Hyperbolic and pseudo-hyperbolic geometric structures

Hyperbolic geometry is the geometry of space of constant negative curvature. Its role was highlighted by a discovery of William Thurston in the 1970s that most of the 3-dimensional shapes can be endowed with a canonical hyperbolic structure, which greatly simplifies their study. It has a cousin, anti-de Sitter geometry, which is the geometry of spacetime of constant negative curvature. In the 1990s Geoffrey Mess brilliantly used Thurston`s ideas to classify 3-dimensional anti-de Sitter spacetimes, which have their applications to quantum gravity theories. Recently, mathematicians started to study general pseudo- hyperbolic geometries, which are geometries of spacetimes with arbitrary dimension of space and of time, and of constant negative curvature. This was motivated by applications to the theory of Lie groups, which deals with the notion of geometry in a broad sense and has wide applications to mathematics and physics. This project focuses on a variety of open questions about pseudo-hyperbolic spacetimes and aims to improve our understanding of their properties. For instance, we will study, how the geometry of anti-de Sitter spacetimes is determined by the intrinsic geometry of its spacelike slices, and will try to understand, when the geometry of pseudo-hyperbolic spacetimes can be non-trivially deformed.