I will talk about a new method of performing a strong coupling expansion for many superconformal field theories in four dimensions, in particular those that are relavant for the AdS/CFT correspondence. I will first explain the methods for the case of N=4 SYM, as well as what calculations can be done (both analytically and numerically) and I will show how they compare with the dual AdS geometry. I will then explain what generalizations are required for other setups and what new field theory calculations can be done with these methods that were not available before.

The chiral symmetry of QCD shows up in the linear Weyl--Wigner mode at short Euclidean distances or at high temperatures. On the other hand, low-lying hadronic states exhibit the nonlinear Nambu--Goldstone mode. An interesting question was raised as to whether the linear realization of the chiral symmetry is asymptotically restored for highly excited states. We address it in a number of ways. On the phenomenological side we argue that to the extent the meson Regge trajectories are observed to be linear and equidistant, the Weyl--Wigner mode is not realized. This picture is supported by quasiclassical arguments implying that the quark spin interactions in high excitations are weak, the trajectories are linear, and there is no chiral symmetry restoration. Then we use the string/gauge duality. In the top-down Sakai--Sugimoto construction the nonlinear realization of the chiral symmetry is built in. In the bottom-up AdS/QCD construction by Erlich et al., and Karch et al. the situation is more ambiguous. However, in this approach linearity and equidistance of the Regge trajectories can be naturally implemented, with the chiral symmetry in the Nambu--Goldstone mode. Asymptotic chiral symmetry restoration might be possible if a nonlinearity (convergence) of the Regge trajectories in an intermediate window of n,J, beyond the explored domain, takes place. This would signal the failure of the quasiclassical picture.

I present new analytic solutions of type IIB supergravity with fully backreacting D7-branes describing the addition of an arbitrary number of flavors to the Klebanov-Tseytlin and Klebanov-Strassler theories. I provide a detailed analysis of the field theory and of the duality cascade which describes its RG flow, Seiberg duality is understood as a large gauge transformation in supergravity. Moreover the string background suggests that the UV behavior is a duality wall.

The lectures will start by reviewing closed topological theories, before moving to more recent work in the open/closed theory on spaces with boundary. From the work of Witten, the topology of the so-called Deligne-Mumford moduli spaces of Riemann surfaces with nodes plays a fundamental role in 2-dimensional topological gravity (known to mathematicians as Gromov-Witten theory). For example, by the work of Kontsevich and Manin, it is seen to underly the Witten-Dijkgraaf-Verlinde-Verlinde equation, and hence is intimately related to the theory of Frobenius manifolds and of integrable systems. Most work on these moduli spaces has been focussed on the case of closed topological field theory. In these lectures, I will explore the moduli spaces, analogous to Deligne-Mumford moduli spaces, which play the corresponding role in the open theory. In this case, the world sheet (Riemann surface) has a boundary, and as a result, the moduli spaces are no longer complex orbifolds, but rather real orbifolds with corners. These moduli spaces may be viewed as an explanation of the way that algebraic structures, such as A-infinity categories, cyclic homology, and the Cardy condition, enter topological field theory in two dimensions. This theory should also have applications to understanding the foundations of string theory.

We consider N=1 supersymmetric quantum chromodynamics (SQCD) with the gauge group U(Nc) and Nc+N quark flavors. Nc flavors are massless, the corresponding squark fields develop (small) vacuum expectation values (VEVs) on the Higgs branch. Extra N flavors are endowed with small (and equal) mass terms. We study this theory through its Seiberg's dual: U(N) gauge theory with Nc +N flavors of dual quark fields plus a gauge-singlet mesonic field M. The original theory is referred to as quark theory while the dual one is termed monopole theory. The suggested mild deformation of Seiberg's procedure changes the dynamical regime of the monopole theory from infrared free to asymptotically free at large distances. We show that, upon condensation of the dual quarks, the dual theory supports non-Abelian flux tubes (strings). Seiberg's duality is extended beyond purely massless states to include light states on both sides. Being interpreted in terms of the quark theory, the monopole-theory flux tubes are supposed to carry chromoelectric fields. The string junctions - confined monopole-theory monopoles - can be viewed as constituent quarks of the original quark theory. We interpret closed strings as glueballs of the original quark theory. Moreover, there are string configurations formed by two junctions connected by a pair of different non-Abelian strings. These can be considered as constituent quark mesons of the quark theory.

We will discuss the nonabelian strings found recently in SUSY QCD and non-SUSY gauge theories with scalars. Their properties will be considered in some details. In particular, their rich worldsheet structure involving localized monopoles will be explained. The dependence on the SUSY breaking parameters will be analysed.

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.