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

这项研究的核心是一个重要的未解决的数学问题，称为Kakeya猜想。这个猜想起源于20世纪20年代，在过去的30年里吸引了数学家的极大兴趣，这是由于与数学和数学物理的不同分支出现了意想不到的和根本的联系。各种形式的猜想关注的是线段族（单位长度，在三维空间中指向不同方向）可以重新排列的程度，以便它们共同占据非常小的空间。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，贝西科维奇证明，无论从哪个族开始，总是可以找到一种排列，其线段所占的总空间为零。一种流行的形式的挂谷猜想指出，虽然这种安排可以很小，就其体积而言，他们必须然而，尽可能大的所谓fractal dimension .Very recently a new approach to problem of this type has been designed，导致近分辨率的某些（所谓的多线性）类似的挂谷猜想。这种方法是基于这样的发现，即当任何给定族中的线段同时滑向原点时，某些量（与分形维数有关）会增加。所提出的研究的目的是开发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，提出了类似的单调性为基础的方法，在纯数学和数学物理的各种中心未解决的问题。

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.