Not every homomorphism between topological groups is continuous. However there exists an increasing set of general conditions under which continuity is ensured, particularly when the domain is completely metrizable and the codomain discrete. I'll give some history of such results as well as detail some advances that have been made recently. Much of the presented work is joint with Greg Conner.

Kubische Hyperflächen sind von besonderem Interesse in der Geometrie. Die Tatsache, dass jede glatte kubische Fläche 27 Geraden enthält ist eines der ältesten Ergebnisse der algebraischen Geometrie. Kubische Hyperflächen der Dimension 3 waren die ersten Beispiele von Varietäten, welche eine endliche dominante Abbildung von einem projektiven Raum auf sich zulassen (also unirational sind), aber selbst nicht birational zu einem projektieren Raum (also rational) sind. Die Geraden auf einer kubischen Hyperfläche vom Grad 4 bilden wiederum eine irreduzible holomorphe symplektische Mannigfaltigkeit (Hyperkählermannigfaltigkeit). In diesem Vortrag werde ich über die Geometrie und die Topologie des Modulraums (klassifizierenden Raums) der kubischen Hyperflächen in Dimension 3 berichten.

Moving surfaces are ubiquitous in many areas of mathematics and the applied sciences.

In this talk I will first introduce some well-known geometric evolution equations, and then proceed to more complicated models that describe the motion of fluids and of materials that can undergo phase transitions.

Energy estimates for time dependent PDE’s are essential for proving well-posedness of initial boundary value problems. In a discrete setting energy estimates are used to investigate stability and accuracy of a discretization. Similarly, both continuous and discrete problems can be considered in Laplace-space. When boundaries are present the Laplace transform technique often yields sharper results then the energy method, concerning well-posedness and stability, but the analysis is more involved. In this talk I will discuss how the Laplace transform technique can also yield sharper accuracy results. As an example, I will consider the second order wave equation, discretized by a high order finite difference method.

Siegel modular forms of degree g are generalizations of the usual elliptic modular forms, the case g=1, and are just as intriguing, but much more difficult to construct. We intend to show how one can use invariant theory to describe such modular forms for degree 2 and 3 explicitly. This is joint work with Fabien Clery and Carel Faber.

Stochastic individual base models, that is, measure valued Markov processes describing the evolution of interacting biological populations, have proven over the last years to be effective models in deriving key features of the theory of adaptive dynamics, such as the canonical equation of adaptive dynamics, the trait substitution sequence and the polymorphic evolution sequence. In this talk I review these models an the diverse emerging scaling limits, and I report on recent progress in applying such models to the modelling of cancer therapies, and in particular to immunotherapy and combination therapies of melanoma, based on experimental data by colleagues from the Bonn university hospital.

Scattered data approximation deals with the problem of producing a function s that in some sense represents some given (typically scattered) data and allows to make predictions at other times/locations/parameter settings. Applications are quite diverse: Surface reconstruction, image compression, numerical solution of PDEs (with their diverse applications), to name just a few. In a scattered data interpolation problem, the interpolant is typically a linear combination of some radial basis functions (RBF). The coefficient vector c ∈ R^N of the interpolant may be computed as the solution of a linear system Bc = y which results from enforcing the interpolation conditions for the given scattered data. While properties of the matrix B obviously depend on the choice of basis functions, several of the most commonly used approaches yield highly ill-conditioned, dense matrices B, resulting in a challenge to solve the linear system Bc = y, and hence to solve the scattered data interpolation problem. This talk deals with these challenges and some possible strategies for the solution of this system Bc = y. In particular, we study the application of techniques from the ℋ-matrix framework both for the approximation of the system matrix B itself as well as for the construction of solvers. ℋ-matrices provide a data-sparse matrix format that permits storage and matrix arithmetic in complexity O(N log^α N) for moderate α. It turns out that several typical sets of basis functions from the (scattered data) literature lead to matrices B that fit into this framework, yielding a cost-effective approximation scheme to be illustrated in this talk.