The interplay between curvature and topology of Riemannian manifolds is among the most fundamental questions in differential geometry. Over the past century, various different approaches have been developed to attack these types of problems. This includes variational techniques based on geodesics and minimal surfaces, as well as the Ricci flow approach pioneered by Richard Hamilton. In this lecture, I will give an overview of the subject, focusing on the case of positive curvature.

We consider variations of the usual voter model, which favor types that are locally less common. Such voter models with selection are dual to systems of branching annihilating random walks that are parity preserving. We consider coexistence of types in the voter models which is related to the survival of particles in the branching annihilating random walk. We find conditions for the uniqueness of a homogeneous coexisting invariant law as well as for convergence to this law from homogeneous and coexisting initial laws. For a particular one dimensional model we also show a complete convergence result for any initial condition. This is based on comparison with oriented percolation of the associated branching annihilating random walk.

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.

After reviewing some reasons why one might be interested in G_2 holonomy compactifications of M theory, I shall review some of the recent results about such vacua, which are mainly due to Atiyah, Maldacena, Vafa, Witten, and myself in various collaborations. Topics covered will mainly center around how various aspects of super Yang-Mills theory can be related to properties of M theory in such spacetime backgrounds. These will include instantons, domain walls and confinement.

This is the fourth 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.

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.

This is the second 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.

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.

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.

Space-based laser interferometers such as LISA may prove spectacularly effective not only in listening to gravitational radiation but also in the direct detection of galactic halo dark matter streaming through the solar system. This talk will describe the basic strategy and touch on some entertaining open possibilities.

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.

We present the construction of a general type of topological sigma-model on generalized Calabi-Yau (GCY) manifolds, which includes the familiar A-model and B-model as special examples. We discuss the deformation theory of generalized complex structures on GCY manifolds from the viewpoint of the topological sigma-model. This investigation leads to an extension of the famous Tian-Todorov theorem to the realm of generalized geometry.

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.

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 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.

This is the sixth 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.

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.

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.

I will report on progress and problems with the classification of four-dimensional N=2 superconformal field theories with two complex-dimensional Coulomb branches.

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.