Symmetry and Self-Similar Structures in Geometry and Topology

几何和拓扑中的对称和自相似结构

• 批准号: 1811824
• 负责人: Wouter Van Limbeek
• 金额: $ 14.16万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-08-15 至 2018-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1811824&HistoricalAwards=false
• 关键词: Symmetry; Self; Similar; Structures; Geometry

Symmetry is a pervasive concept in mathematics, giving rise to beautiful and interesting geometric objects, as well as providing an essential tool in geometry and topology. Classically, research has focused on globally defined symmetries, but recently it has been realized that symmetries of local structures offer vastly more examples and connections to other areas of mathematics. This project will study local symmetry, its applications to the study of self-similar geometric objects and related algebraic problems, and recently discovered geometric constructions of efficient networks (so-called expander graphs).The relationship between the geometry of an object and the algebraic structure of its group of automorphisms has been intensely studied in the past, but in the last decade, progress has been made on a much richer theory of symmetries that only appear after passing to a cover, i.e. are "hidden'" The project aims to study this notion in several contexts, such as Riemannian geometry (especially nonpositive curvature) and geometric structures. Further, rich phenomena are expected when the symmetry arises from self-similarity, i.e. in the case of self-covering manifolds, particularly in situations of geometric interest such as Kaehler or affine geometry. Finally, the project aims to study coarse geometric objects associated to group actions (so-called warped cones), especially in the case of actions with spectral gap, where these yield expander graphs.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.

对称性是数学中普遍存在的概念，产生了美丽而有趣的几何对象，并提供了几何和拓扑中的必不可少的工具。从经典上讲，研究集中在全球定义的对称性上，但是最近已经意识到，本地结构的对称性提供了更多的例子和与其他数学领域的联系。该项目将研究本地对称性，其应用于研究自相似的几何对象和相关代数问题，最近发现了有效网络的几何结构（所谓的扩张器图）。物体的几何形状与其自动化群体的代数结构已经在过去的情况下进行了很多研究，但已经取得了进步，但在过去却取得了进步，但在过去却取得了进步，但它已经取得了进步。仅在传递到封面后出现的对称性，即“隐藏”该项目旨在在几种情况下研究这种概念，例如Riemannian几何形状（尤其是非阳性曲率）和几何结构。此外，当对称性源于自相似性时，即在自覆盖的歧管的情况下，尤其是在诸如Kaehler或Aggine几何形状等几何兴趣的情况下，预期的是丰富的现象。最后，该项目旨在研究与小组动作相关的粗略几何对象（所谓的翘曲锥体），尤其是在具有光谱差距的动作的情况下，这些产量扩展器图。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子和更广泛影响的评估审查审查标准来通过评估来通过评估来获得支持的。