Dynamics on homogeneous spaces and Moduli spaces

齐次空间和模空间上的动力学

• 批准号: 1500677
• 负责人: Amir Mohammadi
• 金额: $ 19.7万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-06-01 至 2017-03-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1500677&HistoricalAwards=false
• 关键词: Dynamics; homogeneous; spaces; Moduli

Dynamical systems is the study of the evolution of systems which are changing over time. As a concrete example one can consider the trajectory of a ball on an ideal table. The table is frictionless and the angle of incidence equals the angle of reflection. A classical mathematical problem is to study the trajectories of a ball when the sides of the table form a polygon - not necessarily a rectangle. There can be different types of trajectories. Some trajectories can be periodic and some can be dense on the table. This simple problem is notoriously difficult. It is very difficult to solve the problem for a particular table, unless it is of a special shape; for example, a rectangle or an equilateral triangle. This leads to the study of families of tables that have similarities. You could for instance, study the family of tables with five sides. Taking the family of tables as a new space it is possible to define a new flow on this space. This point of view turns out to have important consequences. For example, we may now be able to say what happens on "most" tables. This proposal studies dynamical systems by taking a similar point of view.We mainly seek rigidity results where rather weak initial data about an object yields an almost complete classification of the object. The following will be the main objectives: (i) Employing a dynamical approach to study problems in number theory and geometry has proven rather fruitful. However, this approach is often noneffective. We will seek effectivization of the rigidity phenomena for the action of groups generated by unipotent subgroups on homogeneous spaces; these rigidity results have served as one of the main tools in the aforementioned applications. (ii) There is an action of the group of nonsingular, real, two by two matrices on the moduli space of a compact Riemann surface; this is closely related to the asymptotic of the number of periodic trajectories on rational polygonal tables. This proposal seeks generalizations of the recent exciting developments which proved certain rigidity results for this action. (iii) We attempt to investigate dynamics on homogeneous spaces with infinite volume, and on homogeneous spaces arising from local fields of positive characteristic. There are various geometric and number theoretical applications which motivate the study of these spaces.

动力系统是研究随时间变化的系统的演化。 作为一个具体的例子，可以考虑球在理想桌子上的轨迹。台面是无摩擦的，入射角等于反射角。 一个经典的数学问题是研究球的轨迹时，表的两侧形成一个多边形-不一定是矩形。 可以有不同类型的轨迹。 有些轨迹可以是周期性的，有些可以是密集的。 这个简单的问题是出了名的难。 这是非常困难的解决问题的一个特定的表，除非它是一个特殊的形状;例如，一个矩形或等边三角形。 这导致了对具有相似性的表族的研究。 例如，你可以研究有五个边的桌子族。 将表族作为一个新的空间，可以在该空间上定义一个新的流。 这种观点产生了重要的后果。 例如，我们现在可以说“大多数”桌子上发生了什么。 这个建议研究动力系统采取类似的观点。我们主要寻求刚性的结果，相当弱的初始数据的对象产生一个几乎完整的分类的对象。 主要目标如下： (i)采用动力学方法研究数论和几何问题已被证明是卓有成效的。 然而，这种方法往往是无效的。 我们将寻求有效化的刚性现象所产生的单位幂子群在齐次空间上的群的行动，这些刚性的结果已作为上述应用的主要工具之一。 (ii)有一个行动的非奇异的，真实的，两个由两个矩阵的模空间上的一个紧凑的黎曼曲面，这是密切相关的渐近数量的周期轨迹的理性多边形表。 这一建议旨在概括最近令人兴奋的发展，这些发展证明了这一行动的某些刚性结果。 (iii)我们试图研究具有无限体积的齐次空间上的动力学，以及由正特征的局部场引起的齐次空间上的动力学。有各种几何和数论的应用，激励这些空间的研究。