Symmetry in Solvmanifolds and Geometric Evolutions

在这个项目中，主要研究人员建议通过发展Riemannian和Lie结构来寻找给定齐次空间上的高度对称的黎曼度量，从而加深对齐次空间的等距群的理解。所使用的混合技术位于黎曼几何学和几何不变理论的交叉点上，并受到几何进化的推动，如利玛奇流。在存在可传递的幂零等距群的情况下，所提出的方法已被主要研究者成功地用来证明了零流形上的Ricci孤子度量有最大等距群。主要研究人员建议在更一般的情况下发展这种方法，即等距传递群是可解的，以得到类似于Einstein和Ricci孤子度规在解流形上的结果。紧致零流形上的类似问题也将被讨论。由于几个问题可以自然地用几何不变量理论的语言重新表述，这一途径也将被探索。这个项目致力于为给定物体寻找最佳形状的几何基本问题。感兴趣的对象是齐次空间，它是具有每个点看起来与其他点相同的性质的空间。这些空间是许多数学分支的基本例子，一个多世纪以来一直是现代几何学的灵感源泉。虽然人们对均匀空间进行了不懈的探索，但仍有许多根本性的问题没有得到解决。首席研究员将解决寻找首选度规的问题，目的是证明爱因斯坦和利玛窦孤子度规是固定均匀空间上最对称的几何选择，如果它们存在的话。PI将继续他的工作，指导本科生研究和吸引研究生进入研究领域。该项目有可能通过研究可解李群上的Ricci流，为他所在大学的本科生和研究生提供有吸引力和挑战性的机会。PI将继续他的工作，监督本科生的研究，并为他的研究领域吸引研究生。该项目有可能通过研究可解李群上的Ricci流，为他所在大学的本科生和研究生提供有吸引力和挑战性的机会。

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.