Mathematical Sciences: Topics in Topology

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9501701 Burghelea This project pursues research on the topology of the free loop spaces, on the homotopy type of the space of symplectomorphisms and on von Neumann topology. More precisely, one investigates: (1) the analogy between algebraic topology of the free loop spaces and algebraic geometry; (2) applications of string cohomology; (3) new methods to detect nontrivial homotopy groups of the space of symplectomorphisms; and (4) numerical invariants such as the Novikov-Shubin invariant, L2-torsion, and the L2-eta invariant for compact manifolds with infinite fundamental group. This research will provide new developments in areas like algebraic geometry, differential geometry, elliptic operators, and von Neumann algebras. It will broaden the present knowledge in, and (hopefully) will discover new relationships between, such different areas as algebraic geometry and free loop spaces, or algebraic geometry and symplectotopology. It will also answer specific questions (solve ppoblems) of immediate concern in topology, geometry and mathematical physics. ***