Contents

Preface ix

A remark on notation xi

Acknowledgments xii

Chapter 1. Expository Articles 1

§1.1. Quantum mechanics and Tomb Raider 1

§1.2. Compressed sensing and single-pixel cameras 10

§1.3. Soft analysis, hard analysis, and the finite convergence

principle 17

§1.4. The Lebesgue differentiation theorem and the Szemer´edi

regularity lemma 30

§1.5. Ultrafilters, non-standard analysis, and epsilon management 38

§1.6. Dyadic models 57

§1.7. “Math doesn’t suck”, and the Chayes-McKellar-Winn theorem 69

§1.8. Nonfirstorderisability 79

§1.9. Amplification, arbitrage, and the tensor power trick 83

§1.10. The crossing number inequality 94

§1.11. Ratner’s theorems 103

§1.12. Unipotent elements of the Lorentz group, and conic sections 110

§1.13. The Jordan normal form and the Euclidean algorithm 118

§1.14. John’s blowup theorem for the non-linear wave equation 123

§1.15. Hilbert’s Nullstellensatz 130

§1.16. The Hahn-Banach theorem, Menger’s theorem, and Helly’s

theorem 139

vii