Dr. Matthias Winter on his research:
"I am a Senior Lecturer at the Department of Mathematics, Brunel University London in Uxbridge (United Kingdom). My research interests are concerned with Partial Differential Equations, in particular Reaction-Diffusion Systems to model pattern formation in biological systems.
It has been very nice to visit the IADM for two weeks during June 2019. The research activities at the institute are closely related to my own projects. I study spiky patterns. These are steady states or a time-periodic solutions which possess sharp peaks near certain points in a domain but are approximately constant away from these points. In the limit of small diffusivity the diameter of these spikes become smaller and smaller, and finally they shrink to their respective points.
A certain topic which I have been investigating for a few years is the case of systems having as solutions stable spike clusters. Here a spike cluster is a group of spikes which all converge to the same point in some limit. This type of behaviour has been shown for the Gierer-Meinhardt system, a prototype of the so-called class of activator-inhibitor systems. It occurs in the following situations:
(i) two small diffusivities and an inhomogeneous coefficient,
(ii) two small diffusivities and curvature terms coming from the boundary of a two-dimensional domain,
(iii) two small diffusivities and curvature terms coming from a two-dimensional Riemannian manifold.
During this research stay progress has been achieved in subject area (iii). I had some very interesting discussions, above all with Prof. Dr. Guido Schneider and Prof. Dr. Jens Wirth. I would like to thank everyone at the institute for their kind hospitality."
Dr. Matthias Winter, Brunel University London