Table of Contents:
  • Introduction
  • Part 1. Basics. Background material
  • Dynamical background
  • Rotations
  • Group extensions
  • Part 2. Cubes. Cubes in an algebraic setting
  • Dynamical cubes
  • Cubes in ergodic theory
  • The structure factors
  • Part 3. Nilmanifolds and nilsystems. Nilmanifolds
  • Nilsystems
  • Cubic structures in nilmanifolds
  • Factors of nilsystems
  • Polynomials in nilmanifolds and nilsystems
  • Arithmetic progressions in nilsystems
  • Part 4. Structure theorems. The ergodic structure theorem
  • Other structure theorems
  • Relations between consecutive factors
  • The structure theorem in a particular case
  • The structure theorem in the general case
  • Part 5. Applications. The method of characteristic factors
  • Uniformity seminorms on ℓ[infinity]
  • Multiple correlations, good weights, and anti-uniformity
  • Inverse results for uniformity seminorms and applications
  • The comparison method.