# New Contexts for Stable Homotopy Theory

Participation in INI programmes is by invitation only. Anyone wishing to apply to participate in the associated workshop(s) should use the relevant workshop application form.

2nd September 2002 to 20th December 2002

Organisers: Professor JPC Greenlees (Sheffield), Professor HR Miller (MIT), Professor F Morel (Jussieu), Professor VP Snaith (Southampton)

## Programme Theme

The versatility of stable homotopy theory and its associated range of cohomological techniques has made it an important branch of mathematics. Recently there have been several fundamental developments which have been used to solve a number of longstanding questions. This programme aims to consolidate the advances within homotopy theory itself which have led to these results, to open the way to substantial further developments within the subject, to expose a diversity of new applications and to bring practitioners of these subjects into contact with each other and with practising homotopy theorists. The programme will be designed around specialist workshops, but the aim will be to maximize contact between all parties by means of a number of introductory short courses.

Recent developments have culminated in several stable homotopy constructions which are homotopy theoretic enrichments of the category of abelian groups (spectral algebra''). These new stable homotopy categories are useful for studying a wide range of phenomena, from algebraic K-theory and arithmetic to the elliptic cohomology phenomena introduced by Witten. In algebraic geometry the motivic homotopy category of Voevodsky and Morel has already been used in Voevodsky's proof of the Milnor conjecture in algebraic K-theory. In algebraic topology, the work of Hopkins et al on elliptic cohomology has led to geometric applications and topological refinements of modular forms.

y means of a series of workshops, seminars and short courses on the themes of

1. axiomatic and enriched homotopy theory
2. homotopy theory of geometric categories
3. K-theory and arithmetic
4. Elliptic cohomology and chromatic phenomena

it is hoped to promote further applications of the latest stable homotopy theoretic techniques.

Final Scientific Report:
Attachments: