Type: Seminar

Distribution: World

Expiry: 19 Oct 2011

Auth: zhangou@bari.maths.usyd.edu.au

Geometry Seminar : Carberry -- Integrable Systems and Harmonic Maps Speaker: Dr. Emma Carberry (Sydney) Time: Tuesday, October 11th, 12(NOON)--1PM Room: Carslaw 707A --------------------------------------------- Series Title: Integrable Systems and Harmonic Maps Abstract: in this expository series of talks I will give a tour of the theory of harmonic maps from surfaces into Lie groups and symmetric spaces. This subject brings together integrable systems, differential geometry and complex algebraic geometry. It also has connections with mathematical physics, in fact these harmonic map equations are a reduction of the Yang-Mills equations with a change of signature from the reduction that describes Higgs bundles on a Riemann surface. Of particular interest are harmonic maps of tori, which in many cases can be obtained simply by solving ordinary differential equations and whose moduli spaces can be constructed quite explicitly. --------------------------------------------- Lecture 2 (October. 11) Title: Harmonic Maps Via Ordinary Differential Equations Abstract: harmonic maps are by definition the solutions to the Laplace-Beltrami equation, a second order partial differential equation. However there is a subclass of harmonic maps from a surface to a Lie group or symmetric space which can be described by vastly easier means. These maps of "finite-type" are obtained simply by integrating a pair of commuting vector fields on a finite dimensional space and hence by solving ordinary differential equations. This naturally prompts one to find conditions under which a map is of finite type, for which there are quite general results known when the target manifold is compact. --------------------------------------------- Lecture 1 (October 4th) Title: Introduction to Harmonic Maps of Surfaces Into Lie Groups and Symmetric Spaces Abstract: in this talk I will explain some basic facts about harmonic maps, concentrating on the geometrically interesting situation of mapping a surface into a Lie group or symmetric space. In this case the harmonic condition is equivalent to a certain family of connections all having zero curvature, which is the basis for the integrable systems approach to the subject. ---------------------------------------------