New Methods for Smooth Rigidity of Algebraic Actions

动力系统领域起源于微分方程和天体力学。 它研究系统在运动规律下的长期行为。 到目前为止，具有强混沌特性的系统已经得到了很好的理解。这些结果提供了应用到其他领域的数学以及许多领域的科学，如物理学，力学，计算机科学和生物学。 然而，在许多感兴趣的系统中，只能观察到弱混沌行为。 这个项目的目标是开发新的工具来研究这样的系统;然后将这些结果应用于研究数学的其他领域，如数论和表示论。近年来，对高阶不可约代数作用的研究越来越受到人们的关注。 这些作用的许多例子都表现出一系列显著的刚性特性。 这些例子具有很强的混沌特性。 该项目旨在提供具有较少混沌行为的刚性示例。 这个项目将研究一个广泛的代数作用的光滑刚性，特别是对于完全没有双曲性的作用，如抛物作用。目前的工具无法应用于处理这些行动。 本计画的主要研究主题是发展一种结合经典KAM（Kolmogorov-Arnold-Moser）方法与表示论的方法来研究一大类代数作用的刚性行为。新的方法将提供第一个局部刚度的例子抛物线行动，并有可能建立局部刚度的部分双曲行动的几何性质是明显不同于现有的例子。

The field of dynamical systems originated from differential equations and celestial mechanics. It studies the long run behavior of a system subject to laws of motion. So far, systems with strong chaotic properties have been well understood. These results provide applications to other areas of mathematics as well as to many areas of sciences such as physics, mechanics, computer science, and biology. In many systems of interest however, only weak chaotic behaviors can be observed. The goal of this project is to develop new tools to study such systems; and then apply these results to study other areas of mathematics, such as number theory and representation theory. There is a growing interest in recent years for irreducible higher rank algebraic actions. Many examples of these actions exhibit a remarkable array of rigidity properties. These examples possess strong chaotic properties. This project aims at providing rigidity examples with less chaotic behaviors. This project will study smooth rigidity of a broad class of algebraic actions, especially for the actions completely absent of hyperbolicity, like parabolic actions. Current tools fail to be applied to treat these actions. The main research theme in this project is to develop a method that combines classical KAM (Kolmogorov-Arnold-Moser) approach with representation theory to study the rigidity behavior of a broad class of algebraic actions. The new method will afford the first local rigidity examples for parabolic actions; and has the potential to establish local rigidity for partially hyperbolic actions whose geometric properties are distinctly different from existing examples.