As parameters in quantum mechanics are slowly varied, states undergo a non-Abelian holonomy known as Berry phase. I will review the ideas behind the Berry phase and explain how supersymmetry imposes restrictions on the holonomy. The associated non-Abelian gauge connections must obey certain first order equations which are related to the self-dual Yang-Mills equations, specifically the Hitchin equations, tt star equations, and Bogomolnyi equations. I will end by showing how one can build a BPS 't Hooft-Polyakov monopole in the lab.

I will examine the string dual of the recently constructed N = 6 superconformal Chern-Simons theory of Aharony, Bergman, Jafferis and Maldacena (ABJM theory). In particular I will focus on the SU(2)X SU(2) sector. I will derive the full magnon dispersion relation and compare it to recently found results for ABJM theory at weak coupling. Then I will consider finite-size corrections to the energy of string states in the R X S2 X S2 subspace of AdS4 X CP3 with an angular momentum J on CP3 being large, using two different approaches. I will compare the results with the recently proposed all-loop Bethe ansatz of Gromov and Vieira.

We will discuss the missing pieces in the understanding of the effective field theory description of string creation, the S-dual of the Hanany-Witten effect, both in the open and closed string picture. We explain the origin of a crucial bare Chern-Simons term, that used to be added in by hand for consistency. Finally we summarize the remaining unsettled issues, concerning the need to modify the DBI action and the interpretation of the bare term in M-theory.

In this talk I will describe the computation of the four-point function of two weight-2 and two weight-n 1/2-BPS operators at large N and at strong t Hooft coupling, using the supergravity approximation. I will then discuss the result in the light of the AdS/CFT correspondence.

I am going to review, in the context of gauge theories, the anomalous dimension of twist two operators, the cusp anomaly of Wilson loops and how they relate through AdS/CFT to rotating strings in AdS space. Then I'm going to describe in more detail the spiky strings and their properties for large spin which allows us to obtain a relation between the anomalous dimension of twist two operators and the pp-wave anomaly, an anomaly we find when N=4 SYM theory is put in a pp-wave background. Finally I'm going to consider various new open string solutions describing Wilson loops in a pp-wave and in flat space. Based on 0802.2039 and 0804.3438.

First of two lectures. For details, please consult the webpage www.ma.ic.ac.uk (slash) (tilde) rpwt (slash) LMS.html (sorry, triangle server does not allow special characters)

For details, please consult the webpage www.ma.ic.ac.uk (slash) (tilde) rpwt (slash) LMS.html

2006 Fields medalist Okounkov's invited lecture series begins on Monday. Ideally only those who registered would come to the first few Okounkov lectures, as the IMS has limited space. Anyone else who wants to come is of course very welcome, but please stand at the back for the first few (oversubscribed) lectures. After that things will thin out and there should be plenty of room. For details, please consult the webpage www.ma.ic.ac.uk (slash) (tilde) rpwt (slash) LMS.html (sorry, triangle server does not allow special characters)

For details, please consult the webpage www.ma.ic.ac.uk (slash) (tilde) rpwt (slash) LMS.html

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.