Course Descriptions

16:640:510 - Topics in Analysis

Haim Brezis


Sobolev maps with values into the circle

Course Description:

Sobolev functions with values into the real line are very well understood and play an immense role in many branches of Mathematics. By contrast, Sobolev maps with values into the unit circle have been investigated only in recent years.

Such maps occur e.g. in Physics. The Sobolev framework allows maps with singularities such as x/|x| in 2D or line singularities in 3D which appear in physical problems. It turns out that these classes of maps have an amazingly rich structure from the point of view of Analysis, Geometry and Topology.The course will be based on Lecture Notes which are not yet complete but parts will be available to students.

A decent knowledge of standard Sobolev spaces is required; it can be found in my book “Functional Analysis, Sobolev spaces and PDEs” (Springer)



Haim Brezis, Functional Analysis,and PDEs (Springer)


Functional Analysis, Sobolev spaces

Schedule of Sections: