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: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=679577
• 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.***

数学的大部分内容都是关于空间和量的研究，以及它们之间的关系。 这里的空间概念不仅包括我们所熟悉的三维物理空间，而且还包括其中各种形状和图形的配置，以及更高维的空间，以及从这些开始抽象出来的各种空间数学概念。 这里的量的概念不仅包括我们熟悉的计数、有理数和真实的数，而且还包括在空间上逐点变化的数值量，如地球表面的温度，以及在空间中分布的量，如在地球大气中分布的某种气体的量。 函数、测度和分布的数学概念，以及其他各种相关概念，都是为了在数学上表示这些变量和分布量而设计的。 空间的数学概念有许多微妙的不同，例如拓扑空间、流形和簇的概念，每一个概念都有自己的数学分支来研究，例如拓扑学、微分几何和代数几何。 相应地，与不同的空间概念相关联的函数和分布的概念也有许多微妙的不同，这些概念最终以意想不到的方式被应用，例如，在概率和统计学的研究中，描述实验可能结果的空间中的似然分布。 拟议的研究采用范畴论的方法，数学结构的一般研究，以识别和理解所有这些微妙的不同的空间和数量的概念所共有的共同结构特征，并开发一个系统的范畴理论框架，公理和结果，促进了新的和统一的理解结构，产生和表征一般的变量和分布量。 这将有助于使每个不同的数学分支的概念和方法更容易获得其他的从业者，这些结构研究所获得的见解，使新的和有效的数学方法。 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。