The classification of anti de Sitter black holes is an open problem of central importance in holography. In this talk, I will present new advances in classification of supersymmetric solutions to five-dimensional minimal gauged supergravity. In particular, we prove a black hole uniqueness theorem within a ‘Calabi-type’ subclass of solutions with biaxial symmetry. This subclass includes all currently known black hole solutions within this theory.

In this talk we explore structural results about sets with small doubling in k dimensions. We start in the continuous world with a sharp stability result for the Brunn-Minkowski inequality conjectured by Figalli and Jerison and work our way to the discrete world, where we discuss the natural extension: we show that non-degenerate sets in Z^k with doubling close to 2^k are close to convex progressions i.e. convex sets intersected with a sub-lattice. This talk is based on joint work with Peter van Hintum and Hunter Spink.

We will construct several algebras of functions definable in R_{an,\exp} which are stable under parametric integration. 

Given one such algebra A, we will study the parametric Mellin and Fourier transforms of the functions in A. These are complex-valued oscillatory functions, thus we leave the realm of o-minimality. However, we will show that some of the geometric tameness of the functions in A passes on to their integral transforms.