New Directions in Algebraic $K$-Theory

Abstract

This meeting brought together algebraic geometers, algebraic topologists and geometric topologists, all of whom use algebraic K-theory. The talks and discussions involved all the participants. Mathematics Subject Classification (2000): 19xx. Introduction by the Organisers There have been dramatic advances in algebraic K-theory recently, especially in the computation and understanding of negative K-groups and of nilpotent phenomena in algebraic K-theory. Parallel advances have used remarkably different methods. Quite complete computations for the algebraic K-theory of commutative algebras over fields have been obtained using algebraic geometric techniques. On the other hand, the Farrell-Jones conjecture implies results on the K-theory for arbitrary rings. Proofs here use controlled topology and differential geometry. Given the diversity of interests and backgrounds of the 28 participants in our mini-workshop, we encouraged everyone to make their talks accessible to a wide audience and scheduled five expository talks. The opening talk of the conference was an inspiring talk by Charles Weibel, on the work of Daniel Quillen, the creator of higher algebraic K-theory, who died at the end of April. Wolfgang Luck spoke on the Farrell-Jones conjecture. Jim Davis applied the Farrell-Jones conjecture to give a foundational result on algebraic K-theory, showing that geometric techniques have algebraic consequences for the iterated NK-groups. Bjorn Dundas gave a survey of trace methods on algebraic K-theory, focusing on topological 1470 Oberwolfach Report 27/2011 cyclic homology and his new integral homotopy cartesian square. Christian Haesemeyer gave a survey of algebraic K-theory of singularities and new techniques for computing negative K-theory and NK-theory for commutative Q-algebras. The idea of the expository talks worked quite well; it was remarkable how many of the speakers relied on them. The mini-workshop had a full schedule; in addition to the five expository talks there were seventeen research talks. There were computational talks (Teena Gerhardt, Charles Weibel, Daniel-Juan Pineda), foundational talks (Bruce Williams, Lars Hesselholt, Max Karoubi, Guilermo Cortinas, Jens Hornbostel, Andrew Blumberg, Thomas Geisser), applications of ideas from K-theory to geometric topology (Ib Madsen, Frank Connolly, Qayum Khan, Ian Hambleton, Michael Weiss, Wolfgang Steimle), as well as the proof of the Farrell-Jones Conjecture for the group SLn(Z) (Holger Reich). The talk of Charles Weibel was notable since the topic was research done at the workshop. Weibel’s talk connected and compared two different computations of the NKqR groups, one done by algebraic geometry and one done by geometric topology. This was emblematic of a successful implementation of the original goal of the workshop to compare and contrast two powerful but quite distinct approaches to algebraic K-theory.