We will survey some works on differential Harnack type estimates for the Ricci flow. Starting from the work of Li-Yau on the heat equation, Hamilton's matrix estimate for the Ricci flow, the geometric space-time approach (joint with Sun-Chin Chu), a generalization (joint with Dan Knopf), and its recent extension due to Bing Cheng.

A Calabi-Yau orbifold is locally modeled on Cⁿ/G where G is a finite group of SL(n,C). One way to handle this type of orbifolds is to resolve them using a crepant resolution of singularities. We are interested in studying the topology of the crepant resolution. This can be expressed in terms of the finite group G via the McKay Correspondance.

Here are some open problems discussed in the talk.

Griffiths showed how to associate to a smooth projective variety X its intermediate Jacobian J(X), a complex torus which depends holomorphically on X and which encodes a great deal of information about X and especially about the algebraic cycles in it. When X is a Calabi-Yau threefold, the Griffiths intermediate Jacobian occurs as the fiber of a natural integrable system associated to the moduli of X. This leads to numerous connections with string theory, which is typically compactified on such Calabi-Yaus. In this talk we hope to show that this family of intermediate Jacobians (together with various related integrable systems) is a valuable tool for understanding mathematically a wide range of string theory contexts. These include Seiberg-Witten integrable systems, superpotentials and their connection to Hilbert schemes and other moduli problems, and certain aspects of mirror symmetry, large N duality, heterotic/F-theory duality, Fourier-Mukai transforms in the presence of B-fields, and more.

I will discuss what is known about Legendrian submanifolds in R²ⁿ⁺¹ for n > 1. Specifically I will describe Legendrian Contact Homology for such Legendrian submanifolds and produce many examples.

If time permits I will also discuss various intriguing constructions of Legendrian submanifolds.

The geometry of cubic polynomials, and in particular the rationality of cubic hypersurfaces, has been a catalyst for new developments in algebraic geometry for two centuries. The discovery of the irrationality of cubic curves, for example, led to the study of abelian integrals, which was central to much of 19th century mathematics; while the rationality of cubic surfaces in many ways gave birth to the subject of birational geometry. In the 20th century, Clemens' and Griffiths' proof of the irrationality of cubic threefold provided not just the first example of a counterexample to Luroth's theorem in higher dimensions, but a magnificent example of how Hodge theory could be used to settle algebrao-geometric questions.

In this talk, I'd like to review these developments and then to focus on the next, and currently unsettled, case: the rationality of cubic fourfolds. Here evidence obtained by Brendan Hassett and others suggests that cubic fourfolds may also play a pivotal role, providing among other things an answer to the questions of whether rationality is an open or a closed condition in smooth families. I'll discuss the current state of our knowledge, and what is conjectured.

I will describe some recent joint work with Lionel Mason concerning compact surfaces whose geodesics are all simple closed curves. Our approach not only yields completely new proofs of all the main classical results concerning the Riemannian case, but also gives equally strong results concerning general affine connections.

In contrast to previous work on this subject, our approach is twistor-theoretic, and depends fundamentally on the fact that, up to biholomorphism, there is only one complex structure on CP².

Abstract forthcoming

A conjecture of Hodge type will be stated. From this conjecture it would follow that the kernel of the Abel-Jacobi invariant defined by Griffiths is independent of the embedding of the base field into the complex numbers.

In his thesis, Caldararu described twisted Fourier-Mukai transforms for elliptic fibrations. In this talk I will describe how certain holomorphic symplectic manifolds can be deformed to integrable systems, i.e. fibrations by abelian varieties. These are higher dimensional analogues of elliptic K3 surfaces, and twisted Fourier-Mukai transforms once again arise.

Here are some open problems discussed in the talk.

Traditionally the analytic continuation and functional equation for L-functions of automorphic representations are derived from the so-called Whittacker expansion of automorphic forms. After a brief discussion of this approach and the technical problems associated with it, I shall introduce the notion of an automorphic distribution. I shall then describe how automorphic distributions provide an alternative approach to the study of L-functions. This is joint work with Steve Miller.

I will discuss some fully nonlinear PDEs which first arose in the conformal brach of the variational equivalence problem studied by Bryant and Griffiths. They are conformally invariant, and have turned out to be very useful in conformal geometry. I will discuss some recent joint work with Matt Gursky on four-manifolds with positive scalar curvature, and on the existence of metrics with constant Q-curvature.

We introduce the notion of K-correspondence, and show that many Calabi-Yau varieties carry a lot of self-K-isocorrespondences, which furthermore satisfy the property of multiplying the canonical volume form by a constant of modulus different from 1. This leads to the introduction of a modified Kobayashi-Eisenman pseudovolume form, for which we are able to prove many instances of the Kobayashi conjecture.

Here are some open problems discussed in the talk.