Singular Metrics in Kahler Geometry

PI研究的一个中心主题是偏微分方程（PDE）（如广义相对论中的爱因斯坦方程）与研究这些方程的空间的代数或几何性质之间的联系。许多研究都集中在光滑空间上的偏微分方程的研究上，其中局部几何有一个非常简单的模型，但奇点可以在自然界中自然出现，最着名的是黑洞。目前的研究项目集中在此类奇异空间上的偏微分方程的研究。这项研究将在代数几何，奇异空间在现代研究中发挥核心作用，以及在偏微分方程理论的应用。与此同时，奇异空间自然地作为光滑空间族的极限出现，例如，它可以在某些方向上崩溃。通过这种方式，拟议中的研究将为这些家庭的行为提供新的线索。除了继续这些研究项目外，PI还将继续培训博士生和博士后研究人员。此外，PI还将共同组织一个年度夏季研讨会，为本科生，旨在传达几何思想，他们通常不出现在本科课程. PI还将共同组织一个年度桥梁计划，旨在帮助具有不同背景的研究生快速入门，以确保他们的成功。该项目的目标是从不同的角度研究Kahler几何中的奇点。一方面，奇点可以出现在光滑空间序列的极限中，理解这样的极限空间的结构是很重要的。PI将研究极限空间何时可以与奇异Kahler空间相同的问题，建立在Donaldson-Sun以及Liu和PI的工作基础上。特别感兴趣的是具有Ricci曲率边界的Kahler流形的非塌陷极限空间，然而PI也将研究更少了解的塌陷极限空间。在一个相关的方向，PI将研究规范度量，如Kahler-Einstein度量，在奇异复杂的品种，特别是这些度量的行为附近的奇异集。一个很好的理解这种奇异度量的几何将导致应用微分几何技术的代数几何的奇异品种。最后，关于Kahler流形的坍缩理论，PI将研究几乎坍缩的纤维化的规范度量的构建。该奖项反映了NSF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.