Categorical algebra in analysis, geometry, and topology

分析、几何和拓扑中的分类代数

• 批准号: RGPIN-2019-05274
• 负责人: LucyshynWright, Rory
• 金额: $ 1.75万
• 依托单位: Brandon University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758727
• 关键词: Categorical; algebra; analysis; geometry; topology

Much of mathematics is concerned with the study of space and quantity, and the relation of each to the other. The notion of space here comprises not only the familiar three-dimensional physical space that surrounds us but also the configurations of various shapes and figures therein, as well as higher-dimensional spaces, and various mathematical concepts of space that abstract from these beginnings. The notion of quantity here includes not only the familiar counting numbers, rational, and real numbers, but also numerical quantities that vary from point to point over a space, such as the temperature at the surface of the earth, and also quantities that are distributed through space, such as the quantity of certain gas distributed through the earth's atmosphere. The mathematical notions of function, measure, and distribution, as well as various other related notions, were devised to represent mathematically such variable and distributed quantities. There are various subtly different mathematical notions of space, such as the notions of topological space, manifold, and variety, each studied by its own branch of mathematics, such as topology, differential geometry, and algebraic geometry. Correspondingly, there are numerous subtly different notions of function and distribution associated with different notions of space, and these notions end up being applied in unexpected ways, e.g. to describe the distribution of likelihood across a space of possible outcomes of an experiment, as in the study of probability and statistics. The proposed research employs the methods of category theory, a general study of structure in mathematics, to identify and understand the common structural characteristics shared by all these subtly different notions of space and quantity, and to develop a system of category-theoretic frameworks, axiomatics, and results that foster a new and unified understanding of structures that generate and characterize variable and distributed quantities in general.  This will serve to make the notions and methods of each of various branches of mathematics more accessible to the practitioners of the others, and the insights gained by these structural studies enable new and effective methods in mathematics.  Further, these structural studies allow mathematical insights to be more readily transferred to application domains, e.g. in the design of probabilistic programming languages, and they support the effective variation and adaptation of these ideas to that end.

许多数学都与空间和数量的研究以及彼此之间的关系有关。这里的空间概念不仅包括我们周围的熟悉的三维物理空间，还包括其中的各种形状和图形的配置，以及较高的维度空间以及从这些开始中抽象的各种空间的数学概念。这里的数量概念不仅包括熟悉的计数数，有理数和实数，还包括数量数量，这些数量在空间上的点之间变化，例如地球表面的温度以及通过太空分布的数量，例如通过地球大气层分布的某些气体的数量。函数，测量和分布以及其他各种相关注释的数学说明被设计为数学上代表此类变量和分布式数量。空间的数学数字有各种不同的数学笔记，例如拓扑空间，歧管和多样性的注释，每个数学都按其自己的数学分支进行了研究，例如拓扑，差异几何和代数几何形状。相应地，有许多与空间注释相关的功能和分布的次数不同的音符，这些音符最终以意想不到的方式应用，例如描述了在实验可能结果的空间中的可能性分布，例如在概率和统计研究中。拟议的研究员工类别理论的方法是数学结构的一般研究，旨在识别和理解所有这些微妙的空间和数量不同概念共享的共同结构特征，并开发出类别理论框架，方形框架的系统，并促进对产生和表征可变数量和分布量的新结构的结果，并促进了一个新的和统一的理解。这将有助于使其他从业者更容易获得数学的每个分支的注释和方法，并且这些结构研究获得的见解可以使数学方面的新方法有效。     此外，这些结构研究使数学见解更容易转移到应用领域，例如在概率编程语言的设计中，它们支持这些想法的有效变化和适应。