributed. More specifically, in the framework of Gamma-convergence and stochastic two-scale convergence, we analyse how the bulk Dzyaloshinskii-Moriya interaction affects the magnetic properties of composite ferromagnetic materials.

This is a joint work with E. Davoli and J. Ingmanns.

Gaussian random fields (GRFs) are a type of geostatistical model used in  a range of spatial inference problems. In many such contexts data are available at a given spatial scale (or multiple scales), whereas inference or predictions are required at another scale that represents a different spatial configuration. We are in particular interested in downscaling in the context of global to local climate models, where GRFs play an important role, as a small number of parameters can be used to express a wide range of spatial properties. 

The GRF model of interest and the accompanying Bayesian inferential procedure use the INLA-SPDE approach. In this talk I will describe the GRF model, the inference procedure and simulation method and discuss challenges in this situation. 

We prove the optimality of the randomized Milstein algorithm for solving scalar jump-diffusion stochastic differential equations. For this, we first provide a complete error analysis of the randomized Milstein algorithm for approximating scalar jump-diffusion stochastic differential equations under substantially weaker assumptions than those known in the literature. Then we provide lower bounds for randomized schemes assuming the jump-commutativity condition and conclude optimality. Moreover, we give some insight into the multidimensional case and report numerical experiments that support our theoretical findings.

The Skorokhod Embedding Problem (SEP) is to represent a given probability measure as a Brownian motion stopped at a specific stopping time.

It is a classic problem in probability theory but also has numerous applications, most notably in robust mathematical finance.

There are many different solutions to the (SEP) known today all featuring different additional optimality properties. However, most of these solutions are given via existence results and still only a few computable solutions are known.

In this talk we present a simple probabilistic argument to represent two of the most well known solutions, namely the Root and Rost solutions as optimizers of respective optimal stopping problems. Such a representation allows for numerical computation of the solution.

Moreover we can extend the given argument to establish an optimal stopping representation for multiple marginals, an identity which was previously unknown for the Rost solution.