Warm up problem: Prove that no finite collection of convex heptagonal tiles can be used to tile the plane. (we’re not assuming the tiling is periodic. For example, below is a nice picture of a way to tile the plane with finitely many (two in this case) convex quadrilateral tiles).
Actual talk: Small cancellation theory! Random groups! A proof that “the generic group is a Burnside group” (finitely generated, infinite, but with every element finite order) (yikes!) (!) Annnnnd, finally, how these things are all secretly related. Including the warm-up problem.
[reference: other than the warm-up, most of the content of this talk comes from Ghys’ excellent paper “Groupes Aléatoires d’après Misha Gromov”]