The research activities supported by this award are in the field of geometric analysis, and the specific problems under investigation are located at the intersection of three fields: partial differential equations, differential geometry, and mathematical physics. For example, the functional determinant of an elliptic operator is a problem originating in spectral theory and mathematical physics, and the analysis of the particular problem we consider is a variational problem leading to a fourth order elliptic equation. The associated Lagrangian is unbounded, and the existence of solutions and their qualitative properties is highly nontrivial. Similar equations are used to model the properties of thin films. Another set of problems the principal investigator will explore involves certain fully nonlinear equations arising in conformal geometry and the theory of optimal transportation. Higher order regularity for these equations fails when the underlying manifold is negatively curved. While the structural reasons for this lack of regularity are in some sense understood (i.e., how the nonlinearity leads to the failure of certain estimates), there is no geometric description of the nature of singularities.
The interaction of geometry and analysis dates back to at least the eighteenth century, and yet continues to be an important and highly active field of mathematical research. The classical subject of geometry grew out of our desire to understand certain properties of the physical world, and differential geometry was developed to understand the geometry of curved spaces--for example, the curvature of the surface of the earth, or the curvature of space by matter as predicted by general relativity. In the same way that Descartes realized that plane geometry can be studied using algebra, so differential geometry can be studied using techniques from analysis, especially differential equations. The research in this project involves disparate problems from geometry and mathematical physics, but its various aspects are united by the role played by mathematical analysis in their study.
