SM: Arithmetic Groups and Their Applications in Combinatorics, Geometry and Topology

SM：算术群及其在组合学、几何和拓扑中的应用

• 批准号: 1034750
• 负责人: Andrei Rapinchuk
• 金额: $ 1万
• 依托单位: University of Virginia Main Campus
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-04-15 至 2011-03-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1034750&HistoricalAwards=false
• 关键词: SM; Arithmetic; Groups; Their; Applications

A workshop ``Arithmetic Groups and Their Applications in Combinatorics, Geometry and Topology" will be held April 15-18 on the campus of the University of Virginia in Charlottesville. The scientific program of the workshop will be composed of lectures delivered by the leading experts in the area from the US, Brazil, France and Israel, and will highlight the most significant results obtained in the last few years. Special attention will be given to such topics as the analysis of structural properties of arithmetic groups (virtual positivity of the first Betti number, congruence subgroup problem, bounded generation), their applications in algebraic geometry (fake projective planes and fake versions of other important algebraic varieties), in differential geometry and topology (hyperbolic 3-manifolds, isospectral locally symmetricspaces) and in combinatorics (construction of expanders).Arithmetic groups are special groups whose elements are matrices with integral entries. This notion, which can be traced back to the work of Gauss on integral quadratic form, plays a crucial role in many areas of mathematics including algebra and various parts of number theory (e.g., the theory of automorphic forms). In recent years, new applications of the theory of arithmetic groups have emerged in algebraic and differential geometry, Lie theory and combinatorics. The workshop will showcase important results involving arithmetic groups. The focus will be placed on making the methods of the theory of arithmetic groups accessible to researchers, particularly young ones, working in a variety of areas.

“算术群及其在组合学、几何学和拓扑学中的应用”研讨会将于 4 月 15 日至 18 日在夏洛茨维尔弗吉尼亚大学校园举行。研讨会的科学计划将由顶尖专家的讲座组成来自美国、巴西、法国和以色列的领域，并将重点介绍过去几年中取得的最重要成果，并将特别关注算术群的结构特性分析（算术群的虚拟正性）等主题。第一个贝蒂数、同余子群问题、有界生成）、它们在代数几何（假射影平面和其他重要代数簇的假版本）、微分几何和拓扑（双曲 3 流形、等谱局部对称空间）和组合数学中的应用（扩展器的构造）。算术群是特殊的群，其元素是具有整数项的矩阵。这个概念可以追溯到高斯关于积分二次型的工作在数学的许多领域中发挥着至关重要的作用，包括代数和数论的各个部分（例如自守形式理论）。近年来，算术群理论在代数和微分几何、李理论和组合数学中出现了新的应用。研讨会将展示涉及算术组的重要成果。重点将放在使算术群理论的方法可供各个领域的研究人员，特别是年轻的研究人员使用。