The research activities supported by this award are in the field of geometric analysis, and the specific problems posed are 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 considered 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 problem proposed is the construction of Bach-flat metrics on four-manifolds using surgery methods. The Bach tensor arises in general relativity and conformal geometry, and the Bach-flat condition is the Euler-Lagrange equation for a functional on the space of Riemannian metrics which is quadratic in the curvature, and can be viewed as a nonlinear system of partial differential equations with certain invariance properties. The final set of problems we propose involves nonlinear equations arising in conformal geometry and the theory of optimal transportation. We are specifically interested in the construction of singular solutions of these equations as a way to study the existence of complete Riemannian metrics with prescribed curvature conditions.
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 proposal involves disparate problems from geometry and mathematical physics but are united by the role played by mathematical analysis in their study.
