GÖRAN
GUSTAFSSON Lectures in Mathematics
Gigliola Staffilani Abstracts: 

Lecture 1: The many faces of dispersive equations In recent years great progress has been made in the study of dispersive and wave equations. Over the years the toolbox used in order to attack highly nontrivial problems related to these equations has developed to include a variety of techniques from Fourier and harmonic analysis, analytic number theory, math physics, dynamical systems, probability and symplectic geometry. In this talk I will introduce a variety of problems connected with dispersive and wave equations, such as the derivation of a certain nonlinear Schrodinger equations from a quantum manyparticles system, periodic Strichartz estimates, the concept of energy transfer, the invariance of a Gibbs measure associated to an infinite dimension Hamiltonian system and nonsqueezing theorems for such systems when they also enjoy a symplectic structure. Coffee is served between 12.45 and 13.15 outside the lecture hall.
Lecture 2: Strichartz
Estimates: The dance between analytic number theory and harmonic analysis In this second lecture I will zoom into the rich mathematics behind the proofs of Strichartz estimates. As mentioned in the first lecture these estimates are fundamental in order to prove wellposedness of dispersive initial value problems by assuming only boundedness of the mass and energy of the initial data. In recent years, while looking for a proof of Strichartz estimates in the periodic setting, a much deeper connection than previously anticipated has been found between analytic number theory and harmonic analysis. I will describe in general terms this connection throughout the lecture.
Lecture 3: On long time instability of the zero solution to certain Nonlinear
Schrodinger equations The aim of this third lecture is to analyze the long time instability of the zero solution of the 2D periodic defocusing cubic NLS, while outlining the differences between the dynamics on rational versus irrational tori. In particular we will analyze the structure of the resonance sets in these two different set ups. This instability can also be related to well known phenomena such as energy transfer and forward cascades. 

Sponsored by the Göran Gustafsson Foundation 20180508 