Additive Nonsmoothing, the Kakeya Problem, and Fluid Mechanics

自18世纪以来，人们一直在研究流体力学方程。然而，它们的规律性问题还没有完全被理解。一个三维的、不可压缩的、非受迫的粘性流动能够在有限时间内平稳地发展出奇点吗？这个问题是粘土千禧年的问题之一，它的解决方案同样神秘，也同样重要。在调和分析的其他部分，也出现了类似的问题。沿着这些线最著名的问题之一是Kakeya问题：某个欧几里得空间中的集合是否可以包含每个方向的单位线段，但其维度比周围空间低？这个问题与几个数学领域的重要问题有关，这样一个集合的例子可以提供对波动方程和薛定谔方程解的奇异性的洞察。Kakeya问题与加性组合数学领域中的其他问题也有许多相同的特征，该领域的中心问题是，一个产生许多加性方程的数字集合是否总是该集合中某种加性结构的指示，或者可能作为一种恶意的重合而出现。这个项目旨在对所有这些问题中可能出现的巧合进行分类和解释。首先，首席研究员将研究平面内的准地转方程是否会出现奇点。他希望通过研究简化的模型问题来做到这一点，在这个问题中，所有的行动都发生在有限维李群中，有限维李群是保体积微分同胚类的喷射群之一。其次，他想证明三维Kakeya集的Hausdorff维度严格大于2.5。他希望通过尽可能少地利用集合的结构特性来实现这一点。早期的工作在很大程度上利用了结构性质来获得良好的Minkowski-维界；预计较轻的方法将有效地解决Hausdorff维问题。最后，主要研究者试图发展一个好的理论来寻找可加非光滑集的结构部分。他将尝试用多项式Freiman-Ruzsa猜想的含义来扩展早期关于封套大小的工作。

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.