Research in Applied Analysis

The analysis of partial differential equations is a central subject in mathematics, combining the rigor of modern analysis and geometry with the concrete real-world intuition of physics and other sciences.

Partial differential equations (PDEs) model a huge variety of phenomena, and no general theory is known concerning their solvability and the dynamics of their solutions.

PDEs have been used to establish fundamental results in mathematics and are at the heart of whole fields of Mathematics and Physics.
Examples are:
Complex Analysis, Minimal Surfaces, Harmonic Maps, Kähler and Einstein Geometry, Geometric Flows, Hydrodynamics, Elasticity, General Relativity, Electrodynamics, and Non-relativistic Quantum Mechanics.

Probability theory and the theory of PDEs are strongly intertwined, not only by Kolmogorov's equations. And Deligne's work on differential equations with regular singular points on smooth complex varieties yielded a new solution of Hilbert's 21st problem.

We develop mathematical theory for the solution and regularity of classes of nonlinear PDEs and for the description of their dynamics.  
These include parabolic, elliptic, and hyperbolic PDEs, and integro-differential-equations, as well as related stochastic interacting particle systems and their limiting behavior.

We develop and analyze mathematical models for concrete applications, mainly in biology and medicine, covering for example cell motility, chemotaxis, the dynamics of the cellular cytoskeleton, biochemical pathways, pattern formation in microbial colonies and developmental processes, regeneration of model organisms, and epidemiology.

We teach classical and modern techniques for PDEs, thus providing students with a strong and broad mathematical background for their studies in a variety of mathematical disciplines.
back

Abstract:

We introduce and analyze a mathematical model for the regeneration of planarian flatworms. This system of differential equations incorporates dynamics of head and tail cells which express positional control genes that in turn translate into localized signals that guide stem cell differentiation. Orientation and positional information is encoded in the dynamics of a long range wnt-related signaling gradient.
We motivate our model in relation to experimental data and demonstrate how it correctly reproduces cut and graft experiments. In particular, our system improves on previous models by preserving polarity in regeneration, over orders of magnitude in body size during cutting experiments and growth phases. Our model relies on tristability in cell density dynamics, between head, trunk, and tail. In addition, key to polarity preservation in regeneration, our system includes sensitivity of cell differentiation to gradients of wnt-related signals measured relative to the tissue surface. This process is particularly relevant in a small tissue layer close to wounds during their healing, and modeled here in a robust fashion through dynamic boundary conditions.

Keywords: planaria and regeneration; dynamic boundary conditions; wnt-signal; PDE; math biology