1.1. Simplicial sets 5 1.2. Symmetric spectra 6

2. More detailed results 7 2.1. The algebraic geometry of even periodic ring spectra 7

Abstract. We study the structure of the categories of K(n)-local and E(n)local spectra, using the axiomatic framework developed in earlier work of the authors with John Palmieri. We classify localising and colocalising subcategories, and give characterisations of small, dualisable, and K(n)-nilpotent spectra. We give a number of useful extensions to the theory of vn self maps of finite spectra, and to the theory of Landweber exactness. We show that certain rings of cohomology operations are left Noetherian, and deduce some powerful finiteness results. We study the Picard group of invertible K(n)-local spectra, and the problem of grading homotopy groups over it. We prove (as announced by Hopkins and Gross) that the Brown-Comenetz dual of MnS lies in the Picard group. We give a detailed analysis of some examples when n =1 or 2, and a list of open problems.

ABSTRACT. We begin by showing that in a triangulated category, specifying a projective class is equivalent to specifying an ideal I of morphisms with certain properties, and that if I has these properties, then so does each of its powers. We show how a projective class leads to an Adams spectral sequence and give some results on the convergence and collapsing of this spectral sequence. We use this to study various ideals. In the stable homotopy category we examine phantom maps, skeletal phantom maps, superphantom maps, and ghosts. (A ghost is a map which induces the zero map of homotopy groups.) We show that ghosts lead to a stable analogue of the Lusternik–Schnirelmann category of a space, and we calculate this stable analogue for low-dimensional real projective spaces. We also give a relation between ghosts and the Hopf and Kervaire invariant problems. In the case of A ∞ modules over an A ∞ ring spectrum, the ghost spectral sequence is a universal coefficient spectral sequence. From the phantom projective class we derive a generalized Milnor sequence for filtered diagrams of finite spectra, and from this it follows that the group of phantom maps from X to Y can always be described as a lim1 ←− group. The last two sections focus

2. Determinants, differential cocycles and statement of results 5

Suppose C is a category with a symmetric monoidal structure, which we will refer to as the smash product. Then the Picard category is the full subcategory of objects which have an inverse under the smash product in C, and the Picard group Pic(C) is the collection of isomorphism classes of such invertible objects. The

modules over highly structured ring spectra to give new constructions of MUmodules such as BP, K(n) and so on, which makes it much easier to analyse product structures on these spectra. Unfortunately, their construction only works in its simplest form for modules over MU [ 1] ∗ that are concentrated in 2 degrees divisible by 4; this guarantees that various obstruction groups are trivial. We extend these results to the cases where 2 = 0 or the homotopy groups are allowed to be nonzero in all even degrees; in this context the obstruction groups are nontrivial. We shall show that there are never any obstructions to associativity, and that the obstructions to commutativity are given by a certain power operation; this was inspired by parallel results of Mironov in Baas-Sullivan theory. We use formal group theory to derive various formulae for this power operation, and deduce a number of results about realising 2-local MU∗-modules as MU-modules. 1.

Summary. We give a survey of the meaning, status and applications of the Baum-Connes Conjecture about the topological K-theory of the reduced group C ∗-algebra and the Farrell-Jones Conjecture about the algebraic K- and L-theory of the group ring of a (discrete) group G. Key words: K- and L-groups of group rings and group C ∗-algebras, Baum-Connes

Suppose that F is a number field (i.e. a finite algebraic extension of the field Q of rational numbers) and that OF is the ring of algebraic integers in F. One of the most fascinating and apparently difficult problems in algebraic K-theory is to compute the groups KiOF. These groups were shown to be finitely generated by

We construct stable operations Tn : E" ( ) \Gamma! E"(1=n) ( ) for n ? 0 in the version of elliptic cohomology where the coefficient ring E" agrees with the ring of modular forms for SL 2 (Z) which are meromorphic at 1, and Tn restricts to the n th Hecke operator Tn on E" . In the past few years, the idea of elliptic cohomology has emerged from the combined efforts of a variety of mathematicians and physicists, and it is widely expected that it will play as important a role in global analysis and topology as K--theory and bordism have in the past. At present, there is no explicit geometric description of the cohomology theories that arise in this area, although there are several promising ideas which it is hoped will eventually lead to such a description. On the other hand, there are constructions of these theories based upon cobordism theories and for many purposes these seem to be adequate, at least for problems within the realm of stable homotopy theory. In particular, in ...