The Other Monge-Ampère Equation: Geometry and Singularities in the Hyperbolic Setting

The elliptic Monge-Ampère equation is a cornerstone of fully nonlinear PDE theory with connections many areas of math including geometry, analysis, optimal transport, etc. In this talk, we will explore its less studied but equally fascinating sibling: the hyperbolic Monge-Ampère equation.

The hyperbolic version presents a host of unique analytical challenges, as familiar tools from elliptic theory do not readily apply, leading to solutions with limited regularity and the spontaneous formation of singularities. We will investigate this PDE from a more geometric perspective rooted in the classical method of characteristics. This viewpoint provides the key to understanding the equation's structure and its applications, from constructing surfaces of negative Gaussian curvature to modeling phenomena in other disciplines to building numerical methods for solving such equations.

The goal of this talk is to provide an accessible overview of this circle of ideas. This will set the stage for future talks by Ari Bormanis and Maria Deliyianni, focusing on more technical aspects of the theory.