Global Topological and Geometric aspects of AdS/CFT backgrounds.

Aims/Objectives: The main aim of the project is to classify all solutions of the Einstein equations of gravity theories in 10- and 11-dimensions which include in a certain way a Anti-de-Sitter (AdS) subspace. AdS spaces are the Lorentzian analogue of Riemannian hyperbolic spaces. This question has arisen about 40 years ago and has been given a new impetuous because of the
AdS/CFT correspondence. This is a relationship between gravitational and conformal field theories (CFT). This correspondence
relates strong coupling effects in gauge theories with a certain behaviour of AdS backgrounds in gravitational theories. The objective of the PhD project is to classify the special case AdS5 backgrounds. As the dual gauge theories are 4-dimensional, this is the most important case to be considered because 4-dimensional theories are key in our understanding of nature.

Methodology: This is a problem in Mathematical Physics. Although the question is well established and remains open, a new approach has been developed to solve it. The methodology includes the use of Differential Geometry and in particular the theory of Differential Manifolds. Algebraic Topology and Analysis, like Cohomology theory and the Hopf maximum principle, and other techniques for solving differential equations globally on Manifolds will be deployed. Other aspects of manifold theory like Spin Structures and Generalized Geometry will also be employed to investigate the geometry of the solutions.