Volume 169 Issue 1 – January 2009

The `Harder-Narasimhan trace’ and unitarity of the KZ/Hitchin connection: genus 0

Pages 1-39 by Trivandrum Ramakrishnan Ramadas

Regularity of flat level sets in phase transitions

Pages 41-78 by Ovidiu Savin

Existence of Engel structures

Pages 79-137 by Thomas Vogel

Stable manifolds for an orbitally unstable nonlinear Schrödinger equation

Pages 139-227 by Wilhelm Schlag

On Serre’s conjecture for 2-dimensional mod p representations of $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$

Pages 229-253 by Chandrashekhar Khare, Jean-Pierre Wintenberger

Holomorphic curves into algebraic varieties

Pages 255-267 by Min Ru

Finite groups of symplectic automorphisms of K3 surfaces in positive characteristic

Pages 269-313 by Igor Dolgachev, JongHae Keum

Fitting a $C^m$-smooth function to data, I

Pages 315-346 by Charles Fefferman, Bo'az Klartag