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: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=718340
• 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学和拓扑学中已经存在了一百多年。格罗滕迪克和他的追随者在20世纪60年代引入了“地方”一词的现代形式，以证明韦尔定理。这导致了代数几何中计算技术的爆炸，拓扑理论的引入及其在逻辑中的应用，以及计算代数K理论不变量的方法的发展。K理论计算引入了一个新的微妙水平，因为它们涉及非阿贝尔现象，这些现象被编码在起源于代数拓扑学的大型对象结构中。 现代形式的局部同伦理论是由加拿大数学家贾丁和乔亚尔在20世纪80年代中期提出的，作为对K理论和拓扑理论中现存问题的回应。该理论在数学的多个研究领域以及更广泛的数学科学中都有应用。它是代数几何和数论中动机和动机同伦理论研究的基础，也是现代对称同伦理论的背景，它被编码在堆栈和更高的堆栈中。该理论明确出现在经典的稳定同伦理论中，并在数学物理中应用。 本提案的研究主线是探讨大型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拓扑模型，并行处理系统和拓扑数据分析，特别强调开发并行化技术的研究，这些结构。现有的算法计算不变量的利益（并发，集群和拓扑数据分析中的持久同源性的执行路径）从拓扑对象相关联的数据结构。这些数据结构必须相对较小，这些算法是实用的，这个研究计划的目标是找到技术，将现有的算法扩展到有效的计算设备，为更大的对象。 在持续同伦理论的新兴研究领域的背景下，拓扑数据分析中的稳定性特性特别受到关注。稳定性结果表明，如果两个数据集接近，则相关的拓扑数据分析不变量以可测量的方式接近，使用“交织距离”。这是一项地方性原则，目前的目标是使之全球化。 这项工作属于同伦理论及其在数据分析中的应用，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，根据这一建议培训的HQP将非常适合在所有这些部门的工作。这项工作将巩固加拿大在数学和数据分析之间的边界上一个充满活力的新研究领域的领导地位。