Symmetry in Solvmanifolds and Geometric Evolutions

在这个项目中，首席研究员建议通过发展黎曼和李结构来寻找给定齐次空间上的高度对称黎曼度量，从而对齐次空间的等长群有更深的理解。所使用的混合技术位于黎曼几何和几何不变理论的交叉点，并受到像里奇流这样的几何演化的推动。在传递幂零等距群存在的情况下，本研究成功地证明了幂零流形上的Ricci孤子度量具有极大的等距群。首席研究员建议在更一般的情况下发展这种方法，即传递群的等距是可解的，以达到爱因斯坦和里奇在溶剂流形上的孤子度量的类似结果。关于紧零流形的类似问题也将讨论。由于所提出的几个问题可以自然地用几何不变理论的语言重新表述，因此也将探索这一途径。该项目致力于为给定物体寻找最佳形状的几何基本问题。我们感兴趣的对象是齐次空间，这些空间具有这样的性质即每个点看起来都和其他点一样。这些空间是许多数学分支的基本范例，一个多世纪以来一直是现代几何的灵感来源。尽管人们对同质空间进行了孜孜不倦的探索，但仍有许多基本问题没有得到解决。首席研究员将解决寻找首选度量的问题，旨在证明爱因斯坦和里奇孤子度量是固定齐次空间上最对称的几何选择，当它们存在时。PI将继续他的工作，指导本科生的研究和吸引研究生的研究领域。该项目有可能通过研究可解李群上的Ricci流，为他的大学的本科生和研究生提供具有吸引力和挑战性的机会。PI将继续他的工作，指导本科生的研究，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，为他的大学的本科生和研究生提供具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.