Organizing Committee: P. M. Hajac (Warszawa), N. Higson (State College), R. Meyer (Göttingen)
Suppose that G is a connected Lie group and that K is a maximal compact subgroup of G. There is a smooth family of Lie groups Gt, t a real number, such that Gt = G when t is not 0, and such that G0 is the semidirect product group associated to the adjoint action of K on the quotient of the Lie algebra of G by the Lie algebra of K. The group G0 is called a contraction of G, and in a 1975 paper Mackey proposed that, when G is semisimple, the unitary representation theories of G and G0 ought to be analogous to one another. Mackey's proposed analogy is very closely related to the Connes-Kasparov conjecture in C*-algebra K-theory. I shall briefly review this fact, and then examine the analogy from the related, but different, point of view of Harish-Chandra modules and Hecke algebras.
I will survey properties of bivariant K-theory and contrast them with the stable homotopy category. After that, I explain a general framework for doing homological algebra in triangulated categories, which is general enough to apply in the context of bivariant K-theory.
I shall try to explain the connections between linear elliptic partial differential equations and topology that led to the formulation and proof of the famous Atiyah-Singer index theorem. I shall include all the necessary definitions during the talk (or at least informal versions of them) and as a result the lecture will be accessible to all graduate students.
The construction of C*-algebras for badly behaved quotient spaces proceeds via groupoids. Since this groupoid is, in general, only unique up to Morita equivalence, the resulting C*-algebras are also only unique up to Morita equivalence or, equivalently, stable isomorphism. I will illustrate this situation by considering rotation algebras. These describe the quotient space of the real numbers by a dense subgroup with two generators. Various equivalent descriptions of this non-commutative space use rotations of the circle with irrational angle and the Kronecker foliation of the 2-torus.