Termine Angewandte Mathematik Münster

In this talk, I will consider biochemical networks and address the existence of unstable equilibria based only on the network structure. In particular, I will identify minimal sources of instability, so-called unstable cores, and discuss several of their features. For example: an unstable core can be classified in autocatalytic vs nonautocatalytic, and the presence of at least one autocatalytic core characterizes autocatalysis in network. Further, unstable cores serve as predictive tools for understanding the dynamic range of a network, sufficiently determining whether a network admits multiple equilibria or periodic oscillations. The presented approach relies on bifurcation theory and multiscale methods.

We study the (discrete) Phi^4 model on Z^d with algebraically decaying long-range interactions. This model is the natural discrete analogue of the fractional continuous Phi^4 model. We first describe some general properties of the phase transition that the model undergoes. Then, we study the nature of the critical scaling limits of the model, proving in particular the (marginal) triviality of these limits in dimension 3 with well-chosen interactions.

The exclusion process is one of the best-studied examples of an interacting particle system. In this talk, we consider the stationary distribution of asymmetric simple exclusion processes with open boundaries. We project the stationary distribution onto a subinterval, whose size is allowed to grow with the length of the underlying segment. Depending on the boundary parameters for the exclusion process, we provide sufficient conditions such that the projected stationary distribution is close in total variation distance to a product measure. This talk is based on joint work with Evita Nestoridi.

Detecting the boundary of the braintumor glioblastoma multiforme is a challenging task due to its diffusive nature. To estimate the tumor extent at the current timepoint, we derive a traveling wave formulation of a forward-in-time multiphase tumor model in 1D. Arising challenges are the nonlinearity and the degeneracy of the resulting PDE as well as numerically taking the steep gradient into account. Further unknowns in this formulation are the velocity of the traveling wave as well as the corresponding interval length, which will be tackled by different fixed-point formulations. Numerical experiments will be presented to determine the validity of the method.