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KL2MG-interactions

K-theory, L2-invariants, manifolds, groups, and their interactions

 

Principal Investigator

Prof. Dr. Wolfgang Lück
HIM (Hausdorff Research Institute for Mathematics)
Poppelsdorfer Allee 45
53115 Bonn

 

Abstract

Many milestone results in mathematics emerge from interactions and transfer of techniques and methods between dierent areas. I want to attack outstanding problems concerning K-theory, L2-invariants, manifolds and group theory. The time is ripe to use the exciting and profound progress that has been made during the last years in the individual areas to build new bridges, gain new insights, open the door to new applications, and to trigger new innovative activities worldwide lasting beyond the proposed funding period. 

The starting point are the prominent conjectures of Farrell-Jones on the algebraic K- and L-theory of group rings, of Baum-Connes on the topological K-theory of reduced group C-algebras, and of Atiyah on the integrality of L2-Betti numbers.

I intend to analyze and establish the Farrell-Jones Conjecture in other settings such as topological cyclic homology of \group rings" over the sphere spectrum, algebraic K-theory of Hecke algebras of totally disconnected groups, the topological K-theory of Frechet group algebras, andWaldhausen's A-theory of classifying spaces of groups. This has new and far-reaching consequences for automorphism groups of closed aspherical manifolds, the structure of group rings, and representation theory. Recent proofs by the PI of the Farrell-Jones Conjecture for certain classes of groups require input from homotopy theory, geometric group theory, controlled topology and ows on metric spaces, and will be transferred to the new situations. There is also a program towards a proof of the Atiyah Conjecture based on the Farrell-Jones Conjecture and ring theory. Furthermore, I want to attack open problems such as the approximation of L2-torsion for towers of nite coverings, and the relation of the rst L2-Betti number, the cost and the rank gradient of a nitely generated group. I see a high potential for new striking applications of the Farrell-Jones Conjecture and L2-techniques to manifolds and groups.

 

Laufzeit

01.11.2015 - 31.10.2020

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