Singularity formation in Kahler geometry

卡勒几何中奇点的形成

• 批准号: 2304692
• 负责人: Song Sun
• 金额: $ 40.7万
• 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: 2023
• 资助国家: 美国
• 起止时间: 2023-09-01 至 2026-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2304692&HistoricalAwards=false
• 关键词: Singularity; formation; Kahler; geometry

This project focuses on special geometric structures, known as Kahler-Einstein metrics, which represent generalizations of solutions to Einstein's equations for gravity, as wells as structures called instantons, which generalize solutions to Maxwell's equations for electromagnetism. These structures play pivotal roles in modern theoretical physics, particularly in areas like string theory. More specifically, this project is mainly concerned with how these structures can develop defects, known as singularities, and to what extent the resulting object has a predictable shape, when viewed in the large. The successful resolution of these problems will substantially enhance our comprehension of the intricate interactions between various research directions in mathematics. This project will also have broader impacts, as it will give rise to numerous fascinating and interconnected questions offering ample research and training opportunities for graduate students keen on exploring this captivating field. The PI's research will be concentrated on several lines of investigation. Firstly, the PI plans to investigate the structure of complete Calabi-Yau metrics with Euclidean volume growth. The PI, along with J. Zhang, has already achieved a satisfactory structural theory under the assumption of quadratic curvature decay. However, the general scenario poses more significant challenges, necessitating the development of novel technical tools. Secondly, the PI is interested in understanding the collapsing phenomenon in canonical metrics. In this regard, previous collaborative work with R. Zhang has made substantial progress, specifically in the context of 4-dimensional hyperkähler metrics. Extending these findings to other settings holds paramount importance for their applications in geometry. Lastly, the PI will delve into related questions concerning Hermitian-Yang-Mills instantons. While parallel questions can be raised, answering them will require the incorporation of intriguing new techniques.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.

该项目侧重于特殊的几何结构，称为Kahler-Einstein指标，它代表了爱因斯坦的重力方程解决方案的概括，以及称为Instantons的结构，该结构将麦克斯韦方程的解决方案概括为电子磁性。这些结构在现代理论物理学中扮演着关键的角色，尤其是在弦理论等领域。更具体地说，该项目主要关注这些结构如何形成缺陷（称为奇异性），以及在大型中查看所产生的对象在多大程度上具有可预测的形状。这些问题的成功解决将大大增强我们对数学各种研究方向之间复杂相互作用的理解。该项目还将产生更广泛的影响，因为它将引起许多有趣和相互联系的问题，为研究生提供充足的研究和培训机会，热衷于探索这个迷人的领域。 PI的研究将集中在几项投资上。首先，PI计划研究具有欧几里得量增长的完整Calabi-yau指标的结构。 PI与J. Zhang一起在二次咖喱衰变的假设下已经达到了令人满意的结构理论。但是，总体场景面临更大的挑战，需要开发新颖的技术工具。其次，PI有兴趣了解规范指标中的崩溃现象。在这方面，以前与R. Zhang的合作工作已取得了重大进展，特别是在4维HyperKähler指标的背景下。将这些发现扩展到其他设置对其在几何形状中的应用而言至关重要。最后，PI将研究有关Hermitian-Yang-Mills Instantons的相关问题。虽然可以提出并行问题，但回答它们将需要纳入有趣的新技术。该奖项反映了NSF的法定任务，并且使用基金会的知识分子优点和更广泛的影响审查标准，认为通过评估被认为是珍贵的支持。