This past January, I gave a talk outlining the proof of Van der Waerden's theorem, a result in number theory, using a standard recurrence-finding tool from ergodic theory, the Multiple Birkhoff Recurrence Theorem. In the present talk I will use a stronger ergodic tool to prove a stronger number theoretical statement, Szemeredi's theorem in the integers (in the sense that Szemeredi implies Van der Waerden). This proof demonstrates the remarkable versatility of these ergodic tools in proving seemingly unassailable statements in number theory. Time permitting, I will outline the proof of another application of such ergodic tools to mathematics, the equidistribution of real polynomials modulo one.
I'll define everything needed, but a little measure theory and topology would be helpful background material.