Singular Metrics in Kahler Geometry

PI研究的一个中心主题是偏微分方程（PDEs）之间的联系，比如广义相对论中的爱因斯坦方程，以及这些方程所研究的空间的代数或几何性质。许多研究都集中在光滑空间上的偏微分方程的研究上，那里的局部几何有一个非常简单的模型，然而奇点可以在自然界中自然出现，最著名的是黑洞。目前的研究项目主要集中在这些奇异空间上偏微分方程的研究。这项研究将应用于代数几何，其中奇异空间在现代研究中起着核心作用，以及偏微分方程理论。与此同时，奇异空间作为平滑空间族的极限自然出现，例如，平滑空间族可以在某些方向上坍缩。通过这种方式，拟议的研究将为这些家庭的行为提供新的线索。除了进行这些研究项目外，PI还将继续培养博士生和博士后研究人员。此外，PI还将为本科生共同组织一年一度的夏季研讨会，旨在向他们传达在本科课程中通常不会出现的几何思想。PI还将共同组织一个年度桥梁项目，旨在帮助来自不同背景的即将入学的研究生加快速度，以确保他们的成功。该项目的目的是从不同的角度研究Kahler几何中的奇点。一方面，奇异点可能出现在光滑空间序列的极限上，理解这种极限空间的结构是很重要的。在Donaldson-Sun、Liu和PI的工作基础上，PI将研究极限空间何时可以被识别为一个奇异的Kahler空间。特别感兴趣的是具有里奇曲率边界的卡勒流形的非坍缩极限空间，然而PI也将研究鲜为人知的坍缩极限空间。在相关方向上，PI将研究奇异复变上的规范度量，如Kahler-Einstein度量，特别是这些度量在奇异集附近的行为。对这类奇异度量的几何学有一个很好的理解，将导致微分几何技术在奇异变量的代数几何中的应用。最后，关于Kahler流形的坍缩理论，PI将研究在几乎坍缩的颤振上的规范度量的构造。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。

A central theme in the PI's research is the connections between partial differential equations (PDEs) such as Einstein's equations in general relativity, and the algebraic or geometric properties of the spaces on which these equations are studied. Much research has been focused on the study of PDEs on smooth spaces, where the local geometry has a very simple model, however singularities can arise naturally in nature, the most well known being black holes. The current research project focuses on the study of PDEs on such singular spaces. This study will have applications in algebraic geometry, where singular spaces play a central role in modern research, as well as in the theory of partial differential equations. At the same time, singular spaces occur naturally as limits of families of smooth spaces, which could collapse in certain directions for instance. In this way the proposed research will shed new light on the behavior of such families. Aside from pursuing these research projects, the PI will continue training PhD students and postdoctoral researchers. In addition the PI will also co-organize a yearly summer workshop forundergraduate students aimed at conveying ideas in geometry to them which do not typically appear in the undergraduate curriculum. The PI will also co-organize a yearly bridge program aimed at helping incoming graduate students with diverse backgrounds get up to speed, to ensure their success.The objective of the project is to investigate singularities in Kahler geometry from different points of view. On the one hand, singularities can arise in the limit of a sequence of smooth spaces, and it is important to understand the structure of such limit spaces. The PI will study the question of when the limit space can be identified with a singular Kahler space, building on work of Donaldson-Sun as well as Liu and the PI. Of particular interest are non-collapsed limit spaces of Kahler manifolds with Ricci curvature bounds, however the PI will also investigate the much less understood collapsed limit spaces. In a related direction, the PI will study canonical metrics, such as Kahler-Einstein metrics, on singular complex varieties, in particular the behavior of such metrics near the singular set. A good understanding of the geometry of such singular metrics would lead to applications of differential geometric techniques to the algebraic geometry of singular varieties. Finally, in relation to the collapsing theory of Kahler manifolds, the PI will study the construction of canonical metrics on fibrations that are almost collapsed.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.