CURVATURE, SYMMETRY, AND STABILITY IN HOMOGENEOUS SPACES

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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.