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American Institute of Mathematics, Pasadena, California

This workshop, sponsored by AIM and the NSF, will be devoted to exploring the geometric and analytic regularity of fully nonlinear partial differential equations (PDE) arising from geometry and physics. Ever since Yau's solution to the Calabi conjecture, geometric PDEs, such as the complex Monge-Ampère equations, have been extensively studied from both the geometric and analytic perspective. These equations have profound applications across many areas of mathematics, including differential geometry, metric geometry and algebraic geometry. Notable advancements have been made in this field, including the sharp L∞ estimates of Kolodziej, and Hölder continuity for complex Monge-Ampère equations. The goal is to bring together experts in this field to explore open problems and develop new techniques which are expected to yield solutions to some of the long-standing questions.

The main topics of the workshop include: Exploration of the geometry of Kähler metrics in the absence of Ricci curvature lower bound, using analytic estimates on Kähler potentials. Analyzing the connection between stability, subsolutions, and the existence of solutions for a specific class of fully nonlinear PDEs. Hölder continuity of solutions for fully nonlinear PDEs, particularly those of the Hessian type. Geometric and analytic estimates for singular metrics on complex varieties.