Welcome to the Biomathematics group!
Describing and understanding the complexity of biological systems through mathematics is the motivation of the working group. We use methods of partial differential equations, variational calculus and geometric measure theory and investigate fundamental questions about the shape of biological cells, protein distribution on membranes and spatially and temporally coupled reaction-diffusion processes. The focus of our group is on the development and improvement of mathematical methods that enable the understanding of fundamental organizational principles of complex behavior.
Main interests
Free boundary value problems
The coupling of PDEs in two or more phases to geometric quantities on the phase interfaces occurs in a variety of applications in material and life sciences. A rigorous mathematical analysis of such systems requires the use of weak solution concepts and methods of geometric measure theory. An an example of our contributions, we were able to show the existence of weak solutions for a coupled Navier-Stokes Mullins-Sekerka problem for two-phase flows.
Polarization of biological cells
The formation of highly heterogeneous protein distributions is essential for many cell biological processes. Suitable mathematical descriptions are initially in the form of reaction-diffusion systems that take into account the mutual relationship between processes inside the cell and those on the cell membrane. Rigorous asymptotic simplifications allow the derivation of effective models and the characterization of the qualitative behavior of such systems. A particular contribution in this field proves the convergence of certain polarization problems against generalized obstacle problems and rigorously derives polarization criteria.
Variational analysis of curvature energies
Models for biological membranes are often based on Canham-Helfrich curvature energies (with the Willmore functional as a special case). Compared to phase separation energies, the higher order of such energies poses particular challenges. Compactness statements can usually only be achieved in spaces of generalized surfaces. Examples of work in this area are the minimization of the Willmore functional under an confinement condition in an external container, for example motivated by internal membranes in biological cells.
Phase field models
In many models, the transition between different phases is not described by a low-dimensional interfacial surface, but by a thin transition layer. Phase field models from materials science are one classic example, other examples are given by diffuse approximations of geometric energies, as often used in numerical simulations. A crucial question is about asymptotic reductions for vanishing thickness of the transition layer. Using variational methods ($\Gamma$-convergence) and measure-theoretic formulations (varifolds as a generalized concept of surfaces), we were able to prove the convergence of the De Giorgi approximation of the Willmore functional.
Geometric flows
The evolution in the direction of a steepest descent describes a possible dynamic towards states of low energy. In the case of phase separation or curvature energies, for example, this leads to mean curvature and Willmore flows. We have analyzed approximation results for such flows and have studied the behavior under (stochastic) perturbations.
Present and former group members
Professor Dr. Matthias Röger
Head of the group.M.Sc. Jakob Fuchs
Mean curvature flow, curvature energies, scale transitions.
M.Sc. Sascha Knüttel
Ph.D. student, graduated in 2023.Dr. Andreas Rätz
Privatdozent (now at Universität Düsseldorf).M.Sc. Nils Dabrock
Ph.D. student, graduated in 2020.Dr. Carsten Zwilling
Ph.D. student, graduated in 2018.Dr. Keith Anguige
Postdoctoral research fellow.Dr. Stephan Hausberg
Ph.D. student, graduated in 2016.Dr. Martin Heida
Postdoctoral research fellow.Dr. Simona Puglisi
Postdoctoral research fellow.Dr. Luca Lussardi
Postdoctoral research fellow.Dr. Annibale Magni
Postdoctoral research fellow.