PrefaceA remark on notationAcknowledgments Chapter 1 Expository Articles1.1 Dvir's proof of the finite field Kakeya conjecture1.2 The Black-Scholes equation1.3 Hassell's proof of scarring for the Bunimovich stadium1.4 What is a gauge?1.5 When are eigenvalues stable?1.6 Concentration compactness and the profile decomposition1.7 The Kakeya conjecture and the Ham Sandwich theorem1.8 An airport-inspired puzzle1.9 A remark on the Kakeya needle problem Chapter 2 The Poincare Conjecture2.1 Riemannian manifolds and curvature2.2 Flows on Riemannian manifolds2.3 The Ricci flow approach to the Poincare conjecture2.4 The maximum principle, and the pinching phenomenon2.5 Finite time extinction of the second homotopy group2.6 Finite time extinction of the third homotopy group, I2.7 Finite time extinction of the third homotopy group, II2.8 Rescaling of Ricci flows and k-non-collapsing2.9 Ricci flow as a gradient flow, log-Sobolev inequalities, and Perelman entropy2.10 Comparison geometry, the high-dimensional limit, and the Perelman reduced volume2.11 Variation of L-geodesics, and monotonicity of the Perelman reduced volume2.12 k-non-collapsing via Perelman's reduced volume2.13 High curvature regions of Ricci flow and k-solutions2.14 Li-Yau-Hamilton Harnack inequalities and k-solutions2.15 Stationary points of Perelman's entropy or reduced volume are gradient shrinking solitons2.16 Geometric limits of Ricci flows, and asymptotic gradient shrinking solitons2.17 Classification of asymptotic gradient shrinking solitons2.18 The structure of k-solutions2.19 The structure of high-curvature regions of Ricci flow2.20 The structure of Ricci flow at the singular time, surgery, and the Poincare conjecture Bibliography Index