Emerging Topics Working Group

Abstract: Arnold diffusion studies the problem of topological instability in nearly integrable Hamiltonian systems. An important contribution was made my John Mather, who announced a result in two and a half degrees of freedom and developed deep theory for its proof. We describe a recent effort to better conceptualize the proof for Arnold diffusion. Combining Mather's theory and classical hyperbolic methods, we define special cohomology classes called Aubry-Mather type, where each such cohomology is connected to a nearby one for a "residue perturbation" of the Hamiltonian. The question of Arnold diffusion then reduces to the question of finding large connected components of such cohomologies. This is a joint work with Vadim Kaloshin.