The geometry, rigidity and combinatorics of spaces and groups with non-positive curvature feature

具有非正曲率特征的空间和群的几何、刚度和组合

• 批准号: 2305411
• 负责人: Jingyin Huang
• 金额: $ 21.73万
• 依托单位: Ohio State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-09-01 至 2026-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2305411&HistoricalAwards=false
• 关键词: geometry; rigidity; combinatorics; spaces; groups

Groups are fundamental abstract symbolic systems in mathematics arising from different branches of mathematics, physics, chemistry and computer science. For example, groups appear in the study of shapes of geometric objects, crystals and quasi-crystals, structure of roots of polynomials, cryptography, algorithm design etc. The study of finite groups, i.e. groups with finitely many elements, has reached a fairly mature stage, accumulating to a complete classification of finite simple groups. However, most infinite groups are fairly mysterious and hard to understand. In the 1980s, Gromov proposed a geometric approaches to group theory. One idea was to realize the mysterious group as a collection of symmetries of some geometric objects with interesting curvature properties, allowing us to study groups from the viewpoint of geometry. This has evolved into a very active field called geometric group theory. This proposal aims to study problems in the frontier of geometric group theory, and seeks applications to some long-standing problems in topology. The proposal also aims to provide resources for training graduate students and postdocs working in the area at the Ohio State University, with an emphasis on supporting under-represented early career stage mathematicians at OSU working in this filed.This proposal is concerned with rigidity and curvature properties of some infinite discrete groups from the viewpoint of geometric group theory, combined with ideas and techniques from ergodic theory, metric geometry, metric graph theory and combinatorial group theory. The project has two more specific research goals. The first is to make progress on a major conjecture on Artin groups, using a new strategy motivated from ideas in metric graph theory. The second is to understand fundamental forms of rigidity for discrete groups, namely quasi-isometric rigidity and measure equivalence rigidity. The project emphasizes the close connections between these forms of rigidity and the curvature properties of singular metric spaces and groups. Several classes of groups of fundamental importance are studied, including Artin groups, CAT(0) groups, graph products etc.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

群是数学中基本的抽象符号系统，源自数学、物理、化学和计算机科学的不同分支。例如，群出现在几何物体的形状、晶体和准晶体的研究、多项式根的结构、密码学、算法设计等方面。有限群，即有限多个元素的群的研究已经相当成熟。阶段，累积到有限简单群的完整分类。然而，大多数无限群都相当神秘且难以理解。 20 世纪 80 年代，格罗莫夫提出了群论的几何方法。 一个想法是将神秘群实现为一些具有有趣曲率特性的几何对象的对称性集合，使我们能够从几何的角度研究群。这已经发展成为一个非常活跃的领域，称为几何群论。该提案旨在研究几何群论的前沿问题，并寻求在拓扑学中一些长期存在的问题上的应用。该提案还旨在为俄亥俄州立大学该领域的研究生和博士后培训提供资源，重点是支持 OSU 在该领域工作的早期职业阶段数学家。该提案涉及刚性和曲率从几何群论的角度出发，结合遍历理论、度量几何、度量图论和组合群论的思想和技术，研究一些无限离散群的性质。该项目有两个更具体的研究目标。第一个是使用受度量图论思想启发的新策略，在 Artin 群的重大猜想上取得进展。第二个是了解离散群刚性的基本形式，即准等距刚性并测量等效刚性。该项目强调这些刚性形式与奇异度量空间和群的曲率特性之间的密切联系。研究了几类具有根本重要性的群体，包括 Artin 群体、CAT(0) 群体、图形产品等。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。