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。它研究受运动规律支配的系统的长期行为。到目前为止，具有强混沌性质的系统已经被很好地理解了。这些结果为其他数学领域以及物理、力学、计算机科学和生物学等许多科学领域提供了应用。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些系统；然后将这些结果应用于数学的其他领域，如数论和表示理论。近年来，人们对不可约高阶代数作用越来越感兴趣。这些动作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性质。本项目旨在提供具有较少混沌行为的刚性示例。本课题将研究一类代数作用的光滑刚性，特别是完全不存在双曲性的作用，如抛物线作用。当前的工具无法应用于处理这些操作。本项目的主要研究主题是开发一种将经典的KAM （Kolmogorov-Arnold-Moser）方法与表示理论相结合的方法来研究一类代数动作的刚性行为。新方法将提供抛物作用的第一个局部刚度实例；并有可能建立局部刚性的部分双曲作用，其几何性质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.