RUI: Invariant geometric structures and rigidity in hyperbolic dynamics.

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。这个项目属于双曲动力系统领域。这些光滑系统在某些方向上以指数膨胀而在其他方向上以指数收缩为特征。稳定性、刚性和双曲系统的分类是光滑动力学中的核心问题。这些问题已经得到了广泛的研究，其中许多问题在一维不变叶积系统中得到了解决。对于具有高维叶状结构的系统，情况要复杂得多。提出的研究的主要部分是研究高维双曲系统。主要研究者对与局部刚性（即小扰动下共轭的规律性）和全局刚性（即代数模型的光滑共轭的存在性）相关的性质特别感兴趣。研究不变叶理和几何结构的性质是本文研究的重要内容。动力系统领域是数学的一个现代分支，它产生于对物理和机械问题的研究。它的主要目标是描述各种抽象的和现实生活中的系统是如何随着时间的推移而演变的。由于这个原因，动力系统理论具有非常广泛的适用性，从天体力学和流体力学到气象学和社会科学。所谓的双曲系统一直是所谓的“光滑”动力学研究的主要对象之一。这些系统的指数收缩和膨胀产生具有复杂轨道结构的混沌行为。这就形成了一个丰富的理论，应用于应用数学和纯数学的各个领域，包括几何和数论。首席研究员在一个主要为本科生服务的机构工作，这是阿拉巴马州南部地区的主要大学。这个项目会对部门产生积极影响吗？该校的本科数学专业及其硕士课程，以及更广泛的学生群体和周边社区。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).This project lies in the area of hyperbolic dynamical systems. These are smooth systems characterized by exponential expansion in some directions and exponential contraction in the other directions. Stability, rigidity, and classification of hyperbolic systems are among the central issues in smooth dynamics. These problems have been extensively studied, and many of them were solved for systems with one-dimensional invariant foliations. The situation is much more complicated for systems with higher-dimensional foliations. The main part of the proposed research is to study higher-dimensional hyperbolic systems. The principal investigator is particularly interested in properties related to local rigidity (i.e., regularity of conjugacy under a small perturbation) and global rigidity (i.e., existence of a smooth conjugacy to an algebraic model). Investigating properties of invariant foliations and geometric structures plays an important part in the proposed research.The field of dynamical systems is a modern branch of mathematics that grew out of the study of physical and mechanical problems. Its main objective is to describe how various abstract and real-life systems evolve over time. For this reason the theory of dynamical systems has a very wide range of applicability, from celestial mechanics and hydrodynamics to meteorology and social sciences. So-called hyperbolic systems have been one of the main objects of study in what is known as "smooth" dynamics. The exponential contraction and expansion in these systems produce chaotic behavior with complex orbit structure. This results in a rich theory with applications in various areas of applied and pure mathematics, including geometry and number theory. The principal investigator works in a primarily undergraduate-serving institution that is the main university in the southern Alabama region. The project will positively affect the department?s undergraduate mathematics major and its Master's program, as well as the broader student population and the surrounding community.