K-THEORY OLD AND NEW Research group, 21 - 30 April, Warszawa P. F. Baum (State College), P. M. Hajac (Warszawa), M. Karoubi (Paris) ------------------------------------------------------------------------- 21 April 2008 (10:15 - 11:45) Banach Centre, Sniadeckich 8 and 23 April 2008 (10:15 - 14:00) Institute of Mathematics, room 5840, Banacha 2 K-THEORY OF A REAL BANACH ALGEBRA AND ITS COMPLEXIFICATION Real K-theory (periodicity 8) seems more complicated than complex K-theory (periodicity 2). In this talk, we show how to relate these two theories using three ingredients: the real K-theory of Atiyah, Clifford algebras and the theory of homotopy fixed point sets. One philosophy which emerges from these considerations is the following: any "general" theorem on complex Banach algebras extends to real ones. During the lecture, we make this statement more precise and consider its applications to the Lichtenbaum-Quillen and real Baum-Connes conjectures. MAX KAROUBI -------------------------------------------------------------------------- 28 April 2008 (10:15 - 11:45) Banach Centre, Sniadeckich 8 MORITA EQUIVALENCE REVISITED Let k be the co-ordinate algebra of a complex affine variety. A k-algebra is an algebra A over the complex numbers with a given k-module structure. The k-module structure is required to be compatible (in an evident way) with the algebraic operations of A. This talk introduces an equivalence relation on k-algebras called "geometric equivalence". This equivalence allows a tearing apart of strata in the primitive ideal space, while Morita equivalence gives a homeomorphism of primitive ideal spaces. The talk is intended for non-specialists. All the basic definitions will be precisely stated. Examples will be given to show how the new equivalence relation works. An application to the representation theory of reductive p-adic groups will be briefly indicated. There should be an analogous equivalence relation (with analogous properties) for an appropriate class of C* algebras. PAUL F. BAUM ------------------------------------------------------------------------- 29 April 2008 (12:00 - 13:30) Institute of Mathematics, room 5870, Banacha 2 TWISTED K-THEORY OLD AND NEW The purpose of this lecture is to give a historical view of the subject, starting from the work of Atiyah, Bott and Shapiro on Clifford modules. There will be also an exposition of some recent results, essentially in the equivariant case, related to algebraic K-theory and operator algebras. Finally, some applications in various mathematical subjects will be given. MAX KAROUBI -------------------------------------------------------------------------- 30 April 2008 (10:15 - 14:00) Institute of Mathematics, room 5840, Banacha 2 THE BAUM-CONNES CONJECTURE The Baum-Connes conjecture has been an outstanding focal point of noncommutative geometry for over twenty years. This lecture will be devoted to the applications of the Baum-Connes conjecture and its relations to classical topology. PAUL F. BAUM -------------------------------------------------------------------------