Rigidity Phenomena for Higher Rank Abelian Actions

建议编号：DMS-0140513PI：Boris Kalinin摘要建议的研究领域是光滑动力系统和遍历理论。该项目的主要目标是研究“更高阶”的动力系统。特别是，研究者将研究离散的和连续的高阶交换群的作用，它们是光滑流形上微分同胚和流的自然推广。高阶动力系统自然而然地出现在各种几何和代数对象的研究中。这些系统的主要例子包括李群的紧陪集上的自同构和平移的双曲和部分双曲作用。研究人员将利用动力学、解析和群论方法研究这类系统的刚性性质。可能的刚性性质的例子包括变量度量的描述、可测同构的正则性以及与代数模型的光滑同构的存在。动力系统和遍历理论是一个相对较新的数学领域，它研究物理和数学系统随时间的演化，例如行星系统、空气和流体流动。这一领域起源于对微分方程和天体力学的经典研究。动力学和遍历理论将新的数学工具引入到这些物理和力学领域，如研究长期定性行为以及各种分析和概率方法。动力学中的新思想和新概念，如分形学和混沌，不仅戏剧性地影响了这个领域本身，而且从根本上改变了我们对世界的理解。当今动力系统研究的影响甚至延伸到气象学、生物学和计算机科学。

Proposal Number: DMS-0140513PI: Boris KalininABSTRACTThe proposed research lies at the area of smooth dynamicalsystems and ergodic theory. The main goal of the project isto investigate "higher rank" dynamical systems. Inparticular, the investigator will study actions of discreteand continuous higher rank abelian groups, which are naturalgeneralizations of diffeomorphisms and flows on smoothmanifolds. Higher rank dynamical systems appear naturally inthe study of various geometric and algebraic objects. Theprime examples of these systems include hyperbolic andpartially hyperbolic actions by automorphisms andtranslations on compact cosets of Lie groups. Using dynamical,analytic, and group theoretic methods the investigatorwill study rigidity properties of such systems. The examplesof possible rigidity properties include description ofinvariant measures, regularity of measurable isomorphisms,and existence of smooth isomorphisms to the algebraic models.Dynamical systems and ergodic theory is a relatively new fieldof mathematics which studies the evolution of physical andmathematical systems over time, for example planet systems,air and fluid flows. This field originated from the classicalstudies in differential equations and celestial mechanics.Dynamics and ergodic theory introduced new mathematical toolsinto these areas of physics and mechanics, such as the studyof the qualitative behavior in the long run as well as variousanalytic and probabilistic methods. New ideas and concepts indynamics, such as fractals and chaos, have not only affectedthe field itself dramatically, but also fundamentally changedour understanding of the world. The influence of the studiesin dynamical systems nowadays goes as far as meteorology, biology, and computer science.