Mathematical Sciences: Free Loop Space, Automorphisms of Manifolds and Cyclic Homology

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Professor Burghelea will study the topological character of natural transformations of simple geometric objects known as compact smooth manifolds. These arise in modern physical theories as well as in many purely mathematical contexts. Other related matters will also play a lesser role in his project. Specifically, Burghelea will study (1) the homotopy type of diffeomorphisms (resp. symplectomorphisms) of a compact smooth (resp. symplectic) manifold M and the relation to the homology of some associated spaces defined in terms of principal bundles over M. (2) the differential geometry and topology of the free loop space of compact Riemannian manifolds. (3) the determinants of elliptic operators. (4) the bounded concordances of complete metric manifolds. (5) geometric and algebraic problems in cyclic homology.