This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The research activities supported by this grant include the following aspects of the geometric theory for the Monge-Ampere equation: a regularity theory in the first Heisenberg group; regularity of solutions to the linearized Monge-Ampere operator in Euclidean space; and the emerging connections between solutions to the Monge-Ampere equation and quasiconformal mappings with convex potential. Problems related to the quasiconformal and bi-Lipschitz Jacobian problems, the characterization of certain quasiconformal mappings with convex potentials in terms of their Monge-Ampere measure, and the corresponding extensions to the subelliptic case will be investigated.
The Monge-Ampere operator is the nonlinear cousin of the omnipresent Laplace operator, whose enormous importance for physics and engineering has long been well documented. As a nonlinear operator, the Monge-Ampere operator is far more complicated than the Laplacian. However, when acting on convex functions, it produces geometric and measure-theoretic objects that help to clarify its workings. For a long time, the Monge-Ampere operator has played a key role in differential geometry and optimal mass transportation. Somewhat more recently, it has found applications to the design of reflector antennas and to so-called quasiconformal mappings. Another exciting aspect of the Monge-Ampere equation comes from its linearization, which has applications in its own right to Lagrangian models of front formation and fluid dynamics. The linearized Monge-Ampere operator is a challenging object, for it does not fall into the category of the well-known "uniformly elliptic" operators.
