A video of the lecture is available here.

Stated, illustrated, and proved Brouwer's fixed point theorem. Then used it to show that any nxn matrix of positive real numbers has at least one eignevector, and that this eigenvector can be chosen so that its entries are all non-negative.

Then showed a Holywood clip on Nash Equilibria which includes a shot of John Nash's PhD thesis in which he uses Brouwer's theorem to prove "the existence in any game of at least one equilibrium point".