Contents Preface vii A remark on notation viii Acknowledgments viii Chapter 1. Expository articles 1 §1.1. An explicitly solvable nonlinear wave equation 2 §1.2. Infinite fields, finite fields, and the Ax-Grothendieck theorem 7 §1.3. Sailing into the wind or faster than the wind 12 §1.4. The completeness and compactness theorems of first-order logic 20 §1.5. Talagrand’s concentration inequality 35 §1.6. The Szemer´ edi-Trotter theorem and the cell decomposition 42 §1.7. Benford’s law, Zipf’s law, and the Pareto distribution 48 §1.8. Selberg’s limit theorem for the Riemann zeta function on the critical line 59 §1.9. P = NP , relativisation, and multiple-choice exams 68 §1.10. Moser’s entropy compression argument 74 §1.11. The AKS primality test 82 §1.12. The prime number theorem in arithmetic progressions, and dueling conspiracies 87 §1.13. Mazur’s swindle 105 §1.14. Grothendieck’s definition of a group 108 §1.15. The “no self-defeating object” argument 116 v

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