Singularity Formation in Kahler Geometry and Yang-Mills Instantons

卡勒几何和杨米尔斯瞬子中奇点的形成

基本信息

批准号：
2004261
负责人：
Song Sun
金额：
$ 35万
依托单位：
University of California-Berkeley
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-08-01 至 2023-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2004261&HistoricalAwards=false
关键词：
Singularity Formation Kahler Geometry Yang

项目摘要

This project will study singularity formation of some special geometric structures governed by non-linear partial differential equations. These structures are generalizations of solutions to the Einstein’s equation for gravity and Maxwell’s equation for electromagnetism, and play significant roles in modern theoretical physics (for example string theory and M-theory). The solution to these problems will enhance the understanding of deep interaction between various research directions in mathematics. In the meantime, there are many interesting related questions that may serve as research projects for the training of graduate students interested in this area.The PI will study problems in two specific directions. First, the PI would like to study collapsing of Calabi-Yau metrics, and connections with algebraic geometry and moduli compactification. Previous work of the PI and collaborators has yielded a satisfactory description in the case of minimal degenerations and the PI proposes to study the more general situation. This requires exploiting the existing techniques from collapsing theory, and further developing those. Secondly, the PI will study singularity analysis for Yang-Mills instantons. In the case of Hermitian-Yang-Mills connections over Kahler manifolds, recent work of the PI and Xuemiao Chen gives a complete algebro-geometric description of the tangent cones, and the PI wants to push these further by discovering more refined structures, and by investigating the case of G2 instantons. This requires building algebro-geometric framework, as well as extending some of these to the more analytic setting.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.

该项目将研究由非线性偏微分方程控制的一些特殊几何结构的奇点形成，这些结构是爱因斯坦引力方程和麦克斯韦电磁方程解的推广，在现代理论物理（例如弦）中发挥着重要作用。这些问题的解决将增强对数学各个研究方向之间的深层相互作用的理解，同时，还有许多有趣的相关问题可以作为培养感兴趣的研究生的研究项目。这PI 将研究两个具体方向的问题。首先，PI 希望研究 Calabi-Yau 度量的折叠，以及与代数几何和模紧化的联系。最小退化的情况，PI 建议研究更一般的情况，这需要利用崩溃理论的现有技术，并进一步发展这些技术。其次，PI 将研究 Yang-Mills 的奇点分析。就卡勒流形上的 Hermitian-Yang-Mills 连接而言，PI 和 Xuemiao Chen 最近的工作给出了切锥的完整代数几何描述，并且 PI 希望通过发现更精细的结构来进一步推动这些描述，并通过研究 G2 瞬子的情况，这需要建立代数几何框架，并将其中一些扩展到更具分析性的环境。该奖项是 NSF 的法定使命，并已获得批准。通过使用基金会的智力优点和更广泛的影响审查标准进行评估，被认为值得支持。