This mathematics research project by Jean Bourgain is focused on harmonic analysis problems related to spectral theory. A first line of research has to do with the behavior of eigenfunctions of compact manifolds at high energy, with focus on the "simplest model" of the flat tori. In this setting, the eigenfunctions are explicit but nevertheless many of their properties remain conjectural. Bourgain intends to explore further the various moment inequalities which these eigenfunctions are supposed to obey and which are essential to issues in the theory of Schrodinger operators for instance, in particular to control theory. There is by now a large array of methods involved. In low dimension, number theory plays a key role. Input from elliptic curve theory (in 2D), distributional properties of lattice points on spheres and Siegel's mass formula led to new insights that deserve further study. In high dimension, recent progress came jointly with breakthroughs in the theory of oscillatory integral operators. The challenge is to establish uniform estimates on higher moments of the normalized eigenfunctions and some of Bourgain's recent work comes tentatively close to this. The interaction with other fields makes this research particularly stimulating. Spectral theory in Lie groups offers a different problematic and panorama of conjectures. Bourgain will continue research on spectral gaps, one of the central themes with many applications to number theory, mathematical physics and theoretical computer science.
This mathematics research project by Jean Bourgan is in the area of harmonic analysis with a focus on the notion of "spectral gap": such notion is known to have broad inter-disciplinary significance, ranging from pure mathematics to computer science, evolutionary biology and solid state physics. One can cite its relevance to the theory of tilings and quasi-crystals, quantum computation, error correcting codes, transport in inhomogeneous media, to mention a few. Much of Bourgain's past work with various collaborators has to do with elaborating a general framework enabling to prove the existence of spectral gaps. Bourgain will also investigate wave interactions; this problem lies at the heart of the study of solutions of many differential equations from physics and engineering. These solutions are roughly speaking obtained by superposition of elementary harmonics which collective effect obey deep mathematical principles. In many important examples, for instance in the theory of Schrodinger operators, this theory is still far from completely established. Bourgain will focus on some of the main conjectures on the behavior of eigenstates at high energy and the related aspects. Striking advances came from other mathematical areas, such as dynamics and number theory, offering new perspective that deserve further exploration. While the advances are undeniable and several conjectural phenomena can now be justified, there remain many challenges in known and less known territory which Bourgain will investigate through this project.
