Mathematical Sciences: Group Proposal for Research: Topologyand Differential Geometry

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9401714 Browder Research will be carried forward in the areas of cohomology of groups, with applications to group actions on manifolds, and various topics on symplectic manifolds. Among the areas considered are the Euler characteristic of central extensions by Z of groups, such that the resultant group is finite dimensional, the rank of symmetry of products of spheres, i.e. the calculation of the maximal rank of products of Z/p which act effectively (or freely) on products of spheres, and related problems on other manifolds. In symplectic geometry the research will focus on Hamiltonian group actions, with particular attention to questions on the cohomology of the reduced spaces, the symplectic quotient, the ring structure of this cohomology, with applications to moduli spaces of flat vector bundles. Also to be considered are obstructions to the construction of symplectic structures and questions related to the algebraic topology of symplectic manifolds. Manifolds are the generalized spaces in which so much of modern physics and engineering find their natural expression, and whose topological (i.e. continuous) properties have been found to play a crucial role in the expression of these theories. In particular, the symmetry properties or the dynamical properties of these spaces, especially the spaces of mechanics, the so called symplectic spaces, and the dynamics associated to them are basic to understanding many physical and mathematical problems. This project will make its contribution using means from algebraic topology and differential geometry. ***