Metric Differential Geometry and Mathematical Gravity

University Of Miami, Coral Gables FL

Investigators

Abstract

DMS-0104042 Gregory G. Galloway Research under this award will be conducted in the areas of Lorentzian and Riemannian geometry, with applications to General Relativity and String Theory. One of the specific aims of this project is to investigate the global structure of asymptotically locally anti-de Sitter spacetimes, especially in connection with the AdS/CFT correspondence. We continue investigations into the relationship of the topology of an asymptotically locally anti-de Sitter spacetime and that of its conformal boundary-at-infinity, and undertake a study of certain topological and geometrical properties of higher dimensional black holes, and related objects arising in string theory, including certain solitions conjectured to be least energy configurations. Another aim of this project is to continue investigations into spacetime rigidity phenomena associated with the occurence of timelike lines and null lines. Applications of the null splitting theorem will be considered, and a new approach to well-known conjectures concerning the rigidity of the singularity theorems is proposed for study. Modern theories of gravity are geometrical in nature. The gravitational field and other fields, black holes and related objects, may be described and analyzed using geometric methods. In more general terms, this project is concerned with the study of certain features of gravity of current scientific interest from this geometric point of view, utilizing the tools of Riemannian geometry, a mathematical theory of space, and Lorentzian geometry, a mathematical theory of spacetime. These theories provide a method for studying the relationship among three fundamental aspects of the spacetime universe: curvature (i.e., the bending of space or spacetime), topology (i.e., the global shape and complexity of space or spacetime) and causal structure (i.e., the large scale behavior of light rays and light cones).