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.

We discuss how the duality group G controls the higher derivative corrections that arise in the effective action of M-theory compactified on a torus. In particular we show that the perturbative contributions always involve the weights of G and that this is a consequence of the appearance of automorphic forms of G in the complete quantum effective action.

String perturbation theory in the presence of D-branes is usually described in terms of conformal field theory (CFT) on worldsheets with boundaries. In this formalism, closed string scattering amplitudes in the presence of static Dp-branes exhibit infrared divergences for p=0 and p=1, so the worldsheet CFT description breaks down. For p=0, it is known that these divergences are due to D0-brane recoil. A systematic framework to take recoil into account is the worldline formalism, where fixed boundary conditions are replaced by dynamical D0-brane worldlines. In this formalism, the divergences that plague the worldsheet CFT are automatically cancelled in a non-trivial way. The amplitudes derived in the worldline formalism can be reproduced by deforming the CFT with a specific bilocal recoil operator. For p=1, the divergences are due to local recoil of D1-branes, which (classically) end up displaced a finite distance from their original position. The quantum version of this phenomenon can be viewed as a simple geometric manifestation of the absence of spontaneous symmetry breaking in 1+1 dimensions. Through a Dirac-Born-Infeld analysis, it is possible to resum these divergences in a way that yields finite, momentum-conserving amplitudes.

The Kahler metric for chiral matter fields plays a crucial role in the computation of soft supersymmetry breaking terms in string compactifications. Due to its non-holomorphic nature, this is difficult to compute in Calabi-Yau backgrounds. I describe techniques that allow the modular weights of the Kahler metric to be computed in IIB string compactifications. This is done by relating the modular dependence of the Kahler metric to that of the physical Yukawa couplings. I briefly discuss the applications to soft terms and neutrino masses.

In this work we define a new limiting procedure that extends the usual thermodynamics treatment of Black Hole physics, to the supersymmetric regime. This procedure is inspired on equivalent statistical mechanics derivations in the dual CFT theory, where the BPS partition function at zero temperature is obtained by a double scaling limit of temperature and the relevant chemical potentials. In supergravity, the resulting partition function depends on emergent generalized chemical potentials conjugated to the different conserved charges of the BPS solitons. With this new approach, studies on stability and phase transitions of supersymmetric solutions are presented. We find stable and unstable regimes with first order phase transitions, as suggested by previous studies on free supersymmetric Yang Mills theory.

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.

Negative curvature Euclidean (Hyperbolic) spaces have already been used in literature as part of eleven dimensional supergravity solutions. In particular it has been shown that they naturally appear when building expanding cosmological models. In this talk I will present M-theory solutions written as direct products of maximally symmetric spaces with hyperbolic components. These backgrounds break supersymmetry because of global effects but are still stable with respect to small scalar perturbations. They appear as near-horizon geometries for wrapped M5-branes, thus allowing for an intuitive interpretation of their stability.

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.