School of Mathematics

Some geometric mechanisms for Arnold diffusion

Rafael de la Llave
Georgia Tech
April 10, 2018
Abstract: We consider the problem whether small perturbations of integrable mechanical systems can have very large effects.

It is known that in many cases, the effects of the perturbations average out, but there are exceptional cases (resonances) where the perturbations do accumulate. It is a complicated problem whether this can keep on happening because once the instability accumulates, the system moves out of resonance.

A General Shadowing result for normally hyperbolic invariant manifolds and its application to Arnold diffusion

Tere Seara
April 10, 2018
Abstract: In this talk we present a general shadowing result for normally hyperbolic invariant manifolds. The result does not use the existence of invariant objects like tori inside the manifold and works in very general settings.

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.

Arnold diffusion for `complete' families of perturbations with two or three independent harmonics

Amadeu Delshams
April 9, 2018
Abstract: We prove that for any non-trivial perturbation depending on any two independent harmonics of a pendulum and a rotor there is global instability. The proof is based on the geometrical method and relies on the concrete computation of several scattering maps.

Fukaya categories of Calabi-Yau hypersurfaces

Paul Seidel
Massachusetts Institute of Technology; Member, School of Mathematics
April 9, 2018

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.