Rigidity and non-rigidity of commutative algebraic actions

拟议的项目在于动力系统领域，也涉及数论。主要对象是代数行动的环面和其他齐次空间的高秩交换群。这类作用的研究对动力学系统和丢番图近似的发展都有重要贡献。本研究的目的是探讨这些系统的刚性，一个现象的特点是稀缺的不变对象下的行动。主要目标是阐明高阶、双曲性和不可约性，这三个典型的条件是如何使一个动作享受刚性、它们之间的平衡以及定量地影响刚性属性的。动力系统领域研究点的长期轨迹根据某些给定的规则随时间演化。由于起源于牛顿力学定律，该学科在物理学，生物学，经济学和其他科学中有许多应用。拟议的研究将集中在动力系统的算术性质。换句话说，进化的规则可以用一组整数和有理数来表征。通过研究这类系统的动力学行为，人们期望发现数论性质。这项工作还试图加强遍历理论，数论，李群，傅立叶分析和添加剂组合学之间的联系，结合这些数学分支的方法。PI打算在项目期间编写简要说明，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.