CURVATURE, SYMMETRY, AND STABILITY IN HOMOGENEOUS SPACES

均匀空间中的曲率、对称性和稳定性

基本信息

批准号：
1612357
负责人：
Michael Jablonski
金额：
$ 15.8万
依托单位：
University of Oklahoma Norman Campus
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2016
资助国家：
美国
起止时间：
2016-08-01 至 2020-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1612357&HistoricalAwards=false
关键词：
CURVATURE SYMMETRY STABILITY HOMOGENEOUS SPACES

项目摘要

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.

大约 150 年前，当人们意识到我们在高中学到的熟悉的（平坦的）欧几里得几何并不是唯一可以考虑的几何时，几何的基础就受到了动摇。回想起来，考虑非平坦几何形状似乎很自然，因为我们生活的地球当然是圆形的。自从首次发现以来，数学家们研究了具有良好曲率特性的各种模型几何形状，例如圆形球体，并且我们继续提取有关这些模型几何形状的新信息，它们与平面欧几里得几何的区别，以及它们在我们所研究的宇宙中的应用。首席研究员将研究称为齐次爱因斯坦空间的特殊模型几何，以了解更多关于它们的基本属性并致力于它们的分类。该项目的主要目标是推进非紧齐次爱因斯坦和里奇的分类孤子空间。除了解决这些空间是否必然是求解流形的问题之外，PI 将致力于更深入地了解齐次 Ricci 孤子在 Ricci 流及其等距群下的稳定性的特殊性质。为了解决这些问题，PI 打算更完整地开发由 Heber、Lauret 等人发起的利用几何不变量理论研究齐次爱因斯坦度量的程序。