Group Actions, Homogeneous Dynamics, and Number Theory

群作用、齐次动力学和数论

• 批准号: 1700109
• 负责人: Han Li
• 金额: $ 14.4万
• 依托单位: Wesleyan University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-06-15 至 2023-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1700109&HistoricalAwards=false
• 关键词: Group; Actions; Homogeneous; Dynamics; Number

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.

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。探索潜在群体行为的动力学，即研究空间中的点在对称性下如何表现，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。该研究项目的总体目标是推进使用群体行为动力学作为强大工具来研究代数、几何和数论中出现的问题的技术，并更深入地了解这些数学领域之间的联系。本项目将特别关注以下三个方向：（1）发展不定积分二次型的约简理论，并考虑使用动力系统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走和齐次空间中的均分布问题。 (3)将动力学方法融入算术群的经典代数和几何理论中，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开发群、动力学和数论研究的技术。

项目成果

Explicit result on equivalence of rational quadratic forms avoiding primes

有理二次型避免素数等价的显式结果

• DOI: 10.1016/j.jnt.2021.02.006
• 发表时间: 2021-08
• 期刊: Journal of Number Theory
• 影响因子: 0.7
• 作者: Chan, Wai Kiu;Gao, Haochen;Li, Han
• 通讯作者: Li, Han

On realization of isometries for higher rank quadratic lattices over number fields

数域上高阶二次格的等距实现

• DOI: 10.1090/tran/8670
• 发表时间: 2022-07
• 期刊: Transactions of the American Mathematical Society
• 影响因子: 1.3
• 作者: Chan, Wai Kiu;Li, Han
• 通讯作者: Li, Han

On effective equidistribution for quotients of SL(d,ℝ)

关于 SL(d,α) 商的有效均分布

• DOI: 10.1007/s11856-020-1978-z
• 发表时间: 2020-02
• 期刊: Israel journal of mathematics
• 影响因子: 1
• 作者: Aka, M.;Einsiedler, M.;Li, H.;Mohammadi, A.
• 通讯作者: Mohammadi, A.

New bounds in reduction theory of indefinite ternary integral quadratic forms

不定三元积分二次型归约理论的新界限

• 发表时间: 2018-03
• 期刊: Advances in mathematics
• 影响因子: 1.7
• 作者: Li, Han;Margulis, Gregory A
• 通讯作者: Margulis, Gregory A