Analysis and Topology in Interaction

Abstract

The MFO workshop “Analysis and Topology in Interaction”, organised by Ulrich Bunke (Göttingen), Sebastian Goette (Regensburg), Kyoshi Igusa (Brandeis) and Thomas Schick (Göttingen), was held from March 12th through March 18th, 2006.

The aim of this meeting was to reflect on the current state of the interaction between mathematical fields of analysis and topology. It brought together representatives of various mathematical communities ranging from homotopy theory, index theory, global analysis up to mathematical physics. The program of the conference has been a mixture of research and overview talks, sometimes with introductory elements. Topics of the latter kind were the connection of Quantum field theory constructions with twisted K-theory, and motivic groups. There were two more informal introductory lectures on stacks in topological and smooth categories, and the insights given by the study of the Brownian motions into Hodge theory.

The research talks reflected recent developments in the corresponding fields and covered a broad area between topology, geometry, and analysis. The intention of the program was to communicate these developments across the borders of mathematical communities. They stimulated discussions and hopefully future collaborations.

More specifically, the subjects covered by research talks included in particular

• twisted K-theory (explicit constructions and applications, and homotopy theoretic approaches to its calculation);
• index theory on manifolds with boundary or with singularities, and study of corresponding index theorems and eta-invariants; spectral flow;
• algebraic geometry methods in algebraic topology and vice versa (stacks, motivic geometry);
• isomorphism conjectures in K- and L-theory;
• signatures of singular spaces;
• (refined) torsion invariants, in particular for families; signatures for families;
• spectral theory and geometric implications;
• symplectic geometry via topology of the loop space;
• topology of the mapping class group.

Altogether, there were 23 “official” talks and a number of informal presentations (partly extended discussions), and ample free interaction between the participants. One of the evenings was devoted to a problem session; the open problems presented are included in this report.

The conference was attended by 52 participants coming mainly from all over Europe, Northern America and East Asia. Among them, a couple of very young pre-doctoral mathematicians, supported by the EU, had the unique opportunity to participate in such a stimulating event at a very early stage of their career. It is a pleasure to thank the institute for providing a pleasant and stimulating atmosphere.