Mathematisches Forschungsinstitut Oberwolfach Report No. 27/2011 DOI: 10.4171/OWR/2011/27 New Directions in Algebraic K -Theory Organised by James F. Davis (Bloomington) Christian Haesemeyer (Los Angeles) Andrew Ranicki (Edinburgh) Marco Schlichting (Baton Rouge/Warwick) May 15th – May 21st, 2011 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 phe- nomena in algebraic K -theory. Parallel advances have used remarkably different methods. Quite complete computations for the algebraic K -theory of commuta- tive 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 cre- ator of higher algebraic K -theory, who died at the end of April. Wolfgang L¨ uck spoke on the Farrell-Jones conjecture. Jim Davis applied the Farrell-Jones conjec- ture to give a foundational result on algebraic K -theory, showing that geometric techniques have algebraic consequences for the iterated N p K -groups. Bjorn Dun- das gave a survey of trace methods on algebraic K -theory, focusing on topological