Perturbations of smooth group actions and cohomology

光滑群作用和上同调的扰动

• 批准号: 1001884
• 负责人: Danijela Damjanovic
• 金额: $ 13.3万
• 依托单位: William Marsh Rice University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-07-15 至 2014-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1001884&HistoricalAwards=false
• 关键词: Perturbations; smooth; group; actions; cohomology

The central objective of this project is to obtain a better understanding of the local structure of smooth group actions. Lack of dynamical diversity (local rigidity) among small perturbations of a group action is a rare phenomenon and is typically connected to other rigidity phenomena that have been discovered in the past two decades. It happens that some form of infinitesimal (or cohomological) rigidity may lead to local rigidity. In the realm of classical dynamical systems this was proved for Diophantine circle rotations in the 1960s and led to the KAM Theorem. This project is concerned with a general approach to describe local structure based on the cohomological data, even in the absence of cohomological rigidity. The project includes applications to several situations of interest where cohomology is not trivial but is well understood, and to situations where cohomology is simple but there is not enough analytical precision in its description.Understanding small perturbations of dynamical systems has always been one of the imperatives of science, given the fact that complete prediction of system behavior over time can typically be carried out only for mathematically simple systems. The solar system is one example of a natural system that is a small perturbation of a simple system. In this case, a celebrated mathematical theorem (the KAM Theorem, from which the current project draws its inspiration) sheds light on the stability of the system. Group actions, the focal subject of this project, can be viewed as dynamical systems with "multidimensional time," and many systems in biology (neural networks), computer science (multidimensional data storage media), and physics (quasi-crystals) can be modeled more efficiently by mathematical systems that allow for multidimensional time than by classical systems in which time is one-dimensional. This brings about the need to understand small perturbations of such systems. On the other hand, rigidity of local structure is often just one manifestation of a system that is rigid in the large. Others, including rigidity of so-called invariant measures for certain systems with multidimensional time, are closely connected to important problems in number theory.

这个项目的中心目标是获得更好的了解当地的结构顺利集团行动。在群作用的小扰动中缺乏动力学多样性（局部刚性）是一种罕见的现象，通常与过去二十年中发现的其他刚性现象有关。有时候，某种形式的无穷小（或上同调）刚性会导致局部刚性。在经典动力系统领域，这在20世纪60年代针对丢番图圆旋转得到了证明，并引出了KAM定理。本项目关注的是一个通用的方法来描述局部结构的基础上的上同调数据，即使在缺乏上同调刚性。该项目包括应用于几种感兴趣的情况下，上同调不是微不足道的，但很好地理解，以及上同调是简单的，但没有足够的分析精度在其描述的情况。了解动力系统的小扰动一直是科学的当务之急之一，假定通常只能对数学上简单的系统进行系统行为随时间的完全预测。太阳系是自然系统的一个例子，它是一个简单系统的小扰动。在这种情况下，一个著名的数学定理（KAM定理，本项目从中汲取灵感）揭示了系统的稳定性。本项目的重点课题群作用可以被看作是具有“多维时间”的动力系统，生物学（神经网络）、计算机科学（多维数据存储介质）和物理学（准晶体）中的许多系统可以通过允许多维时间的数学系统比时间是一维的经典系统更有效地建模。这就需要了解这种系统的小扰动。另一方面，局部结构的僵化往往只是一个系统整体僵化的表现之一。其他的，包括刚性的所谓不变措施，某些系统的多维时间，是密切相关的重要问题，数论。