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.

The locust olfactory system interfaces with the external world through antennal receptor neurons (ORNs), which represent odors in a distributed, combinatorial manner. ORN axons bundle together to form the antennal nerve, which relays sensory information centrally to the antennal lobe (AL). Within the AL, an odor produces a stimulus-specific temporal progression of neuronal spiking, inspiring the hypothesis that the AL encodes odors through dynamically evolving ensembles of active cells. Such a coding strategy, however, requires higher olfactory centers to integrate a prolonged dynamic profile of AL signals prior to stimulus assessment, a process that is likely to be slow and inconsistent with the generation of quick behavioral responses. Our modeling work has led us to propose an alternate hypothesis: the dynamical interplay of fast and slow inhibition within the locust AL induces transient correlations in the spiking activity of an odor-dependent neural subset, giving rise to a temporal binding code and allowing rapid stimulus detection by downstream elements.

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.

For nearly 25 years Calabi-Yau spaces have played a central role in string theory, yet no explicit metric was known. I will outline ideas pioneered by Donaldson and Yau that lead to such metrics numerically; then extend this approach to solving the hermitian Yang-Mills equation, and also obtain metrics on moduli spaces. Knowledge of these quantities is unavoidable for physical predictions.

We study the conditions under which an algebraic curve can be modelled by a Laurent polynomial that is nondegenerate with respect to its Newton polytope. Such nondegenerate polynomials have become popular objects in explicit algebraic geometry, owing to their connection with toric geometry; however, despite their ubiquity, the intrinsic property of nondegeneracy has not seen much detailed study. We prove that every curve of genus $g \geq 4$ over an algebraically closed field is nondegenerate in the above sense. More generally, let $\mathcal{M}_g^{\textup{nd}}$ be the locus of nondegenerate curves inside the moduli space of curves of genus $g \geq 2$. Then we show that $\dim \mathcal{M}_g^{\textup{nd}} = \min(2g+1,3g-3)$, except for $g=7$ where $\dim \mathcal{M}_7^{\textup{nd}} = 16$; thus, a generic curve of genus $g$ is nondegenerate if and only if $g \geq 4$

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.

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.

T-duality has long been well understood locally via the Buscher rules. Global T-duality in the presence of an arbitrary background is much more involved. It relates backgrounds of different topology and can be seen to map 'regular' or commutative geometries to noncommutative ones. I will give a brief overview of these attempts at studying T-duality and show how T-duality acts very naturally in the context of Hitchin's generalized geometry. I will show that T-duality is an automorphism of the Courant bracket in the most general sense and give an example. If time permits, I will discuss applications to Poisson-Lie T-duality.

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 describe two kinds of corrections to the previously known effective action of heterotic M-theory. The first is due to taking into account the 11d R^4 term. This results in a new contribution to the Kahler potential of the four-dimensional effective theory. The second comes from the flux-generated superpotential. To derive systematically the form of this superpotential, one has to use the approach of group structures. It is essential to consider SU(3), not G_2, structure on the internal 7d manifold. Finally, I will address the issue of how these corrections affect the existence of dS vacua in heterotic M-theory, which were originally found by balancing open membrane instanton effects and gaugino condensation.