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Course Descriptions

16:640:509 - Topics in Analysis

Haim Brezis


Sobolev maps with values into the circle

Course Description:

Sobolev functions with values into R (the real numbers) are well understood
and play an immense role in many branches of Mathematics. By contrast,
the theory of Sobolev maps with values into the unit circle has only been explored in recent years.
Such maps occur e.g. in the asymptotic analysis of the Ginzburg-Landau model (arising in superconductivity). The reason one is interested in Sobolev maps, rather than smooth maps is to allow maps with singularities such as x/|x| in 2-d or line singularities
in 3-d which appear in physical problems. Our focus in this
course is not the Ginzburg-Landau equation per se, but rather the
intrinsic study of the Sobolev space of maps from a domain in R^N, with values into the circle. It turns out that these classes of maps have an amazingly rich
structure. Geometrical and topological effects are already
conspicuous, even in this very simple framework, since the circle has nontrivial topology. On the other hand, the fact that the target space is the circle offers the option to
introduce a lifting,and this raises many interesting questions.


I will share sections of a forthcoming monograph


Sobolev spaces

Schedule of Sections:

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