It is proposed to use microlocal techniques (including Fourier integral operators with complex phase) to study the following topics in global analysis: (A) The Toda PDE, a large N limit of the Toda lattice, in the periodic and non-periodic cases. Topics to be researched include large N estimates relating Toda lattice solutions and solutions to the Toda PDE, and shock formation. (B) Spectral problems for the Laplacian, including the semi-classical asymptotics of the scattering matrix in manifolds with cylindrical ends. (C) Symplectic geometry, via generalized Szego kernels. Specifically, the asymptotics of Kodaira-type embeddings and the relationship between symplectic capacities and geometric quantization will be researched.
The quantum-classical correspondence is a deep feature of Nature, admitting a variety of mathematical manifestations. In general terms, these manifestations take the form of relationships between dynamical systems (systems of ordinary differential equations) and partial differential equations, in suitable asymptotic regimes (the semi-classical limit). The proposed research will investigate some such relationships, in the general areas of completely-integrable dynamical systems, the geometry of the Laplace operator, and differential geometry. A particularly novel aspect of the proposed research is to the Toda PDE, a large N limit of the Toda lattice. The proposed research will result in a deeper understanding of certain non-linear partial differential equations, and of the geometry of phase spaces (symplectic manifolds).
