Homotopy harnessing higher structures

Organisers: Paul Goerss (Northwestern University), John Greenlees (University of Warwick), Stefan Schwede (Rheinische Friedrich-Wilhelms-Universität Bonn), Ulrike Tillmann (University of Oxford)

The last fifteen years have seen an upheaval in algebraic topology. Old problems have been solved using new methods, new methods led to new ideas, new ideas to new problems, and new problems to new  theorems. The result has been a renaissance. 

In retrospect, we can see that this fundamental change occurred as the mathematicians in the field confronted and solved an array of related problems. These included the task of making sense of Witten's insights on the elliptic genus, of finding a provable formulation of Mumford's conjecture, the related development of topological field theories, attempts to make rigorous Morava's program using the  geometry of formal algebraic groups to organize stable homotopy theory, and the solution of the Kervaire invariant problem and subsequent reintegration of equivariant homotopy theory. In each case, there was enormous progress after the introduction and the study of higher homotopical  structure.

This programme will highlight four related themes: the new algebraic topology of  differentiable manifolds, derived representation theory and equivariant homotopy theory, the interplay between arithmetic geometry and stable homotopy theory, and the analysis of  foundations in these new contexts "the homotopy  theory of homotopy theory". The four areas are bound together by the use of higher structures, but they do have slightly different characters: some study objects, others are an analysis of methods. This interplay is intentional: objects inspire methods and methods make objects accessible. The intent is to develop the community of scholars from these diverse, but overlapping, areas of algebraic topology, both through a programme of long-term visitors and a sequence of workshops, one each for the various themes.