I will review the problem of constructing black holes on a braneworld with AdS bulk, and the arguments that a full classical 5D solution will correspond to a quantum corrected 4D black hole. I will show that for negative brane cosmological constant, a Schwarzschild-AdS black string in the bulk can be consistently interpreted as a quantum-corrected black hole on the brane, but the form of the quantum corrections is unlike what we would expect.

We derive the eikonal approximation to high energy interactions in Anti-de Sitter spacetime, resuming in terms of a generalized phase shift, ladder and cross ladder graphs associated to the exchange of a spin j field, to all orders in the coupling constant. Using the AdS/CFT correspondence, the resulting amplitude determines the behavior of the dual CFT four point function for small values of cross ratios, in a Lorentzian regime. We explore the consequences of this result to the dual CFT. In the planar limit this Lorentzian amplitude is dominated by a Regge pole whose nature varies as a function of the 't Hooft coupling. At large coupling, the pole corresponds to graviton exchange in AdS, whereas at weak coupling, the pole is that of the hard perturbative BFKL pomeron. The conformal symmetry of the transverse space E2 is trivially realized on the dual holographic space H3, allowing for a unified description of both weak and strong 't Hooft coupling regimes. The analysis suggests a possible AdS eikonal resummation of multi-pomeron exchanges implementing AdS unitarity, which differs from the usual 4-dimensional eikonal exponentiation.

Sasakian geometry is the odd-dimensional cousin of Kahler geometry. Although the two are very closely related, there are some interesting differences. I will give an introduction to Sasakian, and in particular Sasaki-Einstein, geometry, and review some of the developments in this subject over the last few years. In particular I'll cover explicit constructions, toric Sasakian geometry, and obstructions. Some of these results also provide new insights into Kahler geometry - for example, I'll describe very simple new obstructions to the existence of Kahler-Einstein metrics on Fano orbifolds.

We point out a special corner in the space of Calabi-Yau compactifications where standard-like models tend to emerge. We show how these scenarios are inter-related via a mathematical process of transgression of bundles, a generalisation of the conifold transition. Perhaps our world is special and we live in this oasis within the multitude of vacua.

Two-dimensional conformal field theory can be defined through its correlation functions. These must satisfy certain consistency conditions which arise from the cutting of world sheets along circles or intervals. Generalising what one finds for 2d topological field theories, a solution to these constraints can be obtained from a symmetric special Frobenius algebra in the appropriate braided monoidal category.

Extremal (especially BPS) black holes are of special importance in string theory and AdS/CFT. Their near-horizon geometries provide a precise tool for investigating certain aspects of such black holes. I will begin with a discussion of some general properties of near-horizon geometries. I will then describe some specific applications of these ideas which will include: a classification of near-horizon geometries of asymptotically AdS(5) supersymmetric black holes with two rotational symmetries (of relevance to AdS/CFT), a proof of symmetry enhancement of generic near-horizon geometries in D=4,5 (of relevance to the attractor mechanism), near-horizon geometries in D greater than 5 and finally some open problems.

These will be introductory lectures surveying GW, MNOP and GV invariants -- all different ways of counting curves. For a string theorist this involves seeing the curve as, respectively, the world sheet of a string, a D-brane, or a BPS thingummy. I will describe a 4th way via stable pairs, which in effect means counting D-branes (or stable objects of the derived category, to mathematicians) after a change of stability condition.

We study the near horizon geometry of extremal black holes in five dimensional gauged supergravity using Sen's entropy function formalism. Special attention is paid to the large black hole limit where the near horizon solution exhibits a universal dependence on the rotation. The physical properties of the large black hole solution are shown to agree with predictions from fluid mechanical description of the dual conformal field theory.

We conceive higher bundles with connections entirely Lie algebraically by, essentially, extending the Cartan-Ehresmann definition of a connection from Lie algebras to L-infinity algebras. Using this machinery, we study Chern-Simons n-bundles as obstructions to lifts through String-like extensions of L-infinity algebras.

In this talk I want to describe recent work on geometric (Berry) phases that arise in the quantum theory of certain D-brane states. After briefly reviewing the geometric phase in general theories, I will turn to supersymmetric systems of the type arising in D0-brane configurations in string theory. Interestingly, the resulting Berry phases are related to the four Hopf fibrations. I will show how this fact arises as a consequence of the close relationship between SUSY and the division algebras. The framework of string theory allows to draw some interesting physical consequences from these Berry phases, such as a new interpretation of the Berry phase in terms of gravitational precession and the surprising result that in certain configurations, D0 branes behave as anyons.

These will be introductory lectures surveying GW, MNOP and GV invariants - all different ways of counting curves. For a string theorist this involves seeing the curve as, respectively, the world sheet of a string, a D-brane, or a BPS thingummy. I will describe a 4th way via stable pairs, which in effect means counting D-branes (or stable objects of the derived category, to mathematicians) after a change of stability condition.

Recently Romelsberger made some conjectures about an index for N=1 superconformal quantum field theories applied to certain Seiberg dual theories. Recent joint work by me and Hugh Osborn proves and extends these conjectures by employing certain q-series type identities. I will discuss the superconformal index, how it applies for more general N=1 dual theories and, briefly, the proof of these conjectures.

I give an outline of the program of Shadrin and his collaborators, to give explicit formulas for correlators in topological gravity (with descendants), starting with a Batalin-Vilkovisky algebra. In particular, I will try to explain the relationship with Givental's work on Gromov-Witten invariants. No background is assumed.

These will be introductory lectures surveying GW, MNOP and GV invariants -- all different ways of counting curves. For a string theorist this involves seeing the curve as, respectively, the world sheet of a string, a D-brane, or a BPS thingummy. I will describe a 4th way via stable pairs, which in effect means counting D-branes (or stable objects of the derived category, to mathematicians) after a change of stability condition.

We will discuss the properties of geometries dual to BPS states of N=4 supersymmetric Yang-Mills theory. Despite the non-linearity of the problem, we develop a universal bubbling AdS description of these geometries by focusing on the boundary conditions which ensure their regularity. In the 1/8 BPS case, we find that the S3 cycle shrinks to zero size on a five-dimensional locus inside the six-dimensional base. Enforcing regularity of the full solution requires that the interior of a smooth, generally disconnected five-dimensional surface be removed from the base. The AdS5 x S5 ground state corresponds to excising the interior of an S5, while the 1/8 BPS excitations correspond to deformations (including topology change) of the S5 and/or the excision of additional droplets from the base.

The study of supersymmetric solutions in supergravity have provided invaluable insight into quantum gravity via AdS/CFT correspondence. In this talk I present a new class of intriguing Kahler geometry which appears in the study of AdS solutions from D3 and M2-branes wrapped on 2-cycles. Solving the master equation, which has higher-order couplings in Ricci tensor, we obtain new AdS solutions.

I discuss some aspects of the full moduli space of N=1 supersymmetric gauge theories, focusing on those arising on D-branes at Calabi-Yau singularities. Of particola importance is the full moduli space for one brane, called the master space. I discuss its geometrical properties and its relation with the uncovering of hidden symmetries and the counting of BPS states of the gauge theory.