I will describe some recent results that concern various aspects of the mixing properties of a strong incompressible flow acting together with a diffusion. In particular, we will discuss the short-time decay of solutions of the corresponding initial value problem, asymptotics of the principle Dirichlet eigenvalue and the behavior of the explosion threshold in the Zeldovich problem when the incompressible flow is strong. When the flow is prescribed, the "enhancement" of these characteristics comes from the geometric properties of the flow. We will also show that flows arising from the Stokes-Bousisnesq problems possess these "enhancement" features.

Striking advances in observational cosmology over the past two decades have ushered in a golden era in cosmology, where our focus has turned from what the universe is made of to why it has the form we observe. The leading theory capable of answering such a question, Superstring Theory, does not appear capable of being tested using conventional accelerator-based experiments, forcing us to be more creative in our goal to verify or dismiss it. Focusing on brane inflation as a string theory-inspired model of inflationary cosmology, I will review how the cosmic microwave background (CMB) will provide a deluge of high- precision data into otherwise inaccessible energy scales. These data include possible "Transplanckian" signatures in the power spectrum, indications of variable sound speed or extra dimensions in non- Gaussianity, or constraining the inflation model parameter space using polarization. I then describe how the production of cosmic (super)strings in brane inflation would provide an additional means to verify superstring theory, and which would yield much detailed information about the underlying theory parameters.

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.

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.

This is the fifth of a series of lectures on F-theory. The lectures will present the theory of elliptically fibered Calabi-Yau manifolds in considerable detail, explain how these manifolds are used to produce string vacua by means of the ``F-theory'' construction, and how various properties of these string vacua are determined by the corresponding Calabi-Yau manifolds.

In my talk I shall try to explain the relationship of the Brauer group, derived categories of twisted sheaves, and physics. In particular, I shall attempt to relate an example due to Aspinwall-Morrison-Vafa-Witten to some examples that occur in the study of elliptic Calabi-Yau threefolds without sections.

The talk will be focused on the relation between K theory and M theory and it's generalization to nonzero $B$ field. The main idea is a comparison of the long distance partition functions of the two theories. In the process we will outline a derivation of the (twisted) Atiyah - Hirzebruch spectral sequence from M theory. We will also discuss the relation between $SL(2,Z)$ duality and K theory.

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.

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).

One of the most fundamental unresolved questions in particle physics and cosmology is the origin of the cosmic baryon asymmerty, a necessary ingredient for galaxy formation. There are three known conditions for baryogesis, the Sakharov conditions: (1) Baryon number violation; (2) CP, C violation; (3) (1) & (2) must simulatneously occur out of equilibrium with the rest of the universe. Over the last 25 years it has been a challenge to realize these conditions and the precise number (n/s \sim 6.04 \times 10^{-10}) for baryon asymmetry within the framework of the standard model of particle physics and cosmology. In this seminar we show that the baryon asymmetry can be generated, unifying all three Sakharov conditions, during the epoch of cosmic inflation. The Sakharov conditions are realized in one physical proccess, the quantum generation of elliptically polarized gravitons. We further, show that this mechanism is realized concretely and in a compactification independent manner in superstring theory. We construct a Heterotic model of this inflationary baryogenesis mechanism and observationally constrain(postdict) the string length scale from a cosmological observable. Key to this result is the universality of the model independent axion and the Green-Schwarz mechanism. We report on future work w.r.t a heterotic version of the KKLT mechanism via. geometric transistion.

Motivated by phenomenological applications we consider (warped) compactifications of type IIA string theory on Calabi-Yau fourfolds in the presence of Ramond-Ramond fluxes. The main goal of the talk is a description of low-energy physics, including a new superspace formulation of two-dimensional N = (2,2) dilaton supergravity coupled to matter, and computation of the effective superpotentials induced by fluxes.

We revisit the singular IIB supergravity solution describing M fractional 3-branes on the conifold (hep-th/0002159). Its 5-form flux decreases, which we explain by showing that the relevant N=1 SUSY SU(N+M) x SU(N) gauge theory undergoes repeated Seiberg-duality transformations in which N -> N-M. Far in the IR the gauge theory confines; its chiral symmetry breaking removed the singularity of hep-th/0002159 by deforming the conifold. We propose a non-singular pure-supergravity background dual to the field theory on all scales, with small curvature everywhere if the 't Hooft coupling (g_s M) is large. In the UV it approaches that of hep-th/0002159, incorporating the logarithmic flow of couplings. In the IR the deformation of the conifold gives a geometrical realization of chiral symmetry breaking and confinement. We suggest that pure N=1 Yang-Mills may be dual to strings propagating at small (g_s M) on a warped deformed conifold. We note also that the standard model itself may lie at the base of a duality cascade.

D-branes are usually treated as classical geometric objects in string theory. They can be defined more abstractly in conformal field theory using the boundary state formalism. Some general issues involving this formalism will be discussed and illustrated with examples involving D-branes on both symmetric and asymmetric orbifolds.

General conjectures about the SL(2,Z) modular transformation properties of N=4 super-Yang-Mills correlation functions are presented. It is shown how these modular transformation properties arise from the conjectured duality with IIB string theory on AdS₅ x S⁵. We discuss in detail a prediction of the AdS duality: that N=4 field theory, in an appropriate limit, must exhibit bonus symmetries, corresponding to the enhanced symmetries of IIB string theory in its supergravity limit.

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 AdS/CFT duality buys us a consistent non-perturbative description of gravity (in some class of backgrounds) at the price of manifest spacetime causal structure and locality. In this talk I will describe the bare beginnings of an attempt to understand these issues by studying the duality in Lorentzian signature and showing how states, operators, and classical fields are mapped between the dual theories. I will then describe two examples of classical D-brane and string probes of AdS backgrounds and show that locality in AdS shows up as scale locality for the dual boundary configurations. I will close with some suggestions for what the horizon and singularity of the AdS black hole might mean in the boundary CFT.

D-branes on Calabi-Yau threefolds are interesting both as alternative probes of quantum geometry, and as nontrivial realizations of gauge theories with four supercharges. Of course, these issues are not independent. In this talk I will briefly discuss the former by describing and characterizing D-brane states at the Gepner point in the moduli space of the quintic. I will then use such D-branes to realize d=4 N=1 SUSY gauge theories, and discuss the computation of the superpotential of these theories, particularly for those realized by D6-branes wrapped around special Lagrangian submanifolds of the threefold. I will close with some speculations on the implications of mirror symmetry for these superpotentials.

D-branes on orbifolds with discrete torsion can be defined using projective representations of the orbifold group. We discuss the examples C³ / Z_n × Z_n in detail.