Dr. Khazhgali Kozhasov (Universität Osnabrück)
Singularities of Spectrahedra
It is of interest for applications of algebraic geometry to understand whether some parametrized system of real polynomial equations can exhibit some extremal behavior (for instance, has only real solutions for some values of parameters). For example, a real univariate polynomial has in general non-real roots, but there always exist polynomials with only real roots. I will discuss situations of this kind concentrating on the following problem arising in semi-definite optimization. Given a convex set S described by a linear matrix inequality t Id + x A + y B + z C > 0, one is interested in finding optimal bounds on the number of singular points on the boundary of S in terms of the size n of matrices A, B, C. In a joint work with Taylor Brysiewicz and Mario Kummer we solve this problem completely for n=5, classifying possible outcomes for the number of singular points in a generic case.
Dr. Andreas Focks (Universität Osnabrück)
Models in the Environmental Risk Assessment of Chemicals
In the field of the environmental risk assessment of chemicals, mathematical models have a long tradition of being used to analyse emissions and fate of chemical molecules in the environment, whereas the mathematical modelling of effects of pesticide compounds on non-target species remained for decades on the level of simple statistical regression models. In the last years, however, the development and use of mechanistic eco(toxico)logical models in the regulatory environmental risk assessment of pesticides and plant protection products developed rapidly, and regulatory agencies start to discover the advantages of mechanistic modelling.
My research in the last years was guided by the attempt to combine state-of-art dynamic modelling approaches with questions of applicability and practical relevance of such models for an improvement of the current chemical risk assessment system. The presentation will start with a short flashlight on the current environmental risk assessment system for pesticides in the EU, to describe then the development of toxicokinetic-toxicodynamic (TKTD) models towards now being the first regulatory accepted mechanistic effect modelling approach. The presentation will then outline nature and properties of additional mechanistic effect modelling approaches, such as population models, to end with some ideas for future research.
Dr. Kristina Schubert (Universität Osnabrück)
Fluctuation Results for General Ising Models - Block Spin Ising Models and Random Interactions
Starting from the classical Curie-Weiss model in statistical mechanics, we will consider more general Ising models. On the one hand, we introduce a block structure, i.e. a model of spins in which the vertices are divided into a finite number of blocks and where pair interactions are given according to their blocks. The magnetization is then the vector of magnetizations within each block, and we are interested in its behaviour and in particular in its fluctuations.
On the other hand, we consider Ising models on Erdős-Rényi random graphs. Here, I will also present results on the fluctuations of the magnetization.