Singularities in nonlinear PDEs and geometric flows

Project description

Some nonlinear PDE (or systems of nonlinear PDE) with a small parameter exhibit the formation of line singularities when the small parameter tends to zero. Such PDE are widely used in models from quantum physics and materials science, for example the Ginzburg-Landau type models that describe vortex tubes in superconductors or dislocation lines in crystals. A natural question is how PDE dynamics translates to motion laws for the singularity lines. Typical results are curve shortening flow or the binormal flow that governs vortex ring dynamics in fluids.

While many results are known, other areas of research are wide open. In the project, the student can focus on mixed flows or on how the geometry of a domain drives dynamics on its boundary.