Rigidity of Abelian Actions

拟议的研究是在光滑动力学和遍历理论领域。提出的研究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，各种各样的这类系统表现出一定程度的混沌行为，是非常严格的。这些结果引发了数论和量子力学中一些长期存在的猜想的快速进展。这类系统的刚性与一维时间混沌系统的灵活性形成鲜明对比。这项拟议的研究有两个主要主题。一是进一步研究混沌行为较少的代数多维时间系统的稳定性和刚性：尼氏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阿贝尔作用的局部刚性。在这个方向上，建议的研究涉及KAM理论方法，因此需要详细研究相应的无穷小问题：关于这些行为的第一上同调的描述。第二个主题是研究强部分双曲的非代数系统的存在性和稳定性。这类系统具有特定的不变叶结构，一方面对动作的流形施加限制，另一方面在小扰动下趋于稳健，从而导致动作具有一定程度的稳定性。这一方向的研究导致了强部分双曲多维时间系统的全球分类。动力系统和混沌是在过去几年中蓬勃发展的数学领域，它们与其根源保持着密切的联系，这些根源在于对细胞生物学、纳米技术、气象学和工程学等领域中各种现象的研究。研究自然系统在时间上的演化是当今科学的核心兴趣之一。系统的稳定性问题是自然系统研究中出现的主要问题之一，因为数学模型只是对自然现象的近似。在天体力学中，太阳系的稳定性是一个重要的话题，而KAM理论被证明是更好地理解该系统长期行为的有力工具。具有多维时间的系统出现在量子力学中，其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的数学形式中，其物理性质和产生已经得到了深入的研究。在硬件架构方面，最近探索的场地之一是将经典方法扩展到多维时间(多维调度)。首席调查员将继续鼓励本科生，特别是女性，积极参与研究进程，并努力使他们接触到这项研究的各个方面和应用。这些活动将从赠款中受益。

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.