Let the torus T^2n carry the standard symplectic structure, and a Hamiltonian function H of period 1 in the time variable. By the Arnold Conjecture, proved for the torus by Conley and Zehnder, the Hamiltonian flow has at least 2n+1 orbits of period 1. Conley and Zehnder also proved, under the additional assumption that all period 1 orbits are nondegenerate: If there are only finitely many orbits of period 1, then there are orbits of arbitrarily large minimal (integer) period. We prove this statement also holds in the degenerate case. Thus there are always infinitely many orbits of integer period. This settles a conjecture of Conley for the torus; this conjecture is still open for other compact symplectic manifolds.

I will talk about recent results, obtained in joint work with Rodnianski, concerning conditional pointwise estimates for the curvature tensor of vacuum solutions of the Einstein equations. The assumptions we make allow us to derive, or we simply require, bounds for the energy and flux of curvature across space-like and null hypersurfaces. This allows us to control the geometry of null hypersurfaces and thus to define a `` good'' parametrix for the covariant wave equation. Using this parametrix we can derive pointwise estimates for the curvature tensor. One can regard our results as a General Relativity version of the well known Beale-Kato-Majda removal of singularities theorem in Hydrodynamics.

The lecture will discuss singularities of mean curvature flow. After a review of background material, the focus will be on the case of surfaces of positive mean curvature evolving in 3-space. This is joint work with Toby Colding.

I will present some recent work with Claudio Arezzo concerning the construction of Kahler metrics with constant scalar curvature on the blow up, at isolated points, of a manifold M already endowed with a constant-scalar-curvature Kahler metric. The hypotheses require the vanishing of all holomorphic vector fields on M.

I will discuss a view of Gromov-Witten theory based on analogies to classical topology. Theorems can be proven by localization and degeneration techniques. One outcome is an approach to the Gromov-Witten theory of the Calabi-Yau quintic threefold. The talk represents joint work with D. Maulik.

The talk will describe interaction between nonlinear waves and geometry in General Relativity. It will focus on two examples: the new proof of stability of Minkowski space-time for the Einstein vacuum (and scalar field) equations in harmonic gauge and the Price law for the collapse of a spherically symmetric self-gravitating scalar field

Will discuss the application of methods of singular metrics to problems in algebraic geometry such as Fujita's conjecture of effective freeness and effective very ampleness, effective Matsusaka big theorem, effective Nullstellensatz, deformational invariance of plurigenera, and finite generation of canonical ring.

I will sketch the basic idea of Floer theory, in particular its symplectic version for pairs of Lagrangians and its instanton version for homology 3-spheres. In a joint project with Dietmar Salamon we define a new Floer theory for 3-manifolds with boundary, using the instanton equation with Lagrangian boundary conditions (containing nonlocal conditions on the holonomy). The underlying PDE exhibits some unexpected semi-global behaviour. This can be understood as evidence that the new Floer homology is an intermediate object between the instanton Floer homology and the symplectic Floer homology and can thus be used to prove a conjecture of Atiyah and Floer relating these.