In these lectures, I will introduce the concept of integrability and study its realisation in the context of gauge theory and string theory. In particular, I plan to review recent progress in computing the spectrum of operator dimensions in N=4 supersymmetric Yang-Mills theory and the dual problem of determining the spectrum of string theory on AdS5 x S5.

We will discuss the one-loop anomalous dimensions of Super Yang-Mills operators dual to open strings ending on AdS giant gravitons (AGG) and describe the spin chain description of gauge theory operators. A semi-classical analysis of the Hamiltonian describing the anomalous dimensions of AGG operators will allow us to give a geometrical (stringy) interpretation for the gauge theory parameters. This same analysis will also show evidence for the existence of continuous bands in the Hamiltonian spectrum.

In these lectures, I will introduce the concept of integrability and study its realisation in the context of gauge theory and string theory. In particular, I plan to review recent progress in computing the spectrum of operator dimensions in N=4 supersymmetric Yang-Mills theory and the dual problem of determining the spectrum of string theory on AdS5 x S5.

In these lectures, I will introduce the concept of integrability and study its realisation in the context of gauge theory and string theory. In particular, I plan to review recent progress in computing the spectrum of operator dimensions in N=4 supersymmetric Yang-Mills theory and the dual problem of determining the spectrum of string theory on AdS5 x S5.

This the first of a series of lectures in which I will introduce the concept of integrability and study its realisation in the context of gauge theory and string theory. In particular, I plan to review recent progress in computing the spectrum of operator dimensions in N=4 supersymmetric Yang-Mills theory and the dual problem of determining the spectrum of string theory on AdS5 x S5.

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.

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.