ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

5. Differential Geometry and Global Analysis

1991 MS Classification: 58, 32, 34 Boalch, Philip Paul, University of Oxford, U. K. Symplectic geometry and isomonodromic deformations. Singular flat Gl_{n}(ℂ) connections over ℂP^{1} are viewed in three ways, giving different approaches to their moduli: 1) as spaces of coefficients of linear meromorphic differential systems on a trivial holomorphic vector bundle,

2) using an infinite dimensional C^{∞} viewpoint, and 3) via Stokes matrices (which are the analogue of monodromy matrices for such connections). Intrinsic symplectic structures are exhibited on the spaces in 1) and 2) and the natural map between them is formally shown to be symplectic. These results yield a symplectic description of the equations for isomonodromic deformations (in the sense of Jimbo-Miwa-Ueno) which will also be presented.