Geometry Seminar: Carberry -- Integrable Systems and Harmonic Maps

Geometry Seminar : Carberry -- Integrable Systems and Harmonic Maps

Speaker: Dr. Emma Carberry (Sydney)

Time: Tuesday, October 11th, 12(NOON)--1PM

Room: Carslaw 707A

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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.

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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.

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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.

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