New approaches to central problems in euclidean harmonic analysis and geometric combinatorics

这项拟议研究的核心是一个重要的尚未解决的数学问题，即Kakeya猜想。这个猜想起源于20世纪20年代的S，在过去的30年里引起了数学家们的极大兴趣，因为数学和数学物理的不同分支之间出现了意想不到的基本联系。各种形式的猜想涉及(单位长度的，在三维空间中指向不同方向的)线段族可以被重新排列的程度，以使它们共同占据非常少量的空间。在此重新排列过程中，重要的是所涉及的线段不以任何方式旋转。也许与直觉相反的是，Besicovitch表明，无论从哪个家族开始，总能找到一种线段占用的总空间体积为零的排列。Kakeya猜想的一种流行形式指出，虽然这种排列的体积可以很小，但它们必须在所谓的分维方面尽可能大。最近，人们设计了一种解决这类问题的新方法，导致了Kakeya猜想的某些(所谓的多线性)类似物的近似解析。这种方法是基于这样一种发现，即当任何给定族中的线段同时滑到原点时，某些量(与分维相关)增加。这项研究的目的是发展这种基于单调性的方法，并调查它在多大程度上可以用来建立原始的经典Kakeya猜想及其现代变体。此外，对于纯数学和数学物理中的各种中心未解决问题，也提出了类似的基于单调性的方法。

At the heart of the proposed research is an important unsolved mathematical problem known as the Kakeya conjecture. This conjecture, which originated in the 1920's, has attracted great interest from mathematicians over the last 30 years, due to the emergence of unexpected and fundamental connections with different branches of mathematics and mathematical physics. The various forms of the conjecture concern the extent to which families of line segments (of unit length, and pointing in different directions in three dimensional space) may be rearranged so that collectively they occupy a very small amount of space. In this rearrangement process it is important that the line segments involved are not rotated in any way. Perhaps rather counter-intuitively, it was shown by Besicovitch that, no matter which family one starts off with, an arrangement can always be found for which the total space occupied by the line segments has zero volume. A popular form of the Kakeya conjecture states that although such arrangements can be small in terms of their volume, they must however be as large as possible in terms of their so-called fractal dimension .Very recently a new approach to problems of this type has been devised, leading to the near resolution of certain (so-called multilinear ) analogues of the Kakeya conjecture. This approach is based on the discovery that certain quantities (related to the fractal dimension) increase as the line segments in any given family simultaneously slide to the origin . The purpose of the proposed research is to develop this monotonicity-based approach and investigate the extent to which it may be used to establish the original classical Kakeya conjecture, and its modern variants. Furthermore, similar monotonicity-based approaches to a variety of central unsolved problems in pure mathematics and mathematical physics are proposed.