Fields Notes 17:3 - Page 10

FOCUS PROGRAM Nonlinear Dispersive Partial Differential Equations and Inverse Scattering From July 31 to August 23, 2017, the Fields Institute Focus Program on Nonlinear Dispersive Partial Differential Equations and Inverse scattering brought together specialists in completely integrable systems, inverse scattering, and partial differential equations. The centerpiece of the program was the series of three Coxeter lectures delivered by Percy Deift. Deift began his lectures by describing the structure underlying the integrability of the defocusing nonlinear Schrödinger equation, and after a tour through other topics ranging from Hamiltonian systems to random matrix theory, he concluded his lectures by proposing a working definition of what it means for a problem to be integrable. The lectures drew a large audience including researchers in nearby areas, graduate students, and researchers in tangentially-related areas. me they already know how to do it from the summer school.” During the following two weeks, invited lecturers painted a compelling picture of current research on completely integrable and PDE techniques in dispersive nonlinear waves. Highlights of the first workshop week included lectures by Daniel Tataru and Rowan Killip on derived conserved quantities for the KdV and NLS equations to initial data in rough Sobolev spaces, dramatically increasing the reach of completely integrable methods. In the second week, Adrian Nachman announced a dramatic and far-reaching extension of the solution of the Cauchy problem for the defocusing Davey-Stewartson II equation, joint with his student Idan Regev and Daniel Tataru. Their work combines inverse scattering techniques with deep results in harmonic analysis including new fractional integral estimates and new L 2 -boundedness theorems for pseudo differential operators. Both these results and many others presented at the conference underscore the timeliness of its subject and the importance of bringing researchers in these communities together. The Focus Program began with one-week a summer school for graduate students and early career researchers which developed important background material and described the landscape of integrable systems techniques in dispersive partial differential equations. Some lecturers also emphasized new phenomena and applications. Examples include the lectures of Walter Craig on interacting vortices and the presentation of Patrick Gérard on the integrability of the cubic Szegö equation and the growth of high Sobolev norms in its solutions. One researcher wrote afterwards: “I found it to be a very beneficial meeting. My grad students were also very happy with the summer school. Several times now I have started explaining something to them, but they tell 10 Participants of the Summer School