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.

It is proposed that conformality at the TeV scale be used to solve the hierarchy problem and to restrict fields additional to those of the standard model. This idea provides rigid predictions.

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.

In a remarkable paper, Gopakumar and Vafa have used M-theory to assign BPS invariants to families of holomorphic curves which are believed to yield the Gromov-Witten invariants. In this talk, techniques of algebraic geometry are developed for calculating these invariants which agree with mirror symmetry results in examples. The proper mathematical context for complete generalization of these ideas appears to require geometric invariant theory.

We study the physics of D6 branes wrapped on supersymmetric three-cycles in type IIA string theory. In particular, we argue that although the superpotential in the resulting field theories vanishes to all orders in \alpha', nonperturbative contributions (from disk instantons) should generically be present. We discuss how mirror symmetry can relate these to tree-level sigma model computations in some circumstances. We also present an illustration of the role that background closed string moduli play in the brane worldvolume theory.

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.

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.