Group Actions, Homogeneous Dynamics, and Number Theory

数学中的许多问题都伴随着一组具有所谓群结构的对称性。探索潜在的群作用的动力学，即研究空间中的点在对称性下的行为，最近在证明重要结果和解决数学中长期存在的问题方面发挥了核心作用。本研究课题的主要目的是，将群作用动力学作为研究代数、几何、数论等数学领域的问题的有力工具，并深入理解这些数学领域之间的联系，推进技术的发展。本研究课题将特别关注以下三个方向：（1）发展不定积分二次型的约化理论，并考虑利用动力系统的方法将其推广到数域。(2)根据谱理论和自守形式理论的最新发展，研究李群上的随机游动和齐次空间中的等分布问题。 (3)将动力学方法融入算术群的代数和几何理论，研究算术群生成元的高度界。该项目还旨在寻求新的相互作用，并发展群体，动力学和数论研究中的技术。

Many questions in mathematics come with a set of symmetries that has a so-called group structure. Exploring the dynamics of underlying group actions, namely to study how points in spaces behave under symmetries, has recently played a central role in proving important results and resolving longstanding questions in mathematics. The overall aim of this research project is to advance the technique of using dynamics of group actions as a powerful tool to study questions that arise in algebra, geometry, and number theory, and to gain a deeper understanding of the connections between these mathematical fields.In this project special attention will be given to the following three directions: (1) Develop reduction theory of indefinite integral quadratic forms and consider its generalization to number fields using methods of dynamical systems. (2) Investigate random walks on Lie groups and equidistribution problems in homogenous spaces in light of recent developments in spectral theory and the theory of automorphic forms. (3) Study the height bounds of generators of arithmetic groups incorporating dynamical methods into classical algebraic and geometric theory of arithmetic groups. The project also aims to seek new interactions and develop the techniques in the study of groups, dynamics, and number theory.