Speaker: Doman Takata (Univ. Tokyo)
Title: An infinite-dimensional index theory and the Higson-Kasparov-Trout algebra
Time/Date: 4:45-6:15pm, December 4 (Wed.), 2019
Abstract: According to the Atiyah-Singer index theorem, the analytic index of a Dirac operator on a closed manifold, is determined by the homotopy class of the symbol of the operator. The K-theoretical Poincaré duality generalizes this result in the following sense: For a complete Riemannian manifold M equipped with a proper isometric group action of G, the G-equivariant K-homology of C0(M) (the group consisting of homotopy classes of Dirac operators) is isomorphic to the G-equivariant representable K-theory with a local coefficient of M (the group consisting of homotopy classes of symbols).
In this talk, I will explain my paper arXiv:1811.06811 in which I formulated and proved "(the corollary of) the infinite-dimensional version" of the K-theoretical Poincaré duality using the Higson-Kasparov-Trout algebra. I will also try to explain several technical aspects, because my constructions have a strong infinite-dimensional flavor.