CAREER: Smooth Group Actions - Persistence and Prevalence of Chaotic Behavior

本计画将探讨高阶光滑群作用的几个重要问题：无法从经典动力系统获得的作用的表征（仿射和流）通过经典的建设;在部分双曲高阶行动的空间混沌行为的流行;在何种程度上的整体亚椭圆的sublaplacian保守行动给出了一个一般的设置稳定扰动下;寻找弱刚性高阶作用的新的非代数例子。建立一个桥梁之间的发现代数高阶行动，依赖于分析工具和更动态的方法用于一般平滑的行动是至关重要的，以提高对刚性现象的理解。对于代数高阶行动，该项目将研究几何上获得的上同调障碍物与从表象上的诱导行动中获得的解析障碍物之间的联系。 该战略的一部分是通过侧重于有代表性的例子来回答这些问题。主要研究者的主要兴趣是具有较高等级但缺乏丰富几何或代数结构的群体的行动（例如，阿贝尔群，幂零群，可解群）。动力学在扰动下的持久性是科学中的一个老问题。我们只有在足够简单的系统中才能完全理解未来和过去，这些系统仅仅是观察到的现象的近似。混沌行为最初被认为是一种病理学。然而，这是稳定的来源。研究表明，这是一个健康的心跳，证明了混乱的存在并不奇怪，只要一个人接受的事实，自然界喜欢稳定。对于属于“群体作用”的系统，混沌和稳定性之间的关系甚至更加戏剧性：较弱的混沌行为往往意味着系统的较强稳定性。群体行为可以被认为是具有多维时间的系统。因此，它们是生物学（神经网络）、化学（准晶体）和计算机科学（多维数据存储）中的有用模型。在特定的模型中，混沌以不同的形式出现。本计画的目标是探讨具有不同混沌行为的系统在扰动下保持其动力学性质的条件。这个题目特别适合于向学生介绍动力系统领域的研究。通过学习简单的模型，学生可以培养直觉，了解开放的问题是什么，并通过执行特定的计算为实际研究做出自己的贡献。这就是该项目外联部分的理由，该部分针对高中女生，其目标是在女生智力发展的早期阶段向她们介绍数学研究和对学术生涯的见解。

This project will address several important questions concerning higher-rank smooth group actions: characterization of actions that cannot be obtained from classical dynamical systems (diffeomorphisms and flows) via classical constructions; the prevalence of chaotic behavior in the space of partially hyperbolic higher-rank actions; the extent to which global hypoellipticity of the sublaplacian for conservative actions gives a general set-up for stability under perturbations; the search for new nonalgebraic examples of weakly rigid higher-rank actions. The creation of a bridge between findings on algebraic higher-rank actions that rely on analytic tools and the more dynamical approach used for general smooth actions is crucial for an improved understanding of rigidity phenomena. For algebraic higher-rank actions, the project will study connections between cohomological obstructions obtained geometrically and those obtained analytically from the induced action on representations. Part of the strategy is to work towards answering these questions by focusing on representative examples. The principal investigator's main interest is in actions by groups that have higher rank but that lack rich geometric or algebraic structure (e.g., abelian groups, nilpotent groups, solvable groups).Persistence of dynamics under perturbations is an old question in science. We completely understand the future and the past only for sufficiently simple systems, which are merely approximations of observed phenomena. Chaotic behavior was initially considered to be a pathology. However, it turns out to be a source of stability. Studies showing that it is a healthy heartbeat that demonstrates the presence of chaos are not surprising, provided that one accepts the fact that nature prefers stability. For systems that fall under the heading "group actions" the relation between chaos and stability is even more dramatic: weaker chaotic behavior tends to imply stronger stability for the system. Group actions can be thought of as systems with multidimensional time. As such, they are useful models in biology (neural networks), chemistry (quasi-crystals), and computer science (multidimensional data storage). In particular models, chaos appears in different guises. It is a goal of this project to explore conditions under which systems with diverse chaotic behavior preserve their dynamical properties under perturbations. This topic is especially amenable to introducing students to research in the area of dynamical systems. Through the study of simple models, students can develop intuition, learn what the open problems are, and make their own contribution to the actual research by performing specific computations. This is the rationale for the project's outreach component, which is aimed at high-school girls and the goal of which is to introduce mathematical research and insights into academic careers to female students at an early stage in their intellectual development.