We will generalize the concept of partial and complete observables, intruduced by Rovelli, to constrained systems with an arbitrary number of (first class) constraints and to canonical field theories. Different methods to calculate complete observables (aka Dirac observables) will be presented and applied to examples, moreover their algebraic properties will be considered. Furthermore we will investigate whether it is possible to calculate complete observables in stages. In the case of gravity this will answer the question , whether it is possible to calculate the complete observable with respect to the Hamiltonian constraints, starting from 3-diffeomorphism invariant partial observables.

In the supergravity approximation of the AdS/CFT correspondence correlation fucntions of the strongly coupled gauge theory can be computed by evaluating the on-shell supergravity action with arbitrary Dirichlet boundary conditions for the various fields. It is well known, however, that the on-shell action on the conformal boundary of AdS diverges due to the infinite volume of the bulk. Holographic Renormalization is a method for systematically constructing covariant counter terms which remove the divergences of the on-shell supergravity action for any asymptotically AdS space and it provides an algorithm for calculating renormalized correlation functions of the strongly coupled gauge theory. I will review the main steps involved in this algorithm and present a recently developed Hamiltonian version of the method where the AdS radius plays the role of time in the Hamiltonian formalism. I will then apply this method to calculate correlation functions in the field theory duals of various Poincare and AdS-sliced domain wall solutions of five dimensional gauged supergravity, including the recently discovered Janus background.

Witten's twistor string theory description of N=4 super Yang-Mills has led to much progress in perturbative Yang-Mills theory this year, with accumulating evidence for the localisation of scattering amplitudes in twistor space in various Yang-Mills theories. This talk will focus on the MHV diagram approach, which builds in localisation, and will describe work done at Queen Mary which shows how to define MHV diagrams at loop level. This has been successfully applied to give the general one-loop MHV amplitudes in N=4 and N=1 super Yang-Mills.

Generalised black holes have a horizon given by an arbitrary Einstein manifold. I will describe a criterion for the classical stability of these black holes. Roughly, spherical horizons are stable but lemon-shaped horizons can be unstable. In Anti-de Sitter space, these black holes are dual to gauge theory on a curved background given by the same Einstein manifold. I will argue that the dual thermal field theory effect is a novel phase transition induced by inhomogeneous Casimir pressures and characterised by a condensation of pressure.

In the last year and a half, many new tests of the AdS/CFT correspondence have been achieved by studying a certain class of non-topological solitons corresponding to rotating strings which carry (at least) two large angular momenta. They are dual to CFT operators which are far away from the 1/2 BPS operators of Berenstein, Maldacena and Nastase. We will discuss to which point can these results test the AdS/CFT away from supersymmetry.

I will summarise recent progress in understanding the dynamics of vortices in gauge theories and describe how the vortices can be used as probes to extract information about the strong coupling quantum dynamics of the theory in which its embedded.

This is part of the London topology and geometry seminars

The talk will have two themes. The first is to give an account of recent work of Chen, LeBrun and Weber, proving the existence of a new Einstein metric on a certain 4-manifold. The second will describe a general class of numerical solutions to Calabi's extremal metric equation. The two themes are related by the fact that the metric of Chen, LeBrun and Weber is conformal to an extremal metric, so explicit geometric information can be obtained.