RECENT PROGRESS IN ALGEBRAIC K–THEORY AND ITS RELATIONSHIP WITH TOPOLOGY AND ANALYSIS Mini-Course Notes by Jonathan Rosenberg Brief Outline I. “Assembly, Novikov Conjectures, and Control”. A. Classical applications of the K-theory of group rings B. Assembly and standard examples

k — a regular commutative ring. (Main case of interest, as we shall see, is k = Z.) kG — the group ring over k of a group G. BG — the classifying space of G, a space with contractible universal cover and fundamental group G. Unique up to homotopy equivalence. Has the property that H∗(BG) is the Eilenberg-MacLane homology of G. K(R) — the (non-connective) K-theory spectrum of a ring R. If one ignores negative K-theory, basically K0(R) × BGL(R). Ki(R) — equal by definition to πi(K(R)).