The previous talk in the series is here. The next talk in the series is here.
Suppose you have n points and m lines in the plane. A point-line incidence is a pair (P,L) where P is one of the points and L is one of the lines. How many point-line incidences can there be, given m and n? This question is answered by the Szemerédi-Trotter theorem: the answer, up to a multiplicative constant, is whichever is largest out of m, n, and (mn)2/3. Here I present a beautifully simple proof due to László Székely. It is based on an inequality about the number of crossings there must be if you draw a graph with n vertices and m edges, which is proved by another very nice averaging argument. (The m here is not the same as the first m.) After the introduction, we look at the crossing-number inequality, then deducing the Szemerédi-Trotter theorem.
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.