CURVATURE, SYMMETRY, AND STABILITY IN HOMOGENEOUS SPACES

大约150年前，当人们意识到我们在高中时熟悉的（平坦的）欧几里得几何并不是唯一可以考虑的几何时，几何的基础动摇了。回想起来，考虑非平面几何似乎很自然，因为我们生活的地球当然是圆的。自从他们的第一个发现以来，数学家们已经研究了各种具有良好曲率特性的模型几何，比如圆球，我们继续提取关于这些模型几何的新信息，它们与平面欧几里得几何的区别，以及它们在我们生活的宇宙中的应用。首席研究员将研究被称为齐次爱因斯坦空间的特殊几何模型，以了解更多关于它们的基本性质并努力对它们进行分类。该项目的主要目标是推进非紧、齐次爱因斯坦和里奇孤子空间的分类。除了解决这些空间是否一定是可解流形的问题外，PI还将深入了解齐次Ricci孤子在Ricci流及其等长群下的稳定性。为了解决这些问题，PI打算更完整地发展Heber， Lauret等人提出的使用几何不变量理论研究齐次爱因斯坦度量的计划。

Almost 150 years ago, the foundations of geometry were rocked when it was realized that the familiar (flat) Euclidean geometry that we all learned in high school was not the only possible geometry to consider. In retrospect, it seems natural to consider non-flat geometries as the Earth we live on is, of course, round. Since their first discovery, mathematicians have studied various model geometries with nice curvature properties, such as the round sphere, and we continue to extract new information about these model geometries, how they are distinct from flat Euclidean geometry, and their applications to the universe we live in. The principal investigator will study special model geometries called homogeneous, Einstein spaces to learn more about their basic properties and work towards their classification.The project's main goal is to advance the classification of non-compact, homogeneous Einstein and Ricci soliton spaces. In addition to addressing the question of whether or not these spaces are necessarily solvmanifolds, the PI will work to develop a deeper understanding of the special properties of homogeneous Ricci solitons regarding their stability under the Ricci flow and their isometry groups. To approach these problems, the PI intends to develop more completely the program of using Geometric Invariant Theory to study homogeneous Einstein metrics initiated by Heber, Lauret, et al.