Topology at all scales

所有尺度的拓扑

• 批准号: RGPIN-2019-05401
• 负责人: Nicas, Andrew
• 金额: $ 1.09万
• 依托单位: University of Victoria
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=738250
• 关键词: Topology; scales

Geometry is the study of shapes and spaces. The subject of topology is concerned with those features of geometry which remain unchanged after twisting, stretching or other deformations of a geometrical space. It includes such problems as coloring maps, distinguishing knots and classifying surfaces and their higher dimensional analogs. The goal of "large scale" geometry and topology is to understand those intrinsic features of an unbounded space which remain visible after measurements are taken at increasingly large scales. These ideas have many applications including the qualitative study of dynamical systems, the analysis of very large data sets, and also mathematical problems arising in fundamental modern physics. The proposed research seeks to provide tangible measures of subtle geometric and topological phenomena that could be relevant to these and other potential applications. The key concept of "finite decomposition complexity", which involves fundamental notions of size and distance, and also the geometric and topological theory of "virtual and welded knots and links" together with their intrinsic symmetries will be explored.

几何学是对形状和空间的研究，涉及几何空间扭曲、拉伸或其他变形后保持不变的几何特征，包括着色图、区分结和对表面及其表面进行分类等问题。 “大规模”几何和拓扑的目标是理解无界空间的内在特征，这些特征在越来越大的尺度上进行测量后仍然可见，这些想法有许多应用，包括动力系统的定性研究。该研究旨在提供与这些和其他潜在应用相关的微妙几何和拓扑现象的具体测量“有限分解复杂性”的关键概念。 ，其中涉及尺寸和距离的基本概念，还将探讨“虚拟和焊接结和链接”的几何和拓扑理论及其内在对称性。