New Directions in Homogeneous Dynamics and Diophantine Approximation

这个项目涉及某些代数动力系统及其在数论中的应用。这里的动力系统代表一组抽象的点，以及一个控制点随时间移动的演化定律。这种抽象的动力系统形成了科学和工程中广泛的重要现象模型的基础。事实证明，数学中的许多问题也可以用动力系统的行为来理解，这些问题涉及到用有理数同时逼近真实的数。此外，在这种情况下出现的系统是代数性质的，这使得有可能使用各种各样的复杂工具进行调查。该研究项目旨在推进近似理论中代数动力系统的框架，开发新方法，并在该领域获得具有深远意义的推广结果。研究生和本科生将参与该项目，并介绍数论和动力学的新方法和技术。 这些学生还将在PRIMES计划的框架内监督研究项目，PRIMES计划是一项针对高中生的课后研究计划。近年来，关于数论和动力系统之间的联系的新思想不断涌现，使人们有可能重新思考进一步工作的可能性。本研究计画继续利用这些新的观念与方法，研究齐次动力学与丢番图近似中与轨道渐近性质相关的现象。要使用的动态工具是：定量非发散空间的格子，施密特游戏及其修改，随机游动和积分不等式的复发，措施刚性，有效的混合，和equidistribution属性。

This project deals with certain algebraic dynamical systems and their applications to number theory. A dynamical system here stands for an abstract set of points together with an evolution law that governs the way points move over time. Such abstract dynamical systems form the basis for models of a wide range of important phenomena in science and engineering. It turns out that also many problems in mathematics concerning simultaneous approximation of real numbers by rational numbers can be understood in terms of the behavior of dynamical systems. Furthermore, systems that arise in this context are of algebraic nature, which makes it possible to use a wide variety of sophisticated tools for their investigation. This research project aims to advance the framework of algebraic dynamical systems in approximation theory, develop new methods, and obtain far-reaching generalizations of results in the field. Graduate and undergraduate students will be involved in the project and introduced to new methods and techniques in number theory and dynamics. These students will also supervise research projects within the framework of the PRIMES program, an after-school research program for high school students. During recent years there has been an influx of new ideas concerning connections between number theory and dynamical systems, making it possible to rethink a spectrum of possibilities for further work. This research project continues the study of phenomena in both homogeneous dynamics and Diophantine approximation related to asymptotic properties of orbits using those new ideas and methods. Dynamical tools to be used are: quantitative non-divergence on the space of lattices, Schmidt games and their modifications, recurrence of random walks and integral inequalities, measure rigidity, effective mixing, and equidistribution properties.