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。该项目涉及双曲动力系统领域。这些光滑系统的特征是在某些方向上呈指数扩张，而在另一个方向上呈指数收缩。双曲系统的稳定性、刚性和分类是光滑动力学的中心问题之一。这些问题已经得到了广泛的研究，其中许多问题已经解决了具有一维不变叶的系统。对于具有更高维叶状结构的系统来说，情况要复杂得多。本研究的主要内容是研究高维双曲型系统。主要研究者特别感兴趣的是与局部刚性(即，在小扰动下的共轭正则性)和全局刚性(即，代数模型的光滑共轭的存在性)有关的性质。研究不变叶和几何结构的性质在所提出的研究中起着重要作用。动力系统领域是从物理和力学问题的研究中发展起来的一个现代数学分支。它的主要目标是描述各种抽象的和真实的系统如何随着时间的推移而演变。出于这个原因，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被称为“光滑”动力学的主要研究对象之一。这些系统的指数收缩和指数膨胀产生了具有复杂轨道结构的混沌行为。这导致了丰富的理论，在应用数学和纯数学的各个领域中都有应用，包括几何和数论。首席研究员在一所主要为本科生服务的机构工作，这所机构是阿拉巴马州南部地区的主要大学。该项目将对S系数学本科专业及其硕士点，以及更广泛的学生群体和周围社会产生积极的影响。

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.