The main focus of this introductory lecture is the fascinating interplay between ergodic theory, Ramsey theory and Diophantine analysis.
Ergodic theory has its roots in statistical and celestial mechanics and studies such phenomena as recurrence and uniform distribution of orbits.
Ramsey theory, a branch of combinatorics, is concerned with the phenomenon of preservation of highly organized structures under finite partitions.
Diophantine analysis concerns itself with integer and rational solutions of systems of polynomial equations.
We will start with some examples which demonstrate the usefulness of the ergodic approach to combinatorics and number theory. The discussion will naturally lead us to some fascinating recent developments such as the celebrated Green-Tao theorem on arithmetic progressions in primes. We will conclude by formulating and discussing some natural open problems.