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

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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.