LMS/EPSRC Short Instructional Course
DIFFERENTIAL GEOMETRY, HOMOGENEOUS SPACES AND INTEGRABLE SYSTEMS
University of Durham, 16-20 September 2002

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MARTIN GUEST

GEOMETRY AND INTEGRABLE SYSTEMS


The name integrable system refers to a certain type of differential equation (ordinary or partial) which arises very frequently in pure and applied mathematics and also in physics. These equations have been investigated in an ad hoc manner ever since the beginning of calculus; they were of particular importance in the development of 19th century physics, but it is only rather recently that sufficient mathematical tools have become available for a systematic approach. There is still no precise definition of an integrable system; it is the kind of the thing about which people say you know it when you see it.

A very good modern introduction to the theory of integrable systems is the book Integrable Systems by Hitchin, Segal, and Ward (Oxford University Press, 1999). The book will not be used as a textbook for the course, but if you glance through it you will see how widely integrable systems appear in pure mathematics - in topology, algebraic geometry, differential geometry, analysis - in addition to applied mathematics and physics. Nevertheless, the heart of the subject is concerned with geometry and symmetry. In current mathematical terminology, this means manifolds and Lie groups.

In the course we will look at some examples, and some of the general theory, of integrable systems. The examples will be mainly from differential geometry, especially the theory of surfaces, and the general theory will involve the ideas of connections and curvature and (possibly infinite dimensional) Lie groups. However, prior knowledge of these subjects will not be taken for granted. Students attending the concurrent courses on Differential Geometry and Lie Groups will acquire the necessary background knowledge as the lectures progress.

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Last update: 8 February 2002