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Bergman Kernel Estimates and Spectrum of Complete Riemannian Manifold

完整黎曼流形的伯格曼核估计和谱

基本信息

批准号：
1908513
负责人：
Zhiqin Lu
金额：
$ 34.2万
依托单位：
University of California-Irvine
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-07-01 至 2024-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1908513&HistoricalAwards=false
关键词：
Bergman Kernel Estimates Spectrum Complete

项目摘要

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.

在这个项目中，主要研究者（PI）将研究数学领域中的一些基本问题，称为微分几何。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理解宇宙中起着基础性的作用。例如，弦论，它可能是物理学中的“万有理论”，使用了很多微分几何的工具。PI计划在微分几何的一部分做工作，这是密切相关的弦理论。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的指导下完成论文。更具体地说，PI计划研究Kahler几何和谱理论中的几个重要问题。在Kahler几何，他将继续他的长期项目的分析伯格曼内核及其关系的分析流形与束值部分。在谱理论方面，他将继续研究拉普拉斯算子在完全非紧流形上的微分形式的基本谱，并研究逆谱问题。当PI研究卡-丘模量的几何时，这两部分工作将结合起来。在大多数情况下，卡-丘模量不是完全流形，因此PI需要在对它们进行更多非线性分析之前将谱结果推广到不完全流形上。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估而被认为值得支持。