I will discuss some work in progress with Peter Kronheimer on a variant of Floer's instanton homology dealing with connections that are singular along codimension two submanifolds. As observed by Kronheimer earlier there are some interesting relations with Khovanov homology.

In this lecture, I will give an overview of Ricci flow, focusing on Perelman's work, accessible to non-experts.

We discuss how to extract quantum field theory data from solutions that are asympotically (AdS_p x S_q). We apply this method to general 2-charge fuzzball solutions that is, horizon-free non-singular solutions of IIB supergravity characterized by a number of curves. We propose a precise map that relates any given curve to a specific superposition of R ground states of the D1-D5 system. To test this proposal we compute the holographic 1-point functions associated with these solutions, namely the conserved charges and the vacuum expectation values of chiral primary operators of the boundary theory, and find perfect agreement within the approximations used. All kinematical constraints are satisfied and the proposal is compatible with dynamical constraints.

We consider the hermitian matrix model corresponding to scalar field theory on the fuzzy sphere. Using an expansion of the model which is similar to a high-temperature expansion, we are able to reformulate the model in terms of its eigenvalues. Applying subsequently the saddle point method allows us to extract analytically information on the phase diagram of the theory. Eventually, we can also predict qualitatively the effect of proposed modifications of this theory which are necessary for the theory to be a regularized version of scalar field theory on the plane.

The twistor programme was introduced by Roger Penrose as an approach to quantum gravity in which twistor space should provide the primary geometric background for physics from which space-time should emerge. This talk will review the programme, i.e., the early successes in formulating the self-dual parts of Yang-Mills and gravity on twistor space. It will go on to review the impact of twistor-string theory, in giving at least a perturbative approach to full Yang Mills and conformal gravity, and outline arguments that prove the equivalence between the twistor-string models and the space-time theories. Finally, twistor-string models for Einstein gravity will be reviewed.

The understanding of the nature of spacetime singularity and whether and how it is resolved is one of the most important problem in quantum gravity. Important examples are black hole singularity and cosmological singularity in the big bang. In this talk we will be interested in the later type and an approach to the problem using AdS/CFT correspondence for time-dependent background will be discussed. Our gauge theory results suggest that spacetime singularity is indeed resolved and the mechanism will be discussed.

We discuss the Zamolodchikov-Faddeev algebra for the superstring sigma-model on AdS5 x S5. We find the canonical su(2,2)2 invariant S-matrix satisfying the standard Yang-Baxter and crossing symmetry equations. Its near-plane-wave expansion matches exactly the leading order term recently obtained by the direct perturbative computation. We also show that the S-matrix obtained by Beisert in the gauge theory framework does not satisfy the standard Yang-Baxter equation, and, as a consequence, the corresponding ZF algebra is twisted. The S-matrices in gauge and string theories however are physically equivalent and related by a non-local transformation of the basis states which is explicitly constructed.

Suppose we could calculate the string partition function to all genera using worldsheet methods. What could we learn from this expression about a potentially underlying (non-perturbative) target space description of the theory? We address this question in the context of the open topological A-model.

The first two of a series of seminars and lectures on Ricci Flow and its applications, lecture I at at 1330 and lecture II at 1450 on Mon February 19th. Abstract: This is an introductory two part lecture on the Poincare conjecture, Geometrization and Ricci flow intended for both mathematicians and physicists (assuming some familiarity with the basic notions of differential geometry). Part I: The Poincare conjecture and Thurston's Geometrization Conjecture. We will begin by describing the 3-dimensional Poincare conjecture, a pure topology problem about 3-manifolds. Motivated by analogies with the 2-dimensional case we will see how Thurston brought geometry into 3-dimensional topology, the goal being to describe Thurston's Geometrization Conjecture. To do this we will describe both a little more topology (the sphere and torus decompositions) and a little geometry (a discussion of the 8 different types of homogeneous 3-manifolds). We will then see how the Poincare conjecture follows straightforwardly from the Geometrization Conjecture. Part II: Ricci flow and applications to Geometrization. After a very brief reminder of basic notions of curvature in differential geometry, we introduce the Ricci flow and try to explain why it should be seen as a natural nonlinear heat-type equation which diffuses curvature around a manifold. We will discuss (without proof) some of the very basic analytic results for Ricci flow and discuss the simplest solutions to Ricci flow (e.g. Einstein metrics, Ricci solitons, product metrics). We will describe some of Hamilton's fundamental early work which showed that Ricci flow can be used to geometrize certain 3-manifolds, and discuss why for topological reasons we know that the Ricci flow must usually develop singularities in finite-time. We discuss the general framework in which to analyse singularities of the Ricci flow, the pre-Perelman progress in this theory and what obstructions Perelman needed to overcome to use Ricci-flow (with surgery) to prove the Geometrization Conjecture. (A discussion of Perelman's contributions will be left for another occasion).

I present the generating functions which count the BPS operators in the chiral ring of a N=2 quiver gauge theory that lives on N D3 branes probing an ALE singularity. The difficulty in this computation arises from the fact that this quiver gauge theory has a moduli space of vacua that splits into many branches -- the Higgs, the Coulomb and mixed branches. As a result there can be operators which explore those different branches and the counting gets complicated by having to deal with such operators while avoiding over or under counting. The solution to this problem turns out to be very elegant and is presented in this note. Some surprises with surgery of generating functions arises.

This paper continues the discussion of hep-th/0605038, applying the holographic formulation of self-dual theory to the Ramond-Ramond fields of type II supergravity. We formulate the RR partition function, in the presence of nontrivial H-fields, in terms of the wavefunction of an 11-dimensional Chern-Simons theory. Using the methods of hep-th/0605038 we show how to formulate an action principle for the RR fields of both type IIA and type IIB supergravity, in the presence of RR current. We find a new topological restriction on consistent backgrounds of type IIA supergravity, namely the fourth Wu class must have a lift to the H-twisted cohomology.