Bergman Kernel Estimates and Spectrum of Complete Riemannian Manifold

在这个项目中，原理调查者(PI)将致力于研究被称为微分几何的数学领域中的一些基本问题。虽然微分几何是纯数学的分支，但它所采用的方法和结果在我们对宇宙的理解中起着基础性的作用。例如，弦理论可能是物理学中的“万物理论”，它使用了许多来自微分几何的工具。PI计划做与弦理论密切相关的微分几何部分的工作。PI还将参与许多K-12教育活动，并将深入参与加州大学欧文分校本科生和研究生课程的创建和创新。该奖项将支持研究生在PI的指导下完成他们的论文。更具体地说，PI计划致力于Kahler几何和光谱理论中的几个重要问题。在Kahler几何中，他将继续他的长期项目，即分析Bergman核及其与具有集值截面的流形的关系。在谱论中，他将继续研究完备非紧流形上微分形式的拉普拉斯本质谱，并研究逆谱问题。当PI研究Calabi-Yau模的几何时，这两部分工作将结合在一起。在大多数情况下，Calabi-Yau模不是完全流形，因此PI需要在对其进行更多非线性分析之前将频谱结果推广到不完全流形上。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

In this project the Principle Investigator (PI) will work on some fundamental problems in a field of mathematics called differential geometry. Although differential geometry is branch of pure mathematics, the methods it employs and its results play a fundamental role in our understanding of the Universe. For example, String Theory, which is potentially the "theory of everything" in physics, uses a lot of tools from differential geometry. The PI plans to do work in the part of differential geometry which is closely related to string theory. The PI will also be involved in many K-12 educational activities and will be deeply involved in the creation and innovation of both undergraduate and graduate courses at the University of California at Irvine. The award will support graduate students working on their dissertations under the direction of the PI.More specifically, the PI plans to work on several important problems in Kahler geometry and spectral theory. In Kahler geometry he will continue his long term project on the analysis of the Bergman kernel and its relation to analysis on manifolds with bundle valued sections. In spectral theory, he will continue to study the essential spectrum of the Laplacian on differential forms on complete non-compact manifolds, and study inverse spectral problems. These two parts of work will be combined when the PI studies the geometry of Calabi-Yau moduli. In most cases, Calabi-Yau moduli are not complete manifolds, and therefore the PI needs to generalize the spectrum results onto incomplete manifolds before doing more non-linear analysis on them.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.