Singular Metrics in Kahler Geometry

PI研究的一个中心主题是偏微分方程组(PDE)之间的联系，例如广义相对论中的爱因斯坦方程，以及研究这些方程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组的研究上，在光滑空间中，局部几何有一个非常简单的模型，然而在自然界中可以自然地产生奇点，最著名的是黑洞。目前的研究主要集中在这类奇异空间上的偏微分方程组的研究上。这项研究将在代数几何和偏微分方程组理论中得到应用，在代数几何中，奇异空间在现代研究中扮演着核心角色。同时，奇异空间作为光滑空间族的极限自然出现，例如，光滑空间族可能在某些方向上崩溃。通过这种方式，拟议的研究将为这类家庭的行为提供新的线索。除了从事这些研究项目外，PI还将继续培训博士生和博士后研究人员。此外，国际学生联合会还将联合为本科生举办每年一度的暑期研讨会，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还将联合组织一个年度桥梁项目，旨在帮助不同背景的应届研究生快速掌握，以确保他们的成功。该项目的目标是从不同的角度调查卡勒几何中的奇点。一方面，在光滑空间序列的极限中可能出现奇点，了解这种极限空间的结构是很重要的。PI将研究极限空间何时可以识别为奇异Kahler空间的问题，这是建立在Donaldson-Sun以及Liu和PI的工作基础上的。特别有趣的是具有Ricci曲率界的Kahler流形的非折叠极限空间，然而PI也将研究更少被理解的折叠极限空间。在一个相关的方向上，PI将研究关于奇异复杂变种的规范指标，如卡勒-爱因斯坦指标，特别是这种指标在奇异集附近的行为。对这类奇异度量的几何有一个很好的理解，这将导致微分几何技巧在奇异簇的代数几何中的应用。最后，关于Kahler流形的崩溃理论，PI将研究在几乎崩溃的纤颤上构建正则度量。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.