The equations of fluid mechanics have been studied since the eighteenth century. Nevertheless, the problem of their regularity has not yet been fully understood. Can a three-dimensional, incompressible, unforced viscous flow that starts smoothly develop a singularity in finite time? This question, one of the Clay Millennial Problems, is as mysterious as its solution would be important. In other parts of harmonic analysis, similar questions arise. One of the most famous problems along these lines is the Kakeya problem: Can a set in some Euclidean space contain a unit line segment in every direction but be of lower dimension than the ambient space? This question is connected to important questions in several areas of mathematics, and an example of such a set could provide insight into singularities of solutions to wave equations and Schrodinger equations. The Kakeya problem also shares many features with other problems in the field of additive combinatorics, the field whose central question is whether a set of numbers giving rise to many additive equations is always an indication of some additive structure in the set or can occur as a sort of malevolent coincidence. This project aims to classify and explain the kind of coincidences that can occur in all these problems.
This project takes aim at three specific problems. First, the principal investigator will study whether the quasi-geostrophic equation in the plane develops singularities. He hopes to do this by studying simplified model problems in which all of the action takes place in a finite-dimensional Lie group, one of the jet groups of the class of volume-preserving diffeomorphisms. Second, he would like to show that a Kakeya set in three dimensions has a Hausdorff dimension that is strictly greater than two and a half. He hopes to do this by exploiting as few structural properties of the set as possible. Earlier work used structural properties heavily to obtain good Minkowski-dimension bounds; it is anticipated that a lighter approach will be effective for the problem of Hausdorff dimension. Finally, the principal investigator seeks to develop a good theory for finding structured parts of additively nonsmoothing sets. He will attempt to extend earlier work on the size of capsets with implications for the polynomial Freiman-Ruzsa conjecture.
