Abstract:

In 1917, S. Kakeya posed the following problem: what is the least amount of area in the plane required to rotate a unit needle (a unit line segment) by 180 degrees in the plane? In 1928, A. Besicovitch gave the surprising answer that one could rotate a needle using arbitrarily small area. While this problem (together with more quantitative versions, where the needle has some small thickness delta instead of zero thickness) are now completely solved in two dimensions, the analogous Kakeya problem in higher dimensions has not been fully solved, although there has been much recent progress. In this talk we describe some of these results and how they have applications to other areas of mathematics such as harmonic analysis and PDE.