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.

几何是形状和空间的研究。拓扑的主题与几何特征有关，这些特征在扭曲，拉伸或几何空间的其他变形后保持不变。它包括着色图，区分结和分类表面及其较高尺寸的类似物等问题。 “大规模”几何形状和拓扑的目的是了解无限空间的固有特征，该特征在越来越大的尺度下进行测量后仍然可见。这些想法有许多应用，包括对动态系统的定性研究，非常大的数据集的分析以及在现代物理基本物理学中引起的数学问题。拟议的研究旨在提供可能与这些潜在应用和其他潜在应用相关的微妙几何和拓扑现象的切实测量。涉及大小和距离的基本概念的“有限分解复杂性”的关键概念，以及“虚拟和焊接结和链接”的几何和拓扑理论以及它们的内在对称性。