New Directions in Homogeneous Dynamics and Diophantine Approximation

本课题研究某些代数动力系统及其在数论中的应用。在这里，动力系统代表着一组抽象的点，以及控制点随时间移动的方式的演化规律。这种抽象的动力系统构成了科学和工程中各种重要现象的模型的基础。结果表明，数学中的许多关于有理数同时逼近实数的问题也可以用动力系统的行为来理解。此外，在这种情况下出现的系统是代数性质的，这使得使用各种复杂的工具来研究它们成为可能。本研究项目旨在推进代数动力系统在逼近理论中的框架，发展新的方法，并对该领域的结果进行深远的推广。研究生和本科生将参与这个项目，并学习数论和动力学中的新方法和新技术。这些学生还将在PRIMES计划的框架内监督研究项目，PRIMES计划是一个针对高中生的课后研究计划。近年来，有关数论和动力系统之间的联系的新想法大量涌入，这使得重新考虑进一步工作的一系列可能性成为可能。本研究项目继续使用这些新的思想和方法研究齐次动力学和丢番图近似中与轨道的渐近性质相关的现象。可使用的动力学工具包括：格空间上的定量无散度、Schmidt对策及其修改、随机游动的递归和积分不等式、测度刚性、有效混合和等分布性质。

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.