Singularity formation in Kahler geometry

这个项目的重点是特殊的几何结构，被称为Kahler-Einstein度量，它代表了爱因斯坦引力方程解的推广，以及被称为瞬子的结构，它推广了麦克斯韦电磁学方程的解。这些结构在现代理论物理学中起着关键作用，特别是在弦理论等领域。更具体地说，这个项目主要关注这些结构如何产生缺陷，即奇点，以及在多大程度上产生的物体在大范围内具有可预测的形状。这些问题的成功解决将大大增进我们对数学各个研究方向之间错综复杂的相互作用的理解。该项目还将产生更广泛的影响，因为它将产生许多有趣且相互关联的问题，为热衷于探索这一迷人领域的研究生提供充足的研究和培训机会。私家侦探的研究将集中在几方面的调查。首先，PI计划研究具有欧几里得体积增长的完全Calabi-Yau度量的结构。PI和J. Zhang已经在二次曲率衰减的假设下获得了令人满意的结构理论。然而，总体情况带来了更大的挑战，需要开发新的技术工具。其次，PI感兴趣的是理解规范度量中的坍缩现象。在这方面，之前与R. Zhang的合作工作已经取得了实质性的进展，特别是在4维hyperkähler指标的背景下。将这些发现扩展到其他环境对于它们在几何中的应用具有至关重要的意义。最后，PI将深入研究有关Hermitian-Yang-Mills瞬子的相关问题。虽然可以提出类似的问题，但回答这些问题将需要结合有趣的新技术。该奖项反映了美国国家科学基金会的法定使命，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，被认为值得支持。

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.