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

这个项目将解决与高阶光滑群作用有关的几个重要问题：不能通过经典构造从经典动力系统(微分同态和流)获得的作用的刻画；部分双曲高阶作用空间中混沌行为的盛行；保守作用的子空间的整体亚椭圆性在多大程度上给出了扰动下稳定性的一般设置；寻找弱刚性高阶作用的新的非代数例子。在依赖分析工具的代数高阶作用的研究结果和用于一般光滑作用的更动态的方法之间建立一座桥梁，对于改进对刚性现象的理解至关重要。对于代数高阶作用，该项目将研究几何上获得的上同调障碍与由表示上的诱导作用得到的解析上同调障碍之间的联系。战略的一部分是努力通过关注有代表性的例子来回答这些问题。主要研究人员的主要兴趣是具有较高秩但缺乏丰富的几何或代数结构的群(例如，交换群、幂零群、可解群)的作用。我们只有对足够简单的系统才能完全理解未来和过去，这些系统仅仅是对观察到的现象的近似。混乱的行为最初被认为是一种病理。然而，事实证明，这是一个稳定的来源。研究表明，健康的心跳证明了混乱的存在，这并不令人惊讶，前提是人们接受这样一个事实，即大自然更喜欢稳定。对于属于“群体行动”的系统来说，混沌和稳定性之间的关系更加戏剧化：较弱的混沌行为往往意味着系统具有较强的稳定性。群体行动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)中有用的模型。在特定的模型中，混沌以不同的伪装出现。这个项目的目标是探索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。本主题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，学生可以发展直觉，了解什么是开放问题，并通过具体的计算为实际研究做出自己的贡献。这就是该项目外展部分的理由，该部分针对高中女生，其目标是向处于智力发展早期的女学生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.