Preface
A remark on notation
Acknowledgments
Chapter 1. Expository Articles
1.1. The blue-eyed islanders puzzle
1.2. Kleiner's proof of Gromov's theorem
1.3. The van der Corput lemma, and equidistribution on nilmanifolds
1.4. The strong law of large numbers
1.5. Tate's proof of the functional equation
1.6. The divisor bound
1.7. The Lucas-Lehmer test for Mersenne primes
1.8. Finite subsets of groups with no finite models
1.9. Small samples, and the margin of error
1.10. Non-measurable sets via non-standard analysis
1.11. A counterexample to a strong polynomial Freiman-Ruzsa conjecture
1.12. Some notes on "non-classical" polynomials in finite characteristic
1.13. Cohomology for dynamical systems
Chapter 2. Ergodic Theory
2.1. Overview
2.2. Three categories of dynamical systems
2.3. Minimal dynamical systems, recurrence, and the Stone-Cech compactification
2.4. Multiple recurrence
2.5. Other topological recurrence results
2.6. Isometric systems and isometric extensions
2.7. Structural theory of topological dynamical systems
2.8. The mean ergodic theorem
2.9. Ergodicity
2.10. The Furstenberg correspondence principle
2.11. Compact systems
2.12. Weakly mixing systems
2.13. Compact extensions
2.14. Weakly mixing extensions
2.15. The Furstenberg-Zimmer structure theorem and the Furstenberg recurrence theorem
2.16. A Ratner-type theorem for nilmanifolds
2.17. A Rather-type theorem for SL2(R) orbits
Chapter 3. Lectures in Additive Prime Number Theory
3.1. Structure and randomness in the prime numbers
3.2. Linear equations in primes
3.3. Small gaps between primes
3.4. Sieving for almost primes and expanders
Bibliography
Index
