Pure PGR Seminar - On the Kakeya Conjecture, Abi Linton (Southampton) Seminar

Event details

In 1917, Soichi Kakeya posed the problem of finding a planar domain with the smallest area such that a needle of unit length could be rotated by 180 degrees within the domain. The obvious example of such a set (called a Kakeya needle set) is the unit ball, with area π/4. However, this is certainly not the Kakeya needle set with smallest area, and in fact it was shown by Besicovitch in 1928 that such sets could have arbitrarily small measure. This problem paved the way towards the idea of Kakeya sets – sets containing a unit line segment in each direction – and the conjecture that a Kakeya set in Rn has (Hausdorff) dimension n. To this day, there still has been no full response to this conjecture, but in this talk I will give a short overview of the problem and some of the more recent results.