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16/01/2007, 14:30 — 15:30 — Room P3.10, Mathematics Building

William Goldman, *University of Maryland (U.S.A)*

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Geometry and symmetries of moduli spaces over surfaces

The space of representations of the fundamental group of a surface into a Lie group is a natural object with rich geometry and symmetry. The topology of the surface influences the algebraic structure of the deformation space, with natural families of Hamiltonian flows related to curves on the surfaces. In particular these flows define continuous deformations closely related to the action of the discrete mapping class group of the surface. In the case of the one-holed torus and the four-holed sphere, these deformation spaces are families of affine cubic surfaces with actions of the modular group by polynomial Poisson automorphisms.