What is a circle-valued map made of?

What is a circle-valued map made of?

Mon, 01/11/2010
17:00 		Petru Mironescu (Universite Lyon 1) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
What is a circle-valued map made of?
The maps $ u $ which are continuous in $ {\mathbb R}^n $ and circle-valued are precisely the maps of the form $ u=\exp (i\varphi) $, where the phase $ \varphi $ is continuous and real-valued. In the context of Sobolev spaces, this is not true anymore: a map $ u $ in some Sobolev space $ W^{s,p} $ need not have a phase in the same space. However, it is still possible to describe all the circle-valued Sobolev maps. The characterization relies on a factorization formula for Sobolev maps, involving three objects: good phases, bad phases, and topological singularities. This formula is the analog, in the circle-valued context, of Weierstrass' factorization theorem for holomorphic maps. The purpose of the talk is to describe the factorization and to present a puzzling byproduct concerning sums of Dirac masses.