I will present a modification of the Berkovits twistor string model which gives Einstein supergravity coupled to Yang-Mills, and has a limit in which the gravity modes can be decoupled to give pure gauge theory amplitudes. I will start by reviewing a number of relevant aspects of twistor theory, including special features associated with different space-time signatures, supertwistor space, the Penrose transform, the infinity twistor and Penrose's non-linear graviton construction. I will then review the Witten and Berkovits twistor strings, with emphasis on the latter. The world-sheet formulation of the Berkovits model involves so-called beta-gamma systems, I will describe the symmetries of such systems and their gauging, and explain how the analysis can be applied to the construction of a family of new gauged Berkovits twistor strings which are free from world-sheet anomalies. I will also describe the corresponding spectra in space-time, and show that they give Einstein supergravities instead of the higher derivative conformal supergravities arising in the original twistor string models. The new theories include one with the spectrum of N = 8 supergravity, two theories with the spectrum of N = 4 supergravity coupled to N = 4 Yang-Mills, a family of N greater than 0 models with the spectra of self-dual supergravity coupled to self-dual super-Yang-Mills, and a non-supersymmetric string with the spectrum of self-dual gravity coupled to self-dual Yang-Mills and a scalar. Time permitting, I will discuss what is known about the interactions.

The exact string coupling dependence of higher derivative terms in the type IIA and type IIB effective action is highly constrained by a combination of duality symmetries and by results from perturbative string theory. For example, we show that terms of the form D to the 2k times R to the 4th in type IIA theory should receive no perturbative contributions beyond genus k (k greater than 0). We also propose that the exact modular functions of general type IIB higher derivative terms are determined by a Poisson equation on the fundamental domain of the moduli space.

One of the few known obstructions for a 1-connected 4- manifold to admit a symplectic structure is given by Taubes theorem: if b_+ is greater than two, then such a manifold has a nonvanishing Seiberg--Witten invariant. It is only natural to ask whether the same holds for generalized complex manifolds, as introduced by Hitchin, which are a simultaneous generalization of symplectic and complex manifolds. I will introduce a surgery for generalized complex manifolds whose input is a symplectic 4-manifold containing a symplectic 2-torus with trivial normal bundle and whose output is a 4-manifold endowed with a generalized complex structure exhibiting type change along a 2-torus. I will use this surgery to produce an example of a generalized complex manifold with vanishing Seiberg--Witten invariants and hence which does not admit complex or symplectic structures.

I will talk about the possibilities of using mean curvature flow to deform a Lagrangian submanifold into an area-minimizing Lagrangian (known also as SLag). More precisely, I will show that finite time singularities are unavoidable, i.e., they appear for a large class of stable initial conditions. Then, I will give the optimal theorem regarding the structure of the tangent flow at a singularity.

Among the very first examples of hyper-Kahler manifolds given by Calabi in 1979, there were the cotangent bundles of complex projective spaces, T-star-CPn. Later on, many more examples of hyper-Kahler metrics on cotangent bundles of Kahler manifolds were shown to exist. Finally, Kaledin (1997) and Feix (1999) proved that a real-analytic Kahler metric on a complex manifold M can always be extended to a hyper-Kahler metric in a neighborhood of M in T-star-M. Although these mathematical proofs are rather technical and involved, there exists a streamlined physical construction which leads to the same results and is based on the concept of supersymmetry. As is well-known, four- and five-dimensional N = 2 supersymmetric nonlinear sigma-models possess the property that their target spaces are hyper-Kahler manifolds. The physical construction consists of providing a manifestly N = 2 supersymmetric nonlinear sigma-model whose target space can be shown to be (a neighborhood of the zero section in) the cotangent bundle T-star-M of a Kahler manifold M. This talk will review the salient properties of such supersymmetric nonlinear sigma-models with eight supercharges.

The application of the AdS/CFT correspondence in the calculation of four point functions for chiral primary operators in the large N limit for maximal superconformal gauge theories is reviewed. It is shown how the results satisfy certain constraints, motivated by our understanding from string theory of the spectrum of operators in the large N limit, which allow a construction of the four point function for any half BPS chiral primary operator at large N.

D-branes are considered by the general particle physics community as the arcane inventions of the initiated having no relevance to even semi-realistic QFTs. In this seminar I want to provide compelling evidence that the domain walls of the simplest supersymmetric extension of QCD are D-branes for the confining string of the theory. Conventional D-brane technology then leads to a very intriguing result for the forces between the domain walls which suggests some underlying but unknown magic is at work. (based on hep-th/0505213,0601150)