The previous talk in the series is here. The next talk in the series is here.
Having presented some (but by no means all) of the basic theory of entropy, I give an application to the following problem: let G be a bipartite graph of density c with vertex sets X and Y; prove that if x and x‘ are random vertices of X and y and y‘ are n vertices of Y, then the probability that xyx‘y‘ is a (possibly degenerate) path is at least c3. This is surprisingly tricky to prove by elementary means, though it can be done. Using entropy one can give a very clean and conceptual proof, which can be generalized to prove other statements of a similar kind, all special cases of a famous conjecture of Sidorenko. The argument is due to Balázs Szegedy, and independently to David Conlon, Jeong Han Kim, Choongbum Lee, and Joonkyung Lee.
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.