Topological Rigidity and Dynamics (A12#)

• Subject Area: Mathematics
• Term: from 2011 to 2012
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 5486209

The notion of a manifold models the idea of 'space without singularities'. This project exemplifies that the study of the dynamics of a geometry is necessary even if one is only interested in the classification of 'space' as a static object. A fundamental problem is to classify manifolds within a given homotopy type and to determine their symmetries. The Borel Conjecture says that for aspherical manifolds the homotopy type determines the manifold up to homeomorphism, i.e., conjecturally aspherical manifolds are topologically rigid. Moreover the symmetries, i.e., the groups of homeomorphisms and diffeomorphisms of such manifolds are tractable. Topological rigidity and information about the symmetries is encoded in the so-called Farrell-Jones Conjecture, which plays the central role in this project. The Baum-Connes Conjecture is the analytic counterpart of the Farrell-Jones Conjecture in noncommutative geometry. Proofs of all these conjectures make use of geometries and their dynamics, like for example geodesic ows on Riemannian manifolds. The project intends to prove new cases of the Farrell-Jones Conjecture and hence topological rigidity results. Furthermore the project tries to derive new consequences, to study generalizations of the conjectures and to investigate the relation to the Baum-Connes Conjecture.

• DFG Programme: Collaborative Research Centres
• Subproject of: SFB 647: Space - Time - Matter: Analytic and Geometric Structures
• Applicant Institution: Humboldt-Universität zu Berlin
• Co-Applicant Institution: Freie Universität Berlin
• Project Head: Professor Dr. Holger Reich