We describe a powerful new technique for computing various physical observables in supergravity, without solving any supergravity equations. Applications include gravitational free energies, black hole entropies, and central charges and other CFT quantities of interest in holography. In the talk I will aim to describe the general idea, and give a flavour of some applications.

TBA

I will introduce a family of supersymmetric Chern-Simons-matter theories in d=2+1 dimensions, labelled by a positive integer n, and argue using a number of different dual descriptions that these describe the worldvolume dynamics of M2-branes at a corresponding family of four-fold hypersurface singularities (the n=1 theory being the ABJM theory). The IR properties of these theories and their deformations are also studied.

Sasakian geometry is the odd-dimensional cousin of Kahler geometry. Although the two are very closely related, there are some interesting differences. I will give an introduction to Sasakian, and in particular Sasaki-Einstein, geometry, and review some of the developments in this subject over the last few years. In particular I'll cover explicit constructions, toric Sasakian geometry, and obstructions. Some of these results also provide new insights into Kahler geometry - for example, I'll describe very simple new obstructions to the existence of Kahler-Einstein metrics on Fano orbifolds.