Rigidity of Abelian Actions

阿贝尔行为的刚性

• 批准号: 1004908
• 负责人: Danijela Damjanovic
• 金额: $ 7.26万
• 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: 2009
• 资助国家: 美国
• 起止时间: 2009-08-20 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1004908&HistoricalAwards=false
• 关键词: Rigidity; Abelian; Actions

The proposed research is in the area of smooth dynamics and ergodic theory. The main focus of the proposed research is the study of dynamical systems with multidimensional time. Intensive research during the past two decades proved that wide variety of such systems which display certain degree of chaotic behavior, are remarkably rigid. These results induced fast progress towards some long standing conjectures in number theory and quantum mechanics. Rigidity of such systems stands in sharp contrast with flexibility of chaotic systems with one-dimensional time. There are two main themes within the proposed research. The first is to explore further stability and rigidity of algebraic multidimensional-time systems with less chaotic behavior: existence and local rigidity (i.e. differentiable stability) of partially hyperbolic abelian actions on nilmanifolds, and local rigidity of parabolic abelian actions on certain classes of locally symmetric spaces. In this direction the proposed research involves the KAM theory approach and thus requires a detailed study of the corresponding infinitesimal problem: the description of the first cohomology over these actions. The second theme is to explore the existence and stability properties of non-algebraic systems which are strongly partially hyperbolic. Such systems have specific structure of invariant foliations which on one hand imposes restrictions for the manifold of the action, and on the other hand it tends to be robust under small perturbations thus leading to certain degree of stability for the action. The research in this direction leads towards global classification of strongly partially hyperbolic multidimensional-time systems.Dynamical systems and chaos are the areas of mathematics which have flourished during past years while maintaining a strong connection with their roots which lie in the study of various phenomena in domains like cell biology, nano technology, meteorology and engineering. The studies of evolution of nature systems in time represent one of the core scientific interests today. The problem of stability of systems is one of the main issues which arises in the study of nature systems as mathematical models are merely approximations of the natural phenomena. In celestial mechanics, stability of the solar system is one important topic, and the KAM theory turned out to be a powerful tool towards better understanding of the system's long term behavior. Systems with multi-dimensional time appear in quantum mechanics, where rigidity of such systems is present in the form of uniform distribution of quantum states. Multidimensional-(lattice) time systems also appear in the mathematical formalism for quasicrystals, whose physical properties and generation have been intensively studied. In hardware architecture one of the recently explored venues is extending the classical methodology to multidimensional time (multidimensional scheduling). The principal investigator will continue to encourage undergraduate students, female in particular, to take active part in the research process and will make an effort to expose them to various aspects and applications of this research. These activities will benefit from the grant.

拟议的研究属于平滑动力学和遍历理论领域。 该研究的主要重点是多维时间动力系统的研究。过去二十年的深入研究证明，表现出一定程度混沌行为的各种此类系统都非常僵化。这些结果促使数论和量子力学中一些长期存在的猜想取得了快速进展。 这种系统的刚性与一维时间混沌系统的灵活性形成鲜明对比。拟议的研究有两个主题。第一个是进一步探索具有较少混沌行为的代数多维时间系统的稳定性和刚性：尼尔流形上部分双曲阿贝尔作用的存在性和局部刚性（即可微稳定性），以及某些类局部对称空间上抛物线阿贝尔作用的局部刚性。在这个方向上，所提出的研究涉及 KAM 理论方法，因此需要详细研究相应的无穷小问题：这些行为的第一上同调的描述。 第二个主题是探索强部分双曲非代数系统的存在性和稳定性。 这种系统具有特定的不变叶状结构，一方面对动作的多样性施加限制，另一方面它在小扰动下往往具有鲁棒性，从而导致动作的一定程度的稳定性。这个方向的研究导致了强部分双曲多维时间系统的全局分类。动力系统和混沌是过去几年蓬勃发展的数学领域，同时与其根源在于细胞生物学、纳米技术、气象学和工程学等领域的各种现象的研究保持着紧密的联系。对自然系统随时间演化的研究代表了当今的核心科学兴趣之一。系统稳定性问题是自然系统研究中出现的主要问题之一，因为数学模型仅仅是自然现象的近似。在天体力学中，太阳系的稳定性是一个重要的话题，KAM 理论被证明是一个强大的工具，可以帮助我们更好地理解系统的长期行为。多维时间系统出现在量子力学中，这种系统的刚性以量子态均匀分布的形式存在。多维（晶格）时间系统也出现在准晶体的数学形式中，其物理性质和生成已得到深入研究。在硬件架构中，最近探索的领域之一是将经典方法扩展到多维时间（多维调度）。首席研究员将继续鼓励本科生，特别是女生，积极参与研究过程，并努力让他们接触到这项研究的各个方面和应用。这些活动将受益于赠款。