Volume 162

## This issue of the Annals of Mathematics is dedicated to Armand Borel, 1923–2003.

| From volume 162-1

## Iwasawa’s Main Conjecture for elliptic curves over anticyclotomic $\Bbb Z_p$-extensions

Pages 1-64 by Massimo Bertolini, Henri Darmon | From volume 162-1

## Stable ergodicity of certain linear automorphisms of the torus

Pages 65-107 by Federico Rodriguez Hertz | From volume 162-1

## Well-posedness for the motion of an incompressible liquid with free surface boundary

Pages 109-194 by Hans Lindblad | From volume 162-1

## Geometrization of 3-dimensional orbifolds

Pages 195-290 by Michel Boileau, Bernhard Leeb, Joan Porti | From volume 162-1

## Sharp local well-posedness results for the nonlinear wave equation

Pages 291-366 by Hart F. Smith, Daniel Tataru | From volume 162-1

## Universal bounds for hyperbolic Dehn surgery

Pages 367-421 by Craig Hodgson, Steven P. Kerckhoff | From volume 162-1

## The diameter of the isomorphism class of a Banach space

Pages 423-437 by William B. Johnson, Edward Odell | From volume 162-1

## On the hardness of approximating vertex cover

Pages 439-485 by Irit Dinur, Samuel Safra | From volume 162-1

## The Schrödinger propagator for scattering metrics

Pages 487-523 by Andrew Hassell, Jared Wunsch | From volume 162-1

## The perimeter inequality under Steiner symmetrization: Cases of equality

Pages 525-555 by Miroslav Chlebík, Andrea Cianchi, Nicola Fusco | From volume 162-1

## Einstein metrics on spheres

Pages 557-580 by Charles P. Boyer, Krzysztof Galicki, János Kollár | From volume 162-1

## On fusion categories

Pages 581-642 by Pavel Etingof, Dmitri Nikshych, Viktor Ostrik | From volume 162-2

## On metric Ramsey-type phenomena

Pages 643-709 by Yair Bartal, Nathan Linial, Manor Mendel, Assaf Naor | From volume 162-2

## A new application of random matrices: Ext$(C^*_{\mathrm{red}}(F_2))$ is not a group

Pages 711-775 by Uffe Haagerup, Steen Thorbjørnsen | From volume 162-2

## A resolution of the K(2)-local sphere at the prime 3

Pages 777-822 by Paul Goerss, Hans-Werner Henn, Mark Mahowald, Charles Rezk | From volume 162-2

## The classification of torsion endo-trivial modules

Pages 823-883 by Jon F. Carlson, Jacques Thévenaz | From volume 162-2

## Inverse spectral problems and closed exponential systems

Pages 885-918 by Miklós Horváth | From volume 162-2

## Discreteness of spectrum and positivity criteria for Schrödinger operators

Pages 919-942 by Vladimir Maz'ya, Mikhail Shubin | From volume 162-2

## Hypoellipticity and loss of derivatives (with an Appendix by Makhlouf Derridj and David S. Tartakoff)

Pages 943-986 by Joseph J. Kohn, Makhlouf Derridj, David S. Tartakoff | From volume 162-2

## Sur le théorème de Paley–Wiener d’Arthur

Pages 987-1029 by Patrick Delorme | From volume 162-2

## The density of discriminants of quartic rings and fields

Pages 1031-1063 by Manjul Bhargava | From volume 162-2

## A proof of the Kepler conjecture

Pages 1065-1185 by Thomas C. Hales | From volume 162-3

## Curve shortening and the topology of closed geodesics on surfaces

Pages 1187-1241 by Sigurd B. Angenent | From volume 162-3

## Bilipschitz maps, analytic capacity, and the Cauchy integral

Pages 1243-1304 by Xavier Tolsa | From volume 162-3

## Cabling and transverse simplicity

Pages 1305-1333 by John B. Etnyre, Ko Honda | From volume 162-3

## The two possible values of the chromatic number of a random graph

Pages 1335-1351 by Dimitris Achlioptas, Assaf Naor | From volume 162-3

## Twisted Fermat curves over totally real fields

Pages 1353-1376 by Adrian Diaconu, Ye Tian | From volume 162-3

## Formation of singularities for a transport equation with nonlocal velocity

Pages 1377-1389 by Antonio Córdoba, Diego Córdoba, Marco A. Fontelos | From volume 162-3