Formation of defects in the harmonic map heat flow

The harmonic map heat flow is a model for nematic liquid crystals and also has origins in geometry. We will introduce the model and discuss some of its mathematical properties. In particular, we will focus on the possibility that singularities may develop.

The rate at which singularities develop is investigated in settings with certain symmetries. We use the method of matched asymptotic expansions and identify different scenarios for singularity formation. More specifically, we distinguish between singularities that develop in finite time and those that need infinite time to form.

Finally, we discuss which results can be proven rigorously, as well as some open problems, and we address stability issues (ongoing work with JF Williams).