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

职业生涯：顺利的群体行动——混乱行为的持续存在和普遍存在

• 批准号: 1150210
• 负责人: Danijela Damjanovic
• 金额: $ 40万
• 依托单位: William Marsh Rice University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-07-01 至 2015-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1150210&HistoricalAwards=false
• 关键词: CAREER; Smooth; Group; Actions; Persistence

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.

该项目将解决有关高级平滑组动作的几个重要问题：通过经典结构从经典的动力学系统（差异和流动）中获得的动作表征；在部分双曲更高级作用的空间中混乱行为的普遍性；司板冠军对保守行为的全球性低纤维化在多大程度上为在扰动下的稳定性提供了一般设置；寻找弱刚性高级动作的新的非代码示例。依赖分析工具的代数高级动作的发现与用于一般平稳动作的更具动力的方法之间建立了桥梁，这对于改善对刚性现象的理解至关重要。对于代数高级行动，该项目将在几何学上研究的辅助障碍与从诱导的代表作用进行分析获得的群体学障碍之间的联系。 策略的一部分是通过关注代表性示例来努力回答这些问题。主要研究者的主要兴趣是对等级较高但缺乏丰富几何或代数结构的群体的行动（例如，阿贝尔群体，尼尔氏群体，可解决的群体）。动态下的动态性在科学中是一个古老的问题。我们完全理解未来和过去仅用于足够简单的系统，这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。但是，事实证明它是稳定的来源。研究表明，这是一种健康的心跳，表明混乱的存在并不奇怪，只要人们接受自然更喜欢稳定性的事实。对于属于标题“小组动作”的系统，混乱与稳定之间的关系更加引人注目：较弱的混乱行为倾向于暗示系统的稳定性更强。可以将小组动作视为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）的有用模型。在特别的模型上，混乱以不同的形式出现。该项目的目的是探索各种混乱行为的系统在扰动下保留其动力学特性。该主题尤其适合向学生介绍动态系统领域的研究。通过对简单模型的研究，学生可以发展直觉，了解开放问题是什么，并通过执行特定的计算对实际研究做出自己的贡献。这是该项目的外展部分的基本原理，其目的是针对高中女孩，其目标是在知识发展的早期阶段向女学生介绍数学研究和洞察力。