Multijoints and multilinear duality

Abstract

The joints problem is related to geometric questions at the heart of harmonic analysis. In three dimensions, a joint is a point of intersection of three lines that do not lie within a common plane. Given a collection of lines, one can ask how many joints those lines can form { this is the joints problem. \Duality" describes the relationship between two complementary problems that are distinct but logically equivalent. Any two such problems are said to be dual to one another. The problem that is dual to the joints problem looks to understand geometric properties which abstract on the conventional notion of volume.

Understanding this geometric problem is the aim of this thesis.