Homogeneous spaces with distinguished geometric structures

This project revolves around the study of natural model spaces. For 2,000 years, until a century and a half ago, humanity had one model with which to explore geometry, namely, (flat) Euclidean space. This is the geometry that we all learned in high school. However, to advance science and engineering, we need to use models that incorporate curvature - for example, the Earth is not flat! To this end, the principal investigator (PI) will study special model geometries with symmetry called homogeneous Einstein spaces to investigate their basic properties and work towards their classification. The PI will develop new tools for analyzing these spaces, training a new generation of graduate student mathematicians to go into industry and academia. The PI will continue his outreach to K-12 students, mentoring of undergraduate students through research projects, and organizing of workshops and conferences. The PI will study homogeneous Einstein metrics, continuing the development of tools at the interface of Geometric Invariant Theory and Geometric Analysis. The goal of this program is to classify non-compact, homogeneous Einstein manifolds by developing a list of algebraic invariants which completely characterizes the existence of such Riemannian metrics. Conversely, the PI will use tools from the geometry of Einstein solvmanifolds to further explore and extract deep algebraic properties of these 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.