Abstract: Suppose M is a compact complex manifold. Model theory (a branch of mathematical logic) provides at least two approaches to the study of the complex-analytic subsets of Cartesian powers of M, roughly corresponding to whether one focuses on the real or complex structure on M. We can view M as definable in the structure R_an; that is, as a real globally subanalytic manifold. On the other hand, we can work in the Zariski-type structure CCM where M is the universe and there are predicates for all complex-analytic subvarieties of Cartesian powers of M. The two approaches lead to different notions of a "definable family" of complex-analytic subsets. I will give a geometric characterization, obtained in joint work with Sergei Starchenko in 2008, of when these two notions coincide, in terms of the Barlet or Douady spaces. As a consequence one has that for M Kaehler the two notions coincide.

Abstract: A classical result of Atiyah and Hirzebruch establishes a deep connection between K-theory and cohomology, via the Chern character. The purpose of my talk is to describe this relation in precise terms, and give an overview of its generalizations to the equivariant setting. Along the way we introduce a new class of objects, coming from algebraic geometry, on which many of these classical topological techniques could be successfully applied.