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

Room: 126

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 C_{0}(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.