BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 53, Number 1, January 2016, Pages 57–91 http://dx.doi.org/10.1090/bull/1516 Article electronically published on September 1, 2015 THE NONLINEAR SCHR ¨ ODINGER EQUATION ON TORI: INTEGRATING HARMONIC ANALYSIS, GEOMETRY, AND PROBABILITY ANDREA R. NAHMOD Abstract. The field of nonlinear dispersive and wave equations has under- gone 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 ana- lytic methods. In these lectures we concentrate on the semilinear Schr¨odinger 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 ℓ 2 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. 1. Introduction The nonlinear Schr¨ odinger equation plays an ubiquitous role as a model for dis- persive wave-phenomena in nature. Roughly speaking, dispersion means that when no boundary is present, waves of different wavelengths travel at different phase speeds: long wavelength components propagate faster than short ones. This is the reason why over time dispersive waves spread out in space as they evolve in time, while conserving some form of energy. This phenomenon is called broadening of the wave packet. Dispersive wave-phenomena should be contrasted with transport phenomena where all frequencies move at the same velocity or dissipative phenom- ena (heat equation) in which frequencies gradually taper to zero, that is they do not propagate. The nonlinear Schr¨ odinger equation serves as a mathematical model for the large class of so-called dispersive partial differential equations [1, 67]. It naturally arises in connection to a variety of different physical problems on flat space, tori, and other manifolds. One of them is nonlinear optics in a so-called Kerr medium where one considers electromagnetic waves in a material (e.g., glass fiber) whose time evolu- tion are governed by Maxwell’s equations on R 3 . The nonlinear Maxwell equations however have disparate scales, and understanding their dynamics is a difficult prob- lem. As a first attempt one looks for further simplifications: asymptotic methods then become useful. A natural ansatz is to write the electric field E as a Taylor series whose leading term is a small amplitude wave packet of the form (1.1) Apt, xqe ipξ 0 ¨x´ω 0 tq ` Apt, xqe ´ipξ 0 ¨x´ω 0 tq Received by the editors May 18, 2015. 2010 Mathematics Subject Classification. Primary 42-XX, 35-XX. The author gratefully acknowledges support from NSF grants DMS 1201443 and DMS 1463714. c 2015 American Mathematical Society 57