Group Actions, Homogeneous Dynamics, and Number Theory

数学中的许多问题都与一组所谓的群结构的对称性有关。探索潜在群体行为的动力学，即研究空间中的点在对称下的行为，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。本研究项目的总体目标是将群体行为动力学作为一种强大的工具来研究代数、几何和数论中出现的问题，并对这些数学领域之间的联系有更深入的了解。本课题将着重研究以下三个方向：(1)发展不定积分二次型的约简理论，并考虑用动力系统方法将其推广到数域。(2)结合谱理论和自同构形式理论的最新进展，研究齐次空间中李群上的随机漫步和等分布问题。(3)将动力学方法引入到经典算术群的代数和几何理论中，研究算术群的生成子的高度界。该项目还旨在寻求新的相互作用，并在群体、动力学和数论的研究中发展技术。

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.