Geometric variational problems: regularity vs singularity formation

Yannick Sire (Johns Hopkins University)

Abstract.
I will describe in a very informal way some techniques  to deal with the existence  ( and more qualitatively regularity vs singularity formation) in different geometric problems and their heat flows motivated by (variations of) the harmonic map problem, the construction of Yang-Mills connections or nematic liquid crystals. I will emphasize in particular on recent results on the construction of very fine asymptotics of blow-up solutions via a new gluing method designed for parabolic flows. I’ll describe several open problems and many possible generalizations, since the techniques are rather flexible.