Metric geometry and analysis on Einstein manifolds

度量黎曼几何是现代数学的中心课题。最初的概念可以追溯到1854年6月10日，伯恩哈德·黎曼发表了著名的赫歇尔演讲“Ueber die hypothesen，welche der Geometrie zu Grunde liegen”（关于几何基础上的假设）。这次演讲中的革命性创造深刻地改变了几何学的全球格局。具体来说，黎曼提出了一种新的战略，以推广几何曲面更高的层面，他所谓的Mannigfaltigkeiten（流形）。各种各样的新概念和概念被创造出来：这些包括曲率的概念，它定量地测量空间是如何弯曲的，以及测地线的概念，它是连接流形上两点的长度最小化路径。流形的度量结构的研究，我们现在称之为度量黎曼几何，主要集中在基础空间的整体几何和度量结构之间的相互作用，即如何实现或测量两点之间的距离。本计画主要研究爱因斯坦流形的度量几何，其中度量结构满足广义相对论中的爱因斯坦方程。 PI将把他们的研究与各种级别的培训和指导相结合。 这包括组织夏季研讨会和数学静修黎曼几何;复杂几何和理论物理，并为本科生设计和开发新的研究型课程。本计画研究爱因斯坦流形的退化与定量行为。在与Song Sun的合作中，PI一直致力于研究具有特殊完整性的爱因斯坦流形的坍缩几何，导致该领域的两个重大突破：K3流形上爱因斯坦度量的Gromov-Hausdorff极限的完整分类，以及引力瞬子的渐近模型几何的完整分类。后者可以看作是K3流形上退化Einstein度规的气泡极限。 在此背景下，PI将继续在更高的维度类似的问题，并调查几何结构的退化爱因斯坦度量与通用holonomy在该设置。通过一组合作者，PI还将在更精细的几何和关于完整卡-丘度量的模空间问题方面取得进展。在第三个方向，PI将研究庞加莱-爱因斯坦空间的几何和分析，这起源于数学物理中的AdS/CFT对应。PI将专注于Poincare-Einstein流形上退化算子的奇异行为，Poincare-Einstein度量的几何有限性和定量刚性，以及Poincare-Einstein度量的正则性和退化理论。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估而被认为值得支持。

Metric Riemannian geometry is a central subject in modern mathematics. The original concept dates back to Bernhard Riemann's famous Habilitation lecture "Ueber die hypothesen, welche der Geometrie zu Grunde liegen" (On the hypotheses which lie at the bases of geometry) delivered on 10 June 1854. The revolutionary creations in this lecture profoundly changed the global landscape of geometry. Specifically, Riemann proposed a novel strategy to generalize the geometry of surfaces to higher dimensions which he called Mannigfaltigkeiten (manifolds). A large variety of new notions and concepts were created: these include the notion of curvature which quantitatively measures how a space is curved, and the notion of geodesic which is a length-minimizing path connecting two points on a manifold. The studies of the metric structures of manifolds, what we now call metric Riemannian geometry, primarily focuses on the interplay between the global geometry of the underlying space and the metric structure, namely how the distance between two points can be realized or measured. This project is mainly concerned with the metric geometry of Einstein manifolds where the metric structures satisfy the Einstein equation in the theory of general relativity. The PI will integrate their research with training and mentorship at a variety of levels. This includes organizing summer workshops and mathematical retreats on Riemannian geometry; complex geometry and theoretical physics, and designing and developing new research oriented courses for undergraduate students. This project investigates the degenerations and quantitative behaviors of Einstein manifolds. In joint work with Song Sun, the PI has been working on the collapsing geometry of Einstein manifolds with special holonomy, leading to two major breakthroughs in the field: a complete classification of the Gromov-Hausdorff limits of the Einstein metrics on the K3 manifold, and a complete classification of asymptotic model geometries of gravitational instantons. The latter can be regarded as the bubble limits of the degenerating Einstein metrics on the K3 manifold. Building on this background, the PI will proceed to analogous questions in higher dimensions and investigate geometric structures for the degenerating Einstein metrics with generic holonomy in that setting. With a group of collaborators, the PI will also make advances in more refined geometry and moduli space problems regarding complete Calabi-Yau metrics. In a third direction, the PI will investigate the geometry and analysis of Poincare-Einstein spaces, which originated from the AdS/CFT correspondence in mathematical physics. The PI will focus on the singularity behaviors of degenerate operators on Poincare-Einstein manifolds, geometric finiteness and quantitative rigidity of Poincare-Einstein metrics, as well as regularity and degeneration theory of Poincare-Einstein metrics.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.