Interactions Between Homogeneous Dynamics and Number Theory

这个项目要解决的问题一方面涉及李群齐次空间上的流动的动力学，另一方面涉及丢番图逼近的多维理论。在过去的二十年里，这两个领域之间发现了各种联系，这极大地促进了这两个领域的进步。近年来，作者的研究集中于发展齐次动力学和数论之间的新联系，从而解决了许多重要问题，并为进一步的研究创造了新的方向。研究者将继续他在有界轨道、轨道增长率、带权丢番图逼近和流形上的Khintchine型定理方面的工作。本课题研究代数动力系统及其在数论中的应用。关于有理数同时逼近实数的许多问题可以用某些轨道的行为来表示。在目前的上下文中，动力系统处理的是系统中的点如何在给定的一组微分方程式(或自然定律)支配系统的情况下随时间移动。结果表明，整数方程理论中使用的各种数值近似，只要用动力系统语言表述，就能更好地计算。

ABSTRACTThe problems that are to be addressed in this project involve, on one hand, dynamics of flows on homogeneous spaces of Lie groups, and, on the other hand, the multi-dimensional theory of Diophantine approximations. Various connections between these two fields have been found in the last two decades, which significantly stimulated progress in both fields. During recent years, the proposer's research has been centered on developing new links between homogeneous dynamics and number theory, which has resulted in solving many important problems, as well as in creating new directions for further research. The investigator is to continue his work on bounded trajectories,growth rate of orbits, Diophantine approximation with weights and Khintchine-type theorems on manifolds. This project deals with algebraic dynamical systems and their applications to number theory. Many problems concerning simultaneous approximation of real numbers by rational numbers can be cast in terms of the behavior of certain orbits. Dynamical systems in the present context deal with how points in a system move over time, given a set of differential equations (or laws of nature) governing the system. It turns out that various numerical approximations used in the theory of integer equations can be better calculated once they are phrased in dynamical systems language.