Applications of Homotopy Theory

同伦理论的应用

• 批准号: RGPIN-2020-06461
• 负责人: Jardine, John
• 金额: $ 1.53万
• 依托单位: University of Western Ontario
• 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=741226
• 关键词: Applications; Homotopy; Theory

Local to global methods have been present in Geometry and Topology for over a hundred years. Grothendieck and his followers introduced the modern form of the term "local" in the 1960s, on their way to proving the Weil conjectures. This led to an explosion of calculational technique in Algebraic Geometry, the introduction of Topos Theory and its applications in Logic, and to the development of methods to compute invariants of Algebraic K-theory. The K-theory calculations introduced a new level of subtlety, as they involve non-abelian phenomena which are encoded in large structures of objects which originate in Algebraic Topology. The modern form of local homotopy theory was initiated by the Canadian mathematicians Jardine and Joyal in the mid 1980s in separate but complimentary work, as a response to extant problems in K-theory and Topos Theory. The theory has applications in multiple research areas of Mathematics, and in the Mathematical Sciences more generally. It is the basis for the study of motives and motivic homotopy theory in Algebraic Geometry and Number Theory, and is the context for the modern homotopical theory of symmetries which is encoded in stacks and higher stacks. The theory explicitly appears in classical stable homotopy theory, and in its applications in Mathematical Physics.  The main line of research of this proposal investigates local to global calculational methods in the study of large data structures. This work incorporates existing topological models for both parallel processing systems and topological data analysis, with a particular emphasis on developing parallelization techniques for the study of these structures. Existing algorithms calculate invariants of interest (execution paths in concurrency, clusters and persistent homology in topological data analysis) from topological objects associated to data structures. These data structures must be relatively small for these algorithms to be practical, and the goal of this research program is to find techniques that would extend existing algorithms to effective calculational devices for larger objects.  There is a particular focus on stability properties in Topological Data Analysis, in the context of the emerging research area of persistent homotopy theory. Stability results say that if two data sets are close, then the associated topological data analysis invariants are close in a measurable way, using an "interleaving distance". This is a local principle, and the present aim is to globalize it.This work is in Homotopy Theory with applications to Data Analysis, and is of particular interest to mathematicians and to data scientists. The results of this research will be used in academia, industry, and government, and HQP trained under this proposal will be highly suited for work in all of these sectors. The work will solidify Canada's leadership in a vibrant new research area at the boundary between Mathematics and Data Analytics.

从局部到全局的方法在几何和拓扑学中已经存在了一百多年。Grothendieck和他的追随者在20世纪60年代引入了“本地”一词的现代形式，在他们证明韦尔猜想的过程中。这导致了代数几何中计算技术的爆炸性增长，Topos理论及其在逻辑中的应用的引入，以及计算代数K理论不变量的方法的发展。K理论的计算引入了一个新的微妙水平，因为它们涉及到非阿贝尔现象，这些现象编码在起源于代数拓扑学的对象的大结构中。现代形式的局部同伦理论是由加拿大数学家Jardine和JoYal于20世纪80年代中期在单独但互补的工作中提出的，以回应K-理论和Topos理论中存在的问题。该理论在数学的多个研究领域中都有应用，更广泛地说，在数学科学中也有应用。它是研究《代数几何》和《数论》中动机和动机同伦论的基础，也是现代对称论同伦论的背景，这种对称论是以堆栈和更高的堆栈编码的。这一理论明确地出现在经典的稳定同伦理论及其在数学物理中的应用中。本建议的研究主线是研究大数据结构研究中的局部到全局计算方法。这项工作结合了现有的并行处理系统和拓扑数据分析的拓扑模型，特别强调开发用于研究这些结构的并行化技术。现有算法从与数据结构相关联的拓扑对象计算感兴趣的不变量(并发执行路径、簇和拓扑数据分析中的持久同调)。这些数据结构必须相对较小，才能使这些算法实用，本研究计划的目标是找到将现有算法扩展到更大对象的有效计算工具的技术。在持久同伦理论这一新兴研究领域的背景下，拓扑数据分析特别关注稳定性性质。稳定性结果表明，如果两个数据集接近，则关联的拓扑数据分析不变量以一种可测量的方式接近，使用“交错距离”。这是一个局部原理，目前的目标是将其全球化。这项工作在同伦理论及其应用到数据分析中，数学家和数据科学家特别感兴趣。这项研究的结果将用于学术界、工业界和政府，根据这项建议培训的HQP将非常适合在所有这些部门工作。这项工作将巩固加拿大在数学和数据分析之间充满活力的新研究领域的领导地位。