Harmonic Maps and Loop Groups

WS 2014/15

Prof Dr Christoph Bohle

Lecture

Thursday, 14-16, S11

Overview of the Lecture

The basic goal of the course will be to study how differential geometric problems about harmonic (i.e., “energy minimizing”) maps from (2-real dimensional) surfaces into higher dimensional manifolds can be translated into the language of “loop groups” which are a certain kind of infinite dimensional Lie group.

Some of the content of the course will be based on the book “Harmonic Maps, Loop Groups, and Integrable Systems” by Martin Guest, a preprint of which you can find on his website:

http://www.f.waseda.jp/martin/publications.html

Specifically, we will start out by carefully introducing harmonic maps and how they generalize the idea of a geodesic. Then we will focus on rewriting the harmonic map equations in terms of Maurer-Cartan forms and derive “zero-curvature” equations for a connection which depends on a “spectral” parameter - reviewing all the basic definitions from differential geometry along the way. At this point we may get into some more sophisticated Lie group and Lie algebra theory but again it will all be geared toward students without prior experience in these topics.

Prerequisites

The basic prerequisites are advanced calculus, linear algebra, some basic differential geometry (definition of a manifold, Riemannian metric, etc), and basic complex analysis.

Suggested Literature

TBA