Abstract: 

Over the past 20 years the large data theory of nonlinear Hamiltonian evolution equations has made substantial progress. On the one hand, concentration compactness methods transformed scattering theory, beginning with the work by Bahouri, Gerard ’98 and Kenig, Merle ’05.  On the other hand invariant manifolds, in particular center stable manifolds near the ground state soliton, have shed light on the dynamics of nonlinear wave equations.  In ’21  Jendrej and Lawrie settled the soliton resolution conjecture for equivariant wave maps in continuous times. We will review some of this work. 

Abstract: 

I will discuss a new methodology for proving small data scattering for the nonlinear Schrödinger equation, which avoids the use of Strichartz estimates, and uses instead methods from microlocal analysis.  This methodology is flexible and can in principle be applied to massive wave propagation as in the Klein-Gordon or massive Dirac equations.  This is joint work with Andrew Hassell and Sean Gomes and with Dean Baskin and Moritz Doll.