The principal investigator, in collaboration with Jean Bourgain, has recently created a new set of tools that can successfully address a wide range of problems in the fields of number theory and partial differential equations. Until recently, many of these problems seemed unrelated. In light of his work with Bourgain, these problems are now understood as part of a more general theory that the two call "decoupling." The methods pertain to the field of modern harmonic analysis, a natural framework that allows for the formulation of a general enough theory. This project seeks to enlarge the range of applicability of decouplings, with some high-value targets in sight. A surprising feature of the research is that it removes certain restrictions on frequencies that were thought to be necessary in earlier work. In particular, the old requirement that frequencies have integer coordinates is replaced with the weaker assumption that sufficient spatial separation exists between frequencies. It is expected that the tools that will be developed will be accessible and useful to a large part of the mathematical community.
Decouplings are certain generalizations of the Littlewood--Paley theory in the presence of curvature. The principal investigator's progress in pursuing the line of research related to this subject has relied hitherto on the interplay between multilinear and linear multiscale analysis. He has successfully addressed the case when the relevant manifold is a hypersurface with nonzero Gaussian curvature. He now proposes to develop the optimal decoupling theory for nondegenerate curves. Such a theory has the potential to achieve almost unprecedented applications of harmonic analysis to number theory. One notable example is the resolution of Vinogradov's mean value theorem. There is an interesting related circle of problems for the cone. The fact that it has zero Gaussian curvature poses a new level of difficulty that will most certainly require new ideas. Understanding the cone is part of a more ambitious project that will aim at understanding the decoupling theory for real analytic surfaces. There are further important related questions that remain to be explored, in connection with various restriction theorems and the Kakeya conjectures.
