Preface

  A remark on notation

  Acknowledgments

Chapter 1  Expository Articles

  1.1  Dvir's proof of the finite field Kakeya conjecture

  1.2  The Black-Scholes equation

  1.3  Hassell's proof of scarring for the Bunimovich stadium

  1.4  What is a gauge?

  1.5  When are eigenvalues stable?

  1.6  Concentration compactness and the profile decomposition

  1.7  The Kakeya conjecture and the Ham Sandwich theorem

  1.8  An airport-inspired puzzle

  1.9  A remark on the Kakeya needle problem

Chapter 2  The Poincare Conjecture

  2.1  Riemannian manifolds and curvature

  2.2  Flows on Riemannian manifolds

  2.3  The Ricci flow approach to the Poincare conjecture

  2.4  The maximum principle, and the pinching phenomenon

  2.5  Finite time extinction of the second homotopy group

  2.6  Finite time extinction of the third homotopy group, I

  2.7  Finite time extinction of the third homotopy group, II

  2.8  Rescaling of Ricci flows and k-non-collapsing

  2.9  Ricci flow as a gradient flow, log-Sobolev inequalities, and Perelman entropy

  2.10  Comparison geometry, the high-dimensional limit, and the Perelman reduced volume

  2.11  Variation of L-geodesics, and monotonicity of the Perelman reduced volume

  2.12  k-non-collapsing via Perelman's reduced volume

  2.13  High curvature regions of Ricci flow and k-solutions

  2.14  Li-Yau-Hamilton Harnack inequalities and k-solutions

  2.15  Stationary points of Perelman's entropy or reduced volume are gradient shrinking solitons

  2.16  Geometric limits of Ricci flows, and asymptotic gradient shrinking solitons

  2.17  Classification of asymptotic gradient shrinking solitons

  2.18  The structure of k-solutions

  2.19  The structure of high-curvature regions of Ricci flow

  2.20  The structure of Ricci flow at the singular time, surgery, and the Poincare conjecture

Bibliography

Index