Homogeneous Einstein Spaces

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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.