We revisit the general topic of thermodynamical stability and critical phenomena in black hole physics, analyzing in detail the phase diagram of the five dimensional rotating black hole and the black rings discovered by Emparan and Reall. First we address the issue of microcanonical stability of these spacetimes and its relation to thermodynamics by using the so-called Poincaré (or "turning point") method, which we review in detail. We are able to prove that one of the black ring branches is always locally unstable, showing that there is a change of stability at the point where the two black ring branches meet. Next we study divergence of fluctuations, the geometry of the thermodynamic state space (Ruppeiner geometry) and compute the appropriate critical exponents and verify the scaling laws familiar from RG theory in statistical mechanics. We find that, at extremality, the behaviour of the system is formally very similar to a second order phase transition.

We present and study a new chain of 10-dimensional T duality related solutions and their 11-dimensional parents whose existence had been predicted in the literature based in U duality requirements in 4 dimensions. The first link in this chain is the S dual of the D7-brane. The next link has 6 spatial worldvolume dimensions, it is charged w.r.t. the RR 7-form but depends only on 2 transverse dimensions since the third has to be compactified in a circle and is isometric and hence is similar in this respect to the KK monopole. The next link has 5 spatial worldvolume dimensions, it is charged w.r.t. the RR 6-form but, again, depends only on 2 transverse dimensions since the third and fourth have to be compactified in circles and are isometric and so on for the following links.All these solutions are identical when reduced over the p spatial worldvolume dimensions and preserve a half on the available supersymmetries. Their masses depend on the square of the radii of the isometric directions, just as it happens for the KK monopole. We give a general map of these branes and their duality relations and show how they must appear in the supersymmetry algebra.

We present the most general family of stationary point-like solutions of pure N = 4, d = 4 Supergravity characterized by completely independent electric and magnetic charges, mass, angular momentum and NUT charge plus the asymptotic values of the scalars. It includes, for specific values of the charges all previously known BPS and non-BPS, extreme and non-extreme black holes and Taub-NUT solutions.As a family of solutions, it is manifestly invariant under T and S duality transformations and exhibits a structure related to the underlying special geometry structure of the theory.Finally, we study briefly the black-hole-type subfamily of metrics and give explicit expressions for their entropy and temperature.

We show how the Killing spinors of some maximally supersymmetric supergravity solutions whose metrics describe symmetric spacetimes (including AdS, AdS × S and Hpp-waves) can be easily constructed using purely geometrical and group-theoretical methods. The calculation of the supersymmetry algebras is extremely simple in this formalism.

We show how all known N = 2, d = 4, 5, 6 maximally supersymmetric vacua (Hppwaves and aDS × S solutions) are related through dimensional reduction/oxidation preserving all the unbroken supersymmetries. In particular we show how the N = 2, d = 5 family of vacua (which are the near-horizon geometry of supersymmetric rotating black holes) interpolates between aDS 2 × S 3 and aDS 3 × S 2 in parameter space and how it can be dimensionally reduced to an N = 2, d = 4 dyonic RobinsonBertotti solution with geometry aDS 2 × S 2 and oxidized to an N = 2, d = 6 solution with aDS 3 × S 3 geometry (which is the near-horizon limit of the self-dual string).