This workshop will focus on recent developments in nonlinear elliptic equations and their interactions with differential geometry, Riemannian geometry and complex geometry. Topics will include the study of fully nonlinear elliptic problems which arise for example in the study of hessian equations for hypersurfaces, in complex Kahler geometry or in the prescription of the k-th symmetric functions of the eigenvalues of the Schouten tensor in conformal geometry and its analogue in CR geometry. It will also include the analysis of higher order elliptic equations which arise in affine geometry and conformal geometry. Lecture Schedule with Talk Abstracts SCHEDULE Monday, October 31 9:30 – 10:30 Gang Tian (Princeton Univ.), "Curvature estimate for 4-dimensional Einstein equation" 10:30 – 11:00 Morning Tea – Sixth Floor 11:00 – 12:00 Xiuxiong Chen (Univ. of Wisconsin), "On the Calabi functional in Kahler manifold" 12:00 – 1:00 Jeff Viaclovsky (MIT), “Volume Growth, Curvature Decay, and Critical Metrics” 1:00 – 2:30 Lunch 2:30 – 3:30 Xavier Cabre (ICREA-Univ. Politecnica Catalunya), "Regularity of radial minimizers and extremal radial solutions of semilinear elliptic equations" 3:30 – 4:00 Travel to 60 Evans Hall, UCB 4:10 – 5:10 (At 60 Evans Hall, UCB) Cristian Gutierrez (Temple University), “The Monge-Ampere equation: Overview and Recent Results” Tuesday, November 1 9:30 – 10:30 Richard Schoen (Stanford University), "Blowup issues for the Yamabe equation on high dimensional manifolds" 10:30 – 11:00 Morning Tea – Sixth Floor 11:00 – 12:00 Simon Brendle (Stanford University), "Global convergence of the Yamabe flow in dimension 6 and higher" 12:00 – 1:00 Olivier Druet (ENS Lyons), “The prescribed scalar curvature problem” 1:00 – 2:30 Lunch 2:30 – 3:30 Bernd Amman (Univ. Nancy), "The first Dirac eigenvalue in a conformal class" 3:30 – 4:00 Afternoon Tea – Sixth Floor 4:00 – 5:00 Xiaohua Zhu (Beijing University), “Kahler-Ricci flow and certain complex Monge-Ampere equation" Wednesday, November 2 9:30 – 10:30 Gerhard Huisken (Max Planck Inst. f. Gravitationsphysik), "Rigidity inequalities via inverse mean curvature flow" 10:30 – 11:00 Morning Tea – Sixth Floor 11:00 – 12:00 J. Cheng (Academica Sinica, Taipei), "The mean curvature equation in Pseudohermitian geometry" 12:00 – 1:00 Sophie Chen (Princeton University), “Boundary Estimates for Some Fully Nonlinear PDEs and Applications to Conformal Geometry” 1:00 – 2:30 Lunch 2:30 – 3:30 Qing Han (Univ. of Notre Dame), “Smooth local isometric embedding of surfaces with Gauss curvature changing sign monotonically across a curve” 3:30 – 4:00 Afternoon Tea – Sixth Floor 4:00 – 5:00 Nassif Ghoussoub (Univ. of British Columbia), "Selfdual variational principles for PDEs and evolution equations" Thursday, November 3 9:30 – 10:30 Bruno Franchi (Univ. of Bologna), "Intrinsic submanifolds of Heisenberg group" 10:30 – 11:00 Morning Tea – Sixth Floor 11:00 – 12:00 Joel Spruck (Johns Hopkins Univ.), "Complete hypersurfaces of constant curvature in Hyperbolic space with prescribed asymptotic boundary" 12:00 – 1:00 Refe Mazzeo (Stanford University), “Einstein metrics and higher rank geometry” 1:00 – 2:30 Lunch 2:30 – 3:30 Brian White (Stanford University), "Getting a handle on the helicoid" 3:30 – 4:00 Afternoon Tea – Sixth Floor 4:00 – 5:00 Xujia Wang (Australian National Univ.), “Fully nonlinear partial differential equations and applications in geometry” Friday, November 4 9:30 – 10:30 Luis Caffarelli (Univ. of Texas, Austin), "On homogenization and rates of convergence for random fully nonlinear equations" 10:30 – 11:00 Morning Tea – Sixth Floor 11:00 – 12:00 Alice Chang (Princeton Univ.), "Conformal invariants associated with a smooth measure" 12:00 – 1:00 Rod Gover (Univ. of Auckland) “Conformal Laplacian operators and Q-curvature on Einstein manifolds” 1:00 – 2:30 Lunch 2:30 – 3:30 Jie Qing (UC Santa Cruz), "“On semi-linear conformally invariant PDE” 3:30 – 4:00 Afternoon Tea – Sixth Floor 4:00 – 5:00 Fengbo Hang (Princeton & Michigan State Univ.), “On the higher order Q curvature on the Sphere”