Symmetry in Solvmanifolds and Geometric Evolutions

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In this project, the principal investigator proposes to develop a deeper understanding of isometry groups of homogeneous spaces by evolving Riemannian and Lie structures to look for highly symmetric Riemannian metrics on a given homogeneous space. The hybrid techniques used lie at the intersection of Riemannian geometry and Geometric Invariant Theory and are motivated by geometric evolutions like the Ricci flow. In the presence of a transitive nilpotent group of isometries, the proposed techniques have been successfully employed by the principal investigator to show that Ricci soliton metrics on nilmanifolds have maximal isometry groups. The principal investigator proposes to develop this approach in the more general setting that the transitive group of isometries is solvable to achieve similar results for Einstein and Ricci soliton metrics on solvmanifolds. Similar questions on compact nilmanifolds will also be addressed. As several of the proprosed problems can naturally be rephrased in the language of Geometric Invariant Theory, this avenue will be explored as well.This project is devoted to the fundamental problem in geometry of finding a best shape for a given object. The objects of interest are homogeneous spaces, which are spaces having the property that every point looks the same as every other point. These spaces serve as basic examples across many branches of mathematics and have been a source of inspiration in modern geometry for over a century. Although homogeneous spaces have been tirelessly explored, there are still many fundamental questions that have not been resolved. The principal investigator will address this question of finding preferred metrics, aiming to show that Einstein and Ricci solitons metrics are the most symmetric choice of geometry on a fixed homogeneous space, when they exist.The PI will continue his work, supervising undergraduate research and attracting graduate students for area of research. The project has the potential to provide attractive and challenging opportunities for the undergraduate and graduate students of his university through the study of the Ricci flow on solvable Lie groups. The PI will continue his work, supervising undergraduate research and attracting graduate students for his area of research. The project has the potential to provide attractive and challenging opportunities for the undergraduate and graduate students of his university through the study of the Ricci flow on solvable Lie groups.