Email: G.Carlet@dpmms.cam.ac.uk

After a short review of the classical R-matrix theory we show how to apply it to construct three compatible Hamiltonian structures for the two-dimensional Toda hierarchy.

Email: m.feigin@imperial.ac.uk

I am going to consider generalized quantum CMS problems with trigonometric potentials corresponding to special configurations of hyperplanes with multiplicities which appeared in the investigations of rational problems in the papers by Chalykh, Veselov, and the author. I will define Baker-Akhiezer functions, present difference equations of Ruijsenaars-Macdonald type they satisfy in the spectral parameters, and find the formulas for them. This extends the previous results by Chalykh obtained for trigonometric CMS problems related to root systems and one of the deformations. We also derive geometrical restrictions for a configuration of hyperplanes admitting the Baker-Akhiezer function.

Email: maxim.pavlov@mtu-net.ru

At least three local Hamiltonian structures of the Benney moment chain are found.

Email: bernd@ma.hw.ac.uk

In three space-time dimensions, Einstein's theory of gravity can be formulated as a Chern-Simons theory with the Poincar\'e group P_{2,1} as gauge group. If space-time has the topology R\times S, where S is a two-dimensional surface, the classical phase space of gravity in this formulation is the space of flat P_{2,1} connections on S. The Poisson structure of the phase space can be computed explicitly using methods developed by Fock, Rosly, Alekseev and Malkin. The main technical tool used is the theory of Poisson-Lie groups, of which P_{2,1} is a non-compact but in otherwise very simple example. I will outline the computation of the Poisson structure, and explain how it can be quantised, leading to a rigorous construction of a quantum gravity in 2+1 dimensions. The talk is based on joint work with Catherine Meusburger.

Email: sptsarev@mail.ru

We give a new procedure for generalized factorization and construction of the complete solution of strictly hyperbolic linear partial differential equations or strictly hyperbolic systems of such equations in the plane. This procedure generalizes the classical theory of Laplace transformations of second-order equations in the plane. Examples of integration of non-trivial systems are given.

Email: richard.ward@durham.ac.uk

Periodic instantons (or calorons) are self-dual gauge fields in four dimensions which are periodic in one direction. A special case of this is self-dual fields wich are constant in one direction (these are monopoles in R^3), and a limiting case is where the period becomes infinite (these are instantons in R^4). The first part of the talk describes how the Nahm transform can be used to construct families of caloron/instanton/monopole solutions.

Periodic monopoles (infinite monopole chains) are more difficult to handle, because of divergences. Again there is a Nahm construction, and the second part of the talk will describe what is known about the simplest non-trivial example.

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Last updated 28th February, 2005.