Microlocal Aspects of Geometric Quantization and Mathematical Physics

本文提出用微局部方法（包括复相位的Fourier积分算子）研究全局分析中的下列问题：（A）户田偏微分方程，即户田格点的大N极限，在周期和非周期情况下。 待研究的主题包括大N估计有关户田晶格解决方案和解决方案的户田偏微分方程，和冲击的形成。 (B)Laplacian的谱问题，包括具有圆柱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内核。 具体来说，Kodaira型嵌入的渐近性和辛容量与几何量子化之间的关系将被研究。量子-经典对应是自然界的一个深刻特征，允许各种数学表现。 在一般情况下，这些表现形式之间的关系dynamicalsystems（系统的常微分方程）和偏微分方程，在适当的渐近制度（半经典极限）。 本研究将在完全可积动力系统、拉普拉斯算子几何和微分几何的一般领域中探讨一些这样的关系。 一个特别新颖的方面提出的研究是户田偏微分方程，一个大的N限制的户田晶格。 该研究将使我们对某些非线性偏微分方程和相空间几何（辛流形）有更深的理解。

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).