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KL2MG-interactions

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K theory L 2 invariants manifolds groups and their interactions

Many milestone results in mathematics emerge from interactions and transfer of techniques and methods between different areas. I want to attack outstanding problems concerning K-theory, L^2-invariants, manifolds and group theory.... ver más

31/10/2020

RHEINISCHE FRIEDRI...

2M€

Presupuesto del proyecto: 2M€

Líder del proyecto

RHEINISCHE FRIEDRICHWILHELMSUNIVERSITAT BONN No se ha especificado una descripción o un objeto social para esta compañía.

TRL 4-5

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Financiación concedida El organismo H2020 notifico la concesión del proyecto el día 2020-10-31

ERC-ADG-2014: ERC Advanced Grant

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RHEINISCHE FRIEDRICHWILHELMS...

1.72M€ | Lider

SIDENOR ACEROS ESPECIALES

185.30K€ | Participante

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Duración del proyecto: 63 meses Fecha Inicio: 2015-07-24
Fecha Fin: 2020-10-31

Líder del proyecto

RHEINISCHE FRIEDRICHWILHELMSUNIVERSITAT BONN No se ha especificado una descripción o un objeto social para esta compañía.

TRL 4-5

Presupuesto del proyecto 2M€

Fecha límite de participación Sin fecha límite de participación.

Descripción del proyecto Many milestone results in mathematics emerge from interactions and transfer of techniques and methods between different areas. I want to attack outstanding problems concerning K-theory, L^2-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 L^-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 Fr'echet group algebras, and Waldhausen'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 flows 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 L^2-torsion for towers of finite coverings, and the relation of the first L^2-Betti number, the cost and the rank gradient of a finitely generated group. I see a high potential for new striking applications of the Farrell-Jones Conjecture and L^2-techniques to manifolds and groups.

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