Homogeneous Einstein Spaces

本项目围绕自然模型空间的研究展开。在过去，直到一个半世纪前，唯一研究几何的模型是(平坦的)欧几里德空间--这是我们在高中学到的几何。然而，为了推进科学和工程，我们需要使用包含曲率的模型--例如，地球不是平的，它是圆的！为此，首席研究员(PI)将研究被称为齐次爱因斯坦空间的特殊模型几何，以更多地了解它们的基本性质，并致力于它们的分类。PI将开发新的工具来分析这些空间，并利用研究生的支持来培训新一代数学家应用它们。PI将通过继续开发几何不变量理论的工具以及它们与几何分析工具的交互来对齐次爱因斯坦度量进行调查。本程序的目标是对非紧致齐次爱因斯坦流形进行分类，并确定是否所有这样的空间实际上都是解流形。此外，PI将调查爱因斯坦和利玛窦孤子空间的杰出几何和代数性质。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。

This project revolves around the study of natural model spaces. In the past, until a century and a half ago, the only model whose geometry was investigated was (flat) Euclidean space - this is the geometry that we all learned in high school. However, in order to advance science and engineering, we need to use models that incorporate curvature - for example, the Earth is not flat, it is round! To this end, the principal investigator (PI) will study special model geometries called homogeneous Einstein spaces to learn more about their basic properties and work towards their classification. The PI will develop new tools for analyzing these spaces and utilize the graduate student support to train a new generation of mathematicians in their application. The PI will conduct an investigation into homogeneous Einstein metrics by continuing to develop tools from Geometric Invariant Theory and their interaction with tools from Geometric Analysis. The goal of this program is to classify non-compact, homogeneous Einstein manifolds and determine if all such spaces are, in fact, solvmanifolds. Even further, the PI will investigate the distinguished geometric and algebraic properties of Einstein and Ricci soliton spaces.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.