Rigidity and non-rigidity of commutative algebraic actions

拟议的项目属于动力系统领域，也涉及数论。主要对象是高阶交换群对环面和其他齐次空间的代数作用。对此类作用的研究对动力学系统和丢番图近似的进步做出了重大贡献。本研究旨在研究这些系统的刚性，这种现象的特点是在作用下缺乏不变的对象。主要目标是阐明高阶、双曲性和不可约性这三个动作享受刚性所需的三个典型条件，它们之间的平衡以及定量地影响刚性特性。动力系统领域根据某些给定规则研究点随时间演变的长期轨迹。该学科起源于牛顿的机械定律，在物理学、生物学、经济学和其他科学领域有着广泛的应用。拟议的研究将集中于算术性质的动力系统。换句话说，进化的规则可以用一组整数和有理数来表征。通过研究此类系统的动态行为，人们期望发现数论特性。这项工作还试图通过结合这些数学分支的方法来加强遍历理论、数论、李群、傅里叶分析和加性组合数学之间的联系。 PI 打算在项目期间编写说明性注释，并教授相关主题的研究生课程。

The proposed project lies in the area of dynamical systems and also involves number theory. The main objects are algebraic actions on tori and other homogeneous spaces by higher-rank commutative groups. The studies of such actions have contributed signi cantly to advances both in dynamical sytems and in Diophantine approximations. This research aims to investigate rigidity of these systems, a phenomenon characterized by the scarcity of invariant objects under the action. The main goal is to shed light on how higher-rank, hyperbolicity and irreducibility, the three typical conditions required in order for an action to enjoy rigidity, balance among themselves and quantitatively a ffect rigidity properties.The field of dynamical system studies long-term trajectories of points evolving over time according to certain given rules. Having its origin in Newton's mechanical laws, the subject has found numerous applications in physics, biology, economics and other sciences. The proposed research will focus on dynamical systems of arithmetic nature. In other words, the rules of evolution can be characterized by a set of integers and rational numbers. By studying dynamical behaviors of such systems, one expects to discover number-theoretical properties. This work also seeks to strengthen the connections between ergodic theory, number theory, Lie groups, Fourier analysis and additive combinatorics by combining methods from these mathematical branches. The PI intends to develop expository notes, as well as to teach graduate-level courses on relevant topics during the project.