Microlocal Aspects of Geometric Quantization and Mathematical Physics

本文提出用微局域技术(包括复位相的傅立叶积分算符)来研究全局分析中的下列问题：(A)周期和非周期情况下的Toda晶格的大N极限Toda PDE。要研究的课题包括与Toda晶格解和Toda PDE解有关的大N估计，以及激波形成。(B)拉普拉斯的谱问题，包括具有圆柱形末端的流形中散射矩阵的半经典渐近性。(C)辛几何，通过广义Szego核。具体地说，将研究Kodaira型嵌入的渐近性以及辛容量和几何量子化之间的关系。量子-经典对应是自然界的一个深层次特征，允许多种数学表现形式。一般说来，这些表现形式是动力系统(常微分方程组)和偏微分方程组之间的关系，在适当的渐近机制(半经典极限)下。这项研究将在完全可积动力学系统的一般领域、拉普拉斯算子的几何和微分几何中研究一些这样的关系。一个特别新颖的方面是Toda PDE，它是Toda晶格的一个大的N极限。所提出的研究将导致对某些非线性偏微分方程式和相空间(辛流形)的几何的更深入的理解。

ABSTRACTIt is proposed to use microlocal techniques (including Fourier integraloperators with complex phase) to study the following topics in globalanalysis: (A) The Toda PDE, a large N limit of the Toda lattice, inthe periodic and non-periodic cases. Topics to be researched includelarge N estimates relating Toda lattice solutions and solutions to theToda PDE, and shock formation. (B) Spectral problems for theLaplacian, including the semi-classical asymptotics of the scatteringmatrix in manifolds with cylindrical ends. (C) Symplectic geometry,via generalized Szego kernels. Specifically, the asymptotics ofKodaira-type embeddings and the relationship between symplecticcapacities 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 dynamicalsystems (systems of ordinary differential equations) and partialdifferential equations, in suitable asymptotic regimes (thesemi-classical limit). The proposed research will investigate somesuch relationships, in the general areas of completely-integrabledynamical systems, the geometry of the Laplace operator, anddifferential geometry. A particularly novel aspect of the proposedresearch is to the Toda PDE, a large N limit of the Toda lattice. Theproposed research will result in a deeper understanding of certainnon-linear partial differential equations, and of the geometry of phasespaces (symplectic manifolds).