University of Oxford MATHEMATICAL INSTITUTE LMS -- EPSRC INSTRUCTIONAL CONFERENCE on INTEGRABLE SYSTEMS 14th -- 19th September 1997

Lecturers: N.J. Hitchin, G.B. Segal, R.S. Ward

The course is intended for postgraduate students in the initial stages of their work. The emphasis will be geometric and the lectures will explain some of the connections between the modern theory of integrable systems and other branches of mathematics, and also their central role in recent interactions between mathematics and physics.

Accommodation and meals will be provided at Wadham College, Oxford (a short walk from the Mathematical Institute). Funds are available to meet the accommodation and subsistence costs for EPSRC-supported students (who can reclaim their travel costs from the EPSRC). Limited funds are available to support a small number of other postgraduate students.

For further details and application form, write to:

Dr N.M.J. Woodhouse
The Mathematical Institute
24-29 St Giles'
Oxford OX1 3LB
or email: nwoodh@maths.ox.ac.uk

The course will be built around three lecture courses (5--7 lectures each:

Integrable systems and Riemann surfaces (N.J. Hitchin)

Introduction to Riemann surfaces: abstractly and as algebraic curves. Meromorphic functions and linear equivalence of divisors. Elliptic curves and the Weierstrass function. Line bundles over Riemann surfaces and theta functions.

Spinning top equations. Lax equations. Spectral curves. The eigenspace bundle.

Complex symplectic manifolds. Completely integrable systems. Examples.

Loop groups (G.B. Segal)

Loop groups are interesting from several points of view. These lectures are designed to lead the audience into the theory of integrable systems, and so will focus less on the infinite-dimensional groups in themselves than on their relations with the theory of holomorphic bundles on Riemann surfaces, and with the Riemann-Hilbert problem of finding a holomorphic ordinary differential equation with prescribed monodromies around prescribed singular points.

To get as far as possible with the main ideas, the lectures will concentrate on the loop groups of the unitary and general linear groups. The topics covered will be as follows.

Generalities about loop groups, and the Birkhoff factorization. The restricted Grassmannian of Hilbert space, and the basic representation of the loop group of $U_n$. The $\tau$ function as a matrix element of the basic representation. The simplest examples of Vertex operators, and their relation to the Birkhoff factorization.

Examples of the use of factorization in producing solutions of differential equations and in constructing harmonic maps. A little about isomonodromic deformations, and their relation to $\tau$-functions.

Integrable systems and twistors (R.S. Ward)

Introduction to Integrable Systems. Geometry: connections, curvature, gauge transformations; consistency of linear systems of differential operators; commuting vector fields and Frobenius' theorem. Lax pairs, examples (mechanical systems, solitons, etc). Self-dual Yang-Mills, self-dual Einstein and some of their reductions.

Twistor Constructions. Twistor and minitwistor correspondences for flat space. Vector bundles and self-dual gauge fields; examples (instantons, monopoles, solitons, etc.)

HTML by Brenda Willoughby Tue Apr 22 12:58:57 GMT 1997 Corrections e-mailed to brenda@maths.ox.ac.uk