Fourier methods in ergodic theory and combinatorics
We begin by illustrating a difference between norm and pointwise convergence by proving respectively von Neuman’s mean ergodic theorem and Birkhoff’s pointwise ergodic theorem. Our ultimate goal will be to build the necessary Fourier analytic and number theoretic tools needed to discuss Bourgain’s pointwise ergodic theorem along polynomials orbits. The second part of the course will cover some aspects of Gowers’ norms — the basics of the so-called higher order Fourier analysis — in the context of Gowers’ proof of Szemerédi’s theorem as well as Peluse and Prendiville theory.
No textbook
Measure Theory