Applications of Homotopy Theory

局部同伦理论是通过检查这些物体在小邻域中的行为来研究大型结构的一种理论。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。格罗滕迪克和他的追随者在20世纪60年代为“本地”一词引入了新的含义，以证明韦尔猜想。这个术语的变化导致了几何计算技术的爆发，拓扑理论的引入及其在逻辑中的应用，以及计算代数k理论不变量的一系列方法的发展。k理论计算涉及到一个新的微妙水平，因为它们依赖于非阿贝尔现象，这些现象被编码在起源于代数拓扑的大型物体结构中。******局部同伦理论的现代形式是由Jardine和Joyal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。这个理论现在已经应用于数学的多个部分，更广泛地应用于数学科学。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。该理论出现在经典稳定同伦理论中的椭圆上同伦理论和拓扑模形式中，也出现在等变同伦理论中。******本提案的研究将建立在这些成功的基础上，通过获得对局部同伦理论在传统数学及其应用中可能实现的计算方法的更深入理解。本文给出了数域上经典代数群的上同不变量的计算及其与算术的关系，以及代数曲线的k理论不变量及其模的计算。更抽象地说，我们计划定义和研究由Jardine的循环范畴理论和他关于动力系统同伦理论的工作所提出的etale同伦理论的推广。******在应用方面，并行处理系统的理论行为与更高的对称性密切相关。Jardine使用相干理论找到了一种在并发模型中对执行路径进行分类的算法。该算法只能在较大的结构中局部工作：Jardine计划在大型并行处理模型的研究中使用局部同伦理论的方法，特别强调寻找并行化技术。这是对寻找局部到全局的数据结构分析方法问题的总体攻击的一部分，这些数据结构太大，无法用现有的计算技术进行研究。这个局部到全球的问题是大数据实际分析中最大的问题之一

Local homotopy theory is a study of large structures by examining behaviour in small neighbourhoods on these objects.***Local to global methods have been present in Geometry and Topology for over a hundred years. Grothendieck and his followers introduced new meanings for the term "local" in the 1960s, on their way to proving the Weil conjectures. This variation of a term led to an explosion of calculational technique in Geometry, the introduction of topos theory and its applications in Logic, and the development of an array of methods to compute invariants of algebraic K-theory. The K-theory calculations involved a new level of subtlety, in that they relied on non-abelian phenomena which are encoded in large structures of objects which originate in Algebraic Topology. ******The modern form of local homotopy theory was formulated by Jardine and Joyal in the mid 1980s, as a response to extant problems in K-theory and topos theory. The theory now has applications in multiple parts 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 appears in classical stable homotopy theory in elliptic cohomology theories and topological modular forms, and also in equivariant homotopy theories.******The research of this proposal would build on these successes by acquiring a deeper understanding of calculational methods made possible by local homotopy theory, both in traditional Mathematics and in its applications. The program presented here includes the calculation of cohomological invariants which are associated to classical algebraic groups over number fields and the relation of these invariants with Arithmetic, and the calculation of K-theoretic invariants of algebraic curves and their associated moduli. More abstractly, there is a plan to define and study generalizations of etale homotopy theory which are suggested by Jardine's theory of cocycle categories and by his work on homotopy theories of dynamical systems.******On the applications side, the theoretical behaviour of parallel processing systems is strongly related to higher symmetries. Jardine used coherence theory to find an algorithm that classifies execution paths in concurrency models. This algorithm can only work locally in larger structures: Jardine plans to use the methods of local homotopy theory in the study of large parallel processing models, with a particular emphasis on finding parallelization techniques. This is part of a general attack on the problem of finding local to global methods of analyzing data structures which are too large to study with existing computational techniques. This local to global problem is one of the largest issues in the practical analysis of big data.**