The previous talk in the series is here. The next talk in the series is here.
The sensitivity conjecture was an important and long-standing conjecture in theoretical computer science that had been reduced to the following combinatorial question: let X be a set of 2n-1+1 vertices in the n-dimensional Hamming cube {0,1}n (where two sequences are joined by an edge if they differ in exactly one coordinate) and let G be the subgraph induced by X. How small can the maximum degree of G be? A 1988 paper of Chung, Füredi, Graham and Seymour obtained a lower bound of around log(n) and an upper bound of around √n. The question was which of these was nearer to the truth, the conjecture being that the upper bound was closer.
In July 2019, Hao Huang solved the conjecture with an astonishingly short and simple proof, which I present in full in this video.
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.