Perturbations of smooth group actions and cohomology

该项目的中心目标是更好地了解顺利的集体行动的当地结构。在群体行动的小扰动中缺乏动态多样性(局部刚性)是一种罕见的现象，通常与过去二十年发现的其他刚性现象有关。碰巧的是，某种形式的无穷小(或上同调)刚性可能导致局部刚性。在经典动力系统领域，这一点在20世纪60年代被证明为丢番图圆旋转，并导致了Kam定理。这个项目涉及一种基于上同调数据来描述局部结构的一般方法，即使在没有上同调刚性的情况下也是如此。该项目包括应用于几种感兴趣的情况，其中上同调不是微不足道的，但被很好地理解，以及上同调很简单，但在其描述中没有足够的分析精度。理解动力系统的微小扰动一直是科学的必要条件之一，因为对系统随时间的行为的完整预测通常只能对数学上简单的系统进行。太阳系是自然系统的一个例子，它是一个简单系统的微小扰动。在这种情况下，一个著名的数学定理(Kam定理，本项目的灵感就是从这个定理中得到的)揭示了系统的稳定性。这个项目的重点是群体行动，它可以被视为具有“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)中的许多系统可以用考虑多维时间的数学系统比用经典系统(其中时间是一维的)更有效地建模。这就需要了解这类系统的微小扰动。另一方面，局部结构的刚性往往只是系统整体刚性的一种表现。其他问题，包括具有多维时间的某些系统的所谓不变度量的刚性，与数论中的重要问题密切相关。

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.