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

• 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

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.

该项目将研究一些由非线性偏微分方程控制的特殊几何结构的奇异性形成。这些结构是对爱因斯坦方程的解决方案的概括，而麦克斯韦方程是电磁主义的，并且在现代理论物理学中起着重要作用（例如，弦乐理论和M理论）。解决这些问题的方法将增强对数学各种研究方向之间深层相互作用的理解。同时，有许多有趣的相关问题可以用作研究对该领域感兴趣的研究生的研究项目。PI将在两个特定方向上研究问题。首先，PI希望研究Calabi-yau指标的崩溃，以及与代数几何和模量压实的连接。 PI和合作者的先前工作在最小的退化和PI的建议中提出了令人满意的描述，以研究更一般的情况。这需要从崩溃理论中利用现有技术，并进一步发展。其次，PI将研究Yang-Mills Instantons的奇异性分析。对于Kahler歧管上的Hermitian-Yang-Mills连接，PI和Xuemiao Chen的最新工作对切线锥进行了完整的代数几何描述，PI希望通过发现更精致的结构以及研究G2 Instantons案件来进一步推动它们。这需要建立代数几何框架，并将其中的一些扩展到更分析的环境。该奖项反映了NSF的法定任务，并使用基金会的知识分子优点和更广泛的影响来评估，被认为是宝贵的支持。

项目成果

Geometry of Calabi-Yau Metrics

卡拉比-丘度量的几何

• DOI: 10.1090/noti2454
• 发表时间: 2022
• 期刊: Notices of the American Mathematical Society
• 作者: Sun, Song

No semistability at infinity for Calabi-Yau metrics asymptotic to cones

Calabi-Yau 度量渐近锥体时无无穷远半稳定性

• DOI: 10.1007/s00222-023-01187-4
• 发表时间: 2023
• 期刊: Inventiones mathematicae
• 影响因子: 3.1
• 作者: Sun, Song;Zhang, Junsheng
• 通讯作者: Zhang, Junsheng