The nonlinear Schrödinger equation on tori: Integrating harmonic analysis, geometry, and probability

Abstract

The field of nonlinear dispersive and wave equations has undergone significant progress in the last twenty years thanks to the influx of tools and ideas from nonlinear Fourier and harmonic analysis, geometry, analytic number theory and most recently probability, into the existing functional analytic methods. In these lectures we concentrate on the semilinear Schrödinger equation defined on tori and discuss the most important developments in the analysis of these equations. In particular, we discuss in some detail recent work by J. Bourgain and C. Demeter proving the ℓ2 decoupling conjecture and as a consequence the full range of Strichartz estimates on either rational or irrational tori, thus settling an important earlier conjecture by Bourgain.