Poincarés legacies: pages from year two of a mathematical blog Terence Tao Department of Mathematics, UCLA, Los Angeles, CA 90095 E-mail address: tao@math.ucla.edu To Garth Gaudry, who set me on the road; To my family, for their constant support; And to the readers of my blog, for their feedback and contributions. Contents Preface xi A remark on notation xiv Acknowledgments xiv Chapter 1. Expository articles 1 §1.1. The blue-eyed islanders puzzle 2 §1.2. Kleiner’s proof of Gromov’s theorem 3 §1.3. Dvir’s proof of the finite field Kakeya conjecture 11 §1.4. The van der Corput lemma, and equidistribution on nilmanifolds 16 §1.5. The strong law of large numbers 23 §1.6. The Black-Scholes equation 32 §1.7. Hassell’s proof of scarring for the Bunimovich stadium 44 §1.8. Tate’s proof of the functional equation 48 §1.9. The divisor bound 59 §1.10. What is a gauge? 64 §1.11. The Lucas-Lehmer test for Mersenne primes 86 §1.12. Finite subsets of groups with no fini
