Under what conditions will a subset of the integers contain arithmetic progressions of arbitrary length? Szemeredi's theorem provides a partial answer to this question. In fact, as of 2004, we know that the primes are such a subset, due to the theorem of Green and Tao.
In this talk, I will present an ergodic-theoretic proof of Szemeredi's theorem, which is formally used to prove Green and Tao's result. The punchline is that ergodic theory, a field motivated by problems in statistical mechanics, has something deep to say about number theory. The talk will emphasize the transition from number theoretical questions to the theory of dynamical systems. Along the way, I will discuss several applications of ergodic theory to number theoretic statements: examples include results from Weyl's theory of exponential sums, a simple proof of Van der Waerden's theorem, and the deep results of Hindeman concerning the existence of IP-systems in subsets of the integers of upper banach density.
I will assume no background in ergodic theory, but a basic understanding of [point-set] topology (metrics, compactness) and set theory would be useful.
Lecture notes are available here.