Many classical integrable systems can be written as systems on loop algebras, using the formalism of r-matrices. These systems, in turn, can be viewed as special cases of the generalised Hitchin systems on various moduli spaces of Higgs pairs.
For the group $Gl(N,C)$, these systems all have in common a description in terms of a spectral curve, lying inside a surface, and the Jacobian variety of the curve. One aim of the lectures is to exhibit how the geometry of the surface is an essential feature of these systems, in that they are essentially symmetric products of these surfaces as symplectic varieties. This allows one to explain, for example, the algebro-geometric techniques for integrating these systems.
More generally, it turns out that there is a natural way of describing a class of integrable systems associated to surfaces, of which Hitchin systems constitute the main example, but which also includes, for example, the ``quadratic'' systems due to Sklyanin.
For arbitrary semi-simple Lie groups, one no longer has a discription in terms of Jacobians, but rather in terms of generalised Prym varieties. Again, there is a geometric description, this time not involving surfaces, but more general varieties.
Finally, one need not restrict one's attention to semi-simple groups, and it turns out that considering Hitchin systems for a natural Lie group asssociated to a root system gives a fairly natural geometric setting for the Calogero-Moser system, and clarifies certain mysterious features of these systems.
Lecture 1.
The first lecture will concentrate on the rational and elliptic r-matrix
systems, showing how one can derive fairly explicit separating coordinates for
these systems, from the geometric point of view.
Lecture 2.
The second lecture will
discuss in a more general fashion the Hitchin system, as well as the systems
associated to surfaces.
Lecture 3.
The third lecture will be spent discussing the Calogero-Moser system and
its link to the Hitchin systems.