Mathematical Sciences: Dynamics of Unipotent Translations onHomogeneous Spaces

这个项目涉及对遍历理论的研究。一般说来，遍历理论涉及理解系统的平均行为，这些系统的动力学太复杂或太混乱，以至于无法在微观细节上进行跟踪。在“动力学”的标题下，可以把一组抽象的变换如何作用于光滑空间的现代理论放在下面。通过这种方式，遍历理论在寻求对均匀空间上的流动进行分类的过程中与几何相联系。拉特纳教授是遍历理论的世界领先者。她将继续研究连通李群的幂等子群的“刚性”。特别是，她将试图证明拉古纳坦的一个猜想，该猜想表明每个这样的子群都是“代数的”。这样的结果将对遍历理论、几何和数论的不同领域产生重要影响。

This project involves research in ergodic theory. Ergodic theory in general concerns understanding the average behavior of systems whose dynamics is too complicated or chaotic to be followed in microscopic detail. Under the heading "dynamics" can be placed the modern theory of how groups of abstract transformations act on smooth spaces. In this way ergodic theory makes contact with geometry in its quest to classify flows on homogeneous spaces. Professor Ratner is a world leader in ergodic theory. She will continue her work on the "rigidity" of unipotent subgroups of connected Lie groups. In particular, she will attempt to prove a conjecture of Raghunathan which states that every such subgroup is "algebraic". Such a result would have important implications for the diverse fields of ergodic theory, geometry, and number theory.