Aspects of the AdS/CFT correspondence

The AdS/CFT correspondence is an exact realization of the holographic principle in string theory where all the gravitational physics is conjectured to be encoded at the boundary of the space where the closed strings propagate.
In this thesis we begin by reviewing this duality from first principles: we discuss the large N limit of gauge theories and their relation with string theories and discuss in detail the derivation of the correspondence from D-brane physics and supergravity. We then start with the main research topics of this thesis. Within the framework of AdS/CFT, different sectors of string theory have the potential to model holographically strongly coupled systems in condensed matter physics that enjoy some degree of conformal symmetry. In this thesis we focus on specific systems with non-relativistic scale invariance and construct their geometric duals. The latter are typically AdS plane wave solutions of massive vector theories coupled to Einstein gravity and include the Schrodinger geometries. We discuss how the symmetries of such systems are realised in the gravity models and show how observables such as correlation functions are computed holographically.
We then focus on a different aspect of the AdS/CFT correspondence. Currently, a central problem in holography is to understand how to formulate string theory in asymptotically flat spacetimes in terms of field theories in less dimensions. In this thesis we develop two possible approaches to flat space holography, one based on the flat space limit of AdS/CFT and the other on hyperboloidal foliations of asymptotically Minkowski spaces.