Georgia Institute of Technology College of Sciences

The Parameterization Method is a functional analytic framework for studying invariant manifolds such as stable/unstable manifolds of periodic orbits and invariant tori. This talk will focus on numerical applications such as computing manifolds associated with long periodic orbits, and computing periodic invariant circles (manifolds consisting of several disjoint circles mapping one to another, each of which has an iterate conjugate to an irrational rotation). I will also illustrate how to combine Automatic Differentiation with the Parameterization Method to simplify problems with non-polynomial nonlinearities.

SGSW is a third level specialization of Navier-Stokes (via Boussinesq, then Semi-Geostrophic), and it accurately describes large-scale, rotation-dominated atmospheric flow under the extra-assumption that the horizontal velocity of the fluid is independent of the vertical coordinate. The Cullen-Purser stability condition establishes a connection between SGSW and Optimal Transport by imposing semi-convexity on the pressure; this has led to results of existence of solutions in dual space (i.e., where the problem is transformed under a non-smooth change of variables). In this talk I will present recent results on existence and weak stability of solutions in physical space (i.e., in the original variables) for general initial data, the very first of their kind. This is based on joint work with M. Feldman (UW-Madison).

We establish global well-posedness and scattering for the cubic Dirac equation for small data in the critical space. The theory we develop is the Klein-Gordon counterpart of the Wave Maps / Schroedinger Maps theory. This is joint work with Sebastian Herr.

Many conservative PDE models can be written in a Hamiltonian form. They include Euler equations in fluids, Vlasov models for plasmas and galaxies, ideal MHD for plasmas, Gross–Pitaevskii equation for superfluids and Bose-Einstein condensates, and various water wave models (KDV, BBM, KP, Boussinesq systems etc). I will describe some dynamical problems of these models, from a more unifying point of view by using their Hamiltonian forms. They include: stability/instability of coherent states (steady solution, traveling waves, standing waves etc.), invariant manifolds near unstable states, and inviscid and enhanced damping in fluids and plasmas. It is a topic course that will be taught in the fall.