Doctoral School Mathematics and Scientific Computing

Abstract: In r-neighbour bootstrap percolation, an initially infected set of vertices in a graph spreads the infection to uninfected vertices whenever  the latter have ≥r infected neighbours. We are interested in the probability that a  randomly selected initial set spreads the infection throughout the graph. This has been studied on lattice-like graphs, e.g. for r=2 Balogh and Bollobás identified a threshold function in the hypercube [2]^d. A modified version known as Froböse percolation has also been studied, particularly  on the grid [n]². In this talk we present a threshold function for Froböse percolation in the hypercube [2]^d.

Abstract: This talk gives a gentle introduction to the main tasks of graph drawing through the lens of combinatorial optimization. We exemplify the usefulness of this viewpoint in the study of the geometric k-colored crossing number. This parameter, denoted  cr_k(G) for a given graph G, is a measure of how well a graph can be visualized on k Euclidean planes simultaneously. For small k, we present upper bound improvements to the geometric k-colored crossing number of complete graphs.

Abstract: We suggest a dependence coefficient between a categorical variable and some general variable taking values in a metric space. We derive important theoretical properties and study the large sample behaviour of our suggested estimator. Moreover, we develop an independence test which has an asymptotic χ² distribution if the variables are independent and prove that this test is consistent against any violation of independence. The test is also applicable to the classical K-sample problem with possibly high- or infinite-dimensional distributions. We discuss some extensions, including a variant of the coefficient for measuring conditional dependence.

Abstract: In quantitative dynamic positron emission tomography, tissue kinetics are estimated from tracer concentration time series. This leads to a nonlinear parameter identification problem that typically relies on costly and time-consuming blood sampling. We show that for standard two-tissue compartment models the kinetic parameters are uniquely identifiable from image data alone under reasonable assumptions in an idealized noiseless setting. Numerical experiments with a regularization approach support the analytical result in an application example.

Abstract: The Container loading problem deals with placing cuboid shaped boxes into containers such that the utilized volume is maximized and constraints are satisfied (e.g. rules about how boxes may be stacked, about weight distribution, or the unloading order). We then demonstrate how to apply Monte Carlo Tree Search to this problem. This is a heuristic algorithm combining random sampling with tree-based exploration, leveraging iterative simulations to guide decision-making, making it particularly suitable for complex packing scenarios.

Abstract: An odd matching of a graph G is a matching of G except for a single isolated vertex v. A flip in an odd matching adds an edge between v and another vertex w, in the process removing the edge that was previously incident to v. We study the following questions: (1) Can any matching be transformed into any other via flips; (2) If yes, how many flips does it take; and (3) how do we find the minimum number of required flips.

Abstract: In the fair division problem one must distribute a set of indivisible goods to a set of agents, each of whom may value the goods differently, with the goal of giving each agent equal utility.  Further, we work in the case where the problem must be solved several times sequentially, and consider the cumulative solutions.  We show that there are polynomial time algorithms for finding equitable allocations in some special cases, and also present a generalizable framework for investigating this problem.

Abstract: In this talk, we consider a space-time finite element method for the time-dependent Stokes system. While classical approaches rely on time-stepping schemes, we propose a fully space-time variational formulation in the Bochner setting. This allows for a unified treatment of spatial and temporal discretization. We further present numerical results on arbitrary and unstructured space-time meshes, which demonstrate the flexibility and effectiveness of the proposed method

Abstract: Malle’s conjecture is a central question in arithmetic statistics, predicting an asymptotic formula for the number of number fields with a prescribed Galois group and bounded height. Recently Gundlach introduced a version of Malle‘s conjecture counting by a system of multi-heights. In this work, we establish an asymptotic in Gundlach’s conjecture for the dihedral group of order 8, D₄. We also explain the role of the leading constant as a product of local densities.

Abstract: In materials science one analyzes 3D micro-CT scans of paper with the goal of building stochastic models replicating the material's microstructure. A central task in this setting is to assess whether the material is invariant under rotations and reflections, which is essential for stochastic models. We approach this topic via 2D function-valued random fields, and introduce a novel test for isotropy.