I will describe recent advances in the Pugh-Shub stable ergodicity theory for partially hyperbolic diffeomorphisms. In particular, I consider two "competing" methods to show that a given partially hyperbolic diffeomorphism is stably ergodic (i.e., it is ergodic along with any of its sufficiently small perturbations). One of them relates the problem to to the global estimates of the action of the system along its central direction while another one deals with a more delicate estimates using Lyapunov exponents in the central direction.

The surface of a common donut is called a "torus." While it is a nice geometric object, it is
reasonable to wonder, "So What!" The purpose of this lecture is to illustrate the muscle power
of mathematics by showing how the mathematics of even this simple object explains so many mysteries that occur in everyday life and in our society. Indeed, one example (should time permit) even explains why the US House of Representatives has 435 seats.
*Pizza and soda served!