Dynamical Approaches in Number Theory - TCC module

Starting with Furstenberg's ergodic theoretic proof of Szemeredi's theorem in 1977, a rich story of cross-pollination of ideas between ergodic theory, multiplicative number theory and additive combinatorics developed. This course will cover some of the highlights in this story and the connections between them. Topics covered, depending on time and students' interest, may include

The development of uniformity seminorms in ergodic theory paralleling Gowers norms in additive combinatorics.
The Green-Tao theorem that the primes contain arbitrarily long arithmetic progressions.
The subsequent program to study linear equations in primes, and how it required an understanding of the behaviour of certain special dynamical systems (nilsystems).
Sarnak's influential conjecture on Mobius orthogonality and the relation with an older conjecture of Chowla.
The special case of Chowla's conjecture proved by Tao, the relation with Erdos' discrepancy problem, and the relation with Host-Kra uniformity of multiplicative functions.
Frantzikinakis and Host's extension of Green and Tao's work, showing that aperiodic multiplicative functions are Gowers uniform, and how this connects to the problem of partition regularity of Pythagorean triples.
The "dynamical prime number theorem" of Bergelson and Richter, which led to a new elementary proof of the classical prime number theorem.
Infinite sumsets in sets of positive density and in the primes.