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