The previous talk in the series is here. The next talk in the series is here.
A line in 𝔽pn is a set of the form {x+ty: t=0,1,…,p-1}. We call y the direction of the line. How small can a subset A of 𝔽pn be if it contains at least one line in every direction? This question can be regarded as a combinatorial version of the famous Kakeya problem, which is still unsolved. It too was open for a number of years, until in 2008 Zeev Dvir discovered an astonishingly short proof that used simple facts about polynomials in n variables. The proof is presented in full here.
This video was produced by Tim Gowers as part of his Part III course at the University of Cambridge. Printed notes for this course are available here.