Intuitively, renormalization group flow is irreversible, since one would expect the number of massless degrees of freedom of a theory to decrease as one goes to progressively larger distance scales. This has been proven rigorously in two dimensions (the "c-theorem"), but is otherwise an open problem. I will discuss recent efforts to prove an analogous theorem in four dimensions for supersymmetric theories, for which we can make exact statements not usually accessible in their non-supersymmetric counterparts.

It is known that many random matrix models have a dual description in terms of a topological string theory. Moreover, recently, Ooguri, Strominger and Vafa have shown that one can calculate certain properties of N=2 supersymmetric black holes in four dimensions using a topological string theory. In this talk, based on hep-th/0410141, I connect these two developments, and discuss what matrix models can teach us about black holes.

The dynamics of D branes moving in certain string theory backgrounds can be used to learn about supersymmetric field theories in several dimensions. In this talk I will center on the realization of four-dimensional N=2 and N=1 gauge field theories, by using two types of backgrounds. The first (`brane configurations') involves D branes in the presence of NS fivebranes, whereas the second involves D branes sitting at singular points. A wide variety of conformal theories with marginal couplings, and of finite N=1 theories arises from these constructions. I will also emphasize the equivalence of different constructions by T duality.

I will discuss the statistical analysis of flux vacua with special properties such as (no-scale) supersymmetry and discrete symmetries. Unlike generic flux vacua, enumeration of such special vacua can require number-theoretic methods.

The low energy effective action of the type IIB superstring is given by IIB supergravity. First stringy corrections involve terms with six extra derivatives and can be obtained in an elegant way using a superfield formalism. This talk will be concerned with some recent progress in understanding the IIB superfield. In particular, some non-renormalisation properties of D3-brane supergravity solutions will be demonstrated. The last part of the talk will address higher derivative terms involving three-form fluxes.

We discuss aspects of the renormalization group flow of quiver gauge theories in two dimensions, with little or no supersymmetry, in the spirit of earlier work of Zamolodchikov on non-supersymmetric Landau-Ginzburg theory. By using a variety of techniques including analysis of chiral symmetries, identification of operators, and analysis of perturbative gauge theory diagrams, we show that these theories can (at least) be fine-tuned to flow to orbifold conformal field theory in the infrared. This provides a linear sigma model description of string backgrounds with no spacetime supersymmetry, whose applications and limitations we begin to explore.

The talk will begin with a review of some recent progress in non-supersymmetric string theory, motivating a careful study of such backgrounds. Many non-supersymmetric closed string theories have negative modes (``tachyons''). I will describe an approach to understanding the result of tachyon condensation using configurations in string theory of effectively negative but finite tension (such as orientifold planes and their S-duals).

To define a consistent perturbative geometric heterotic compactification one must specify not only a Calabi-Yau manifold M but also a bundle E on M. The bundle E is required to satisfy a subtle constraint known as ``stability,'' which depends upon the Kähler form. This dependence upon the Kähler form is highly nontrivial---the Kähler cone splits into subcones, with a distinct moduli space of bundles in each subcone---and has long been overlooked by physicists. In this talk we shall describe this behavior and its physical manifestation.

Supersymmetry can severely constrain higher derivative terms in the effective action of both Yang-Mills theories and string theories. I will explain how to obtain these constraints.

In the spirit of hep-th/0309223, we investigate the N=(2,2) NLSM with U(N) gauge fields living on the boundary of the worldsheet. A boundary fermion mechanism is used to rewrite the trace over the path ordered exponential as a local Lagrangian. This allows us to explore the supersymmetric boundary conditions leading to constraints for non-abelian D-branes.

We study four N=1 SU(N)^6 gauge theories, with bi-fundamental chiral matter and a superpotential. In the infrared, these gauge theories all realize the low-energy world-volume description of N coincident D3-branes transverse to the complex cone over a del Pezzo surface dP_3 which is the blowup of P^2 at three generic points. Therefore, the four gauge theories are expected to fall into the same universality class--an example of a phenomenon that has been termed "toric duality." However, little independent evidence has been given that such theories are infrared-equivalent. In fact, we show that the four gauge theories are related by the N=1 duality of Seiberg, vindicating this expectation. We also study holographic aspects of these gauge theories. In particular we relate the spectrum of chiral operators in the gauge theories to wrapped D3-brane states in the AdS dual description. We finally demonstrate that the other known examples of toric duality are related by N=1 duality, a fact which we conjecture holds generally.

We study the conformal field theory describing the extreme low-energy excitations of parallel D3-branes located at a singular point of the transverse space. For quotient singularities such a description is known. Using the fact that partial resolutions of quotient singularities contain other kinds of singular points, as well as a mapping of the moduli space of a singularity onto the parameter space of the corresponding field theory, we compute the worldvolume field theory for branes at more general singularities. We compare our results to the predictions of an extended version of Maldacena's AdS/CFT correspondence as presented last week by D. Morrison.

Abstract: Gopakumar and Vafa have proposed a formula for the Gromov-Witten invariants of Calabi-Yau 3 folds in terms of numbers of BPS states. Part of this formula is explained by higher genus multiple covers and collapsed contributions in Gromov-Witten theory. The mathematical analysis leads to a direct generalization of Gopakumar and Vafa's formula to an integral structure for arbitrary 3 folds. I will discuss a mathematical approach to these questions via integrals over the moduli space of stable curves.

I will talk about geometric transitions on the moduli space of four dimensional, N=1 supersymmetric, string compactifications. In particular, I shall discuss some recent results on geometric transitions of Calabi-Yau's in the presence of branes and flux.

Some of the old problems in algebraic geometry, as well as relatively new problems in the theory of quantization, were solved using topological sigma models. The sigma models describe maps of a manifold M to a target space X. It is very well-known that no sensible theory exists when the dimension of M is greater than two. In my talk I will try to argue in favor of the existence of an interesting theory of maps in the case where M is a four-dimensional Riemannian manifold and X is a classifying space of some compact Lie group (or its finite-dimensional model). To get there we will need to introduce & develop certain aspects of Donaldson theory and higher-dimensional analogues of Whitman hierarchies. No knowledge of Donaldson theory or Whitman hierarchies is necessary.

I will discuss some action functionals constructed by Hitchin which lead to manifolds of special holonomy in six and seven dimensions, and explain how these actions could be related to a topological version of M-theory which deals with variations of G2 structures.

Work over the past few years by Mark Gross, Ilya Zharkov, and Wei-Dong Ruan has led to a clear conjectural picture for the structure of generic supersymmetric T^3 fibrations on Calabi-Yau threefolds. We will explain this picture, show how mirror symmetry would be a natural consequence of it, and discuss other possible applications.

Work over the past few years by Mark Gross, Ilya Zharkov, and Wei-Dong Ruan has led to a clear conjectural picture for the structure of generic supersymmetric T^3 fibrations on Calabi-Yau threefolds. We will explain this picture, show how mirror symmetry would be a natural consequence of it, and discuss other possible applications.

The ``horizon'' of N coincident branes in Minkowski space is the unit sphere in the transverse directions; in a certain scaling limit, string or M-theory in the presence of branes is dual to a compactification on the product of an anti-de Sitter space with the horizon. We study the analogous limit when N branes are placed at a singular point, so that the horizon becomes the so-called ``link'' of the singularity, and is no longer a sphere. A similar scaling argument leads to a natural extension of Maldacena's celebrated ``AdS/CFT correspondence conjecture'' to this situation. The conformal field theories in question have less supersymmetry than the cases studied by Maldacena, with the amount being determined by the Killing spinors on the horizon manifold. An important mathematical tool is the relationship -- derived some years ago by C. Bär -- between Killing spinors on a manifold and covariantly constant spinors on the cone over that manifold.

We present an exact construction of the D0-brane as a soliton on the worldvolume of a BPS D2-brane in the presence of a large B field. We match the mass and spectrum of fluctuations of this soliton to the D0-D2 conformal field theory. This spectrum includes a tachyon: the endpoint of tachyon condensation represents the ``disappearance'' of the D0-brane, and all modes associated with it, as it dissolves in the D2-brane. This construction thus sheds some light on issues related to tachyon condensation.

A surprising anomaly in diffeomorphisms acting on the normal bundle of the M-theory fivebrane remained a puzzle for almost two years. We present a cancellation mechanism based on a careful treatment of the fivebrane by smoothing it out through coupling to gravity which results in a relation between antisymmetric tensor gauge transformations and diffeomorphisms and leads to a complete cancellation of the anomalies. This formalism is used for derivation of the R-symmetry anomalies of the A_N (0,2) tensor-multiplet theories and for understanding the role of anomalies in the AdS/CFT correspondence. The Kaluza-Klein origin of Chern-Simons terms in lower-dimensional supergravity theories is displayed via modification of the Freund-Rubin ansatz. The results imply the existence of interesting 1/N corrections in the AdS/CFT correspondence. A simple derivation of black hole entropy in d=4, N=2 compactifications of M-theory is presented.