The Kakeya needle problem asks: is it possible to rotate a unit needle in the plane using an arbitrarily small amount of area? The answer is known to be yes, but analogous problems in higher dimensions (where one now seeks to find sets of small dimension that contain line segments in each direction) remain open, and are related to many other important conjectures in harmonic analysis, PDE, and even number theory and computer science. There have been many partial results on this problem, using such diverse techniques as geometric measure theory, incidence combinatorics, additive combinatorics, and PDE; more recently, algebraic geometry, and even algebraic topology have been used to obtain new breakthroughs in this subject. We will discuss many of these new developments in this talk.

[This is another of Professor Tao's Clay-Mahler Lectures carried by Access Grid from Macquarie University. Please note the room.]