The award will fund research on the development of important analytical tools for the analysis of partial differential equations. One topic is analysis in rough geometries. The projects to be studied include controlling the regularity of the metric tensor on a manifold with boundary given curvature bounds on the manifold and its boundary and further investigations of elliptic systems on Lipschitz domains. Another class of problems concerns spectral theory and evolution equations. The difference in the behavior of the Laplace operator on Euclidean space and on compact manifolds will be studied. This will include studies of the Gibbs' and Pinsky phenomena and their analogues for nonlinear Schroedinger equations.
This research impacts differential geometry, complex and harmonic analysis as well as partial differential equations. This research has applications in mathematical physics and diffraction theory. Professor Taylor is very well-known for his texts, publications and lectures, so this research will undoubtedly be influential in the development of these topics.
