We apply this result to establish the existence of diffusing orbits in a large class of nearly integrable Hamiltonian systems. Our approach relies on successive applications of the so called `scattering map' along homoclinic orbits to a normally hyperbolic invariant manifold.
Consider a Calabi-Yau manifold which arises as a member of a Lefschetz pencil of anticanonical hypersurfaces in a Fano variety. The Fukaya categories of such manifolds have particularly nice properties. I will review this (partly still conjectural) picture, and how it constrains the field of definition of the Fukaya category.
The problems come in two flavors.
Extrinsic Flavor: Given a point cloud in R^N sampled from an unknown probability density, how can we decide whether that probability density is concentrated near a low-dimensional manifold M with reasonable geometry? If such an M exists, how can we find it? (Joint work with S. Mitter and H. Narayanan)