Objective The development of a mathematical framework for the unified description of all fundamental forces of nature is one of the biggest challenges in mathematical physics and is, therefore, an important objective of European fundamental research.In order to describe gravity together with the other forces, such a framework will have to unite and generalize the differential geometrical methods of general relativity and the operator algebraic methods of quantum theory. One of the promising candidates for this is non-commutative geometry.However, its relation to classical gravity and curvature is still an open problem. The project objective is to construct and study non-commutative geometry as induced by curvature on Riemann manifolds, in collaboration with A. Weinstein (Berkeley) and P. Schupp (IU Bremen). Due to the uncertainty principle, a quantum particle in curved spacetime is always subject to tidal forces even if it moves on a geodesic.Analogous to the motion in a magnetic field, the quantization of the motion in tidal forces yields, in the strong field limit, non-commutative coordinates. The main idea is that, locally, the motion in tidal forces can be reinterpreted as free motion on this non-commutative geometry.For a mathematically rigorous implementation of this idea, the Hamiltonian formulation of geodesic motion is to be linearized, the local symplectic structure to be quantized, leading to a Lie algebroid description of a bundle of local non-commutative geometries.The global non-commutative geometry is described in the integrable case by the convolution algebra of the corresponding Lie groupoid, in the non-integrable case by the star product realization of the corresponding Poisson algebra.These general constructions are to be computed explicitly for the examples of Schwarzschild and Robertson-Walker spacetimes. The representation theory of the resulting algebras will be studied and compared with black hole and cosmological phenomenology. Fields of science natural sciencesphysical sciencesrelativistic mechanicsnatural sciencesmathematicspure mathematicstopologysymplectic topologynatural sciencesphysical sciencesastronomyastrophysicsblack holesnatural sciencesmathematicspure mathematicsalgebranatural sciencesmathematicspure mathematicsgeometry Keywords Lie algebroids Riemann curvature deformation quantization geodesics noncommutative geometry quantum groups representation theory symplectic geometry Programme(s) FP6-MOBILITY - Human resources and Mobility in the specific programme for research, technological development and demonstration "Structuring the European Research Area" under the Sixth Framework Programme 2002-2006 Topic(s) MOBILITY-2.2 - Marie Curie Outgoing International Fellowships (OIF) Call for proposal FP6-2002-MOBILITY-6 See other projects for this call Funding Scheme OIF - Marie Curie actions-Outgoing International Fellowships Coordinator JACOBS UNIVERSITY BREMEN GGMBH EU contribution No data Address Campus ring 1 750 561 Bremen Germany See on map Links Website Opens in new window Total cost No data Participants (1) Sort alphabetically Sort by EU Contribution Expand all Collapse all UNIVERSITY OF CALIFORNIA, BERKELEY United States EU contribution No data Address 970 evans hall #3840 Berkeley See on map Links Website Opens in new window Total cost No data