Jean Bourgain

IBM von Neumann Professor, School of Mathematics

March 23, 2015

Decoupling inequalities in harmonic analysis permit to bound the Fourier transform of measures carried by hyper surfaces by certain square functions defined using the geometry of the hyper surface. The original motivation has to do with issues in PDE, such as smoothing for the wave equation and Strichartz inequalities for the Schrodinger equation on tori. It turns out however that these decoupling inequalities have surprizing number theoretical consequences,on which we will mainly focus. They include new bounds for the number of integral solutions to certain diagonal systems of polynomial equations and mean value theorems relevant to bounding exponential sums and the zeta function! In particular we make some further progress towards the Lindelof hypothesis using the Bombieri-Iwaniec method.