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.

数学的大部分内容涉及空间和数量以及彼此之间的关系。这里的空间概念不仅包括我们周围熟悉的三维物理空间，还包括其中各种形状和图形的配置。 ，以及高维空间，以及从这些开始抽象出来的各种数学概念。这里的数量概念不仅包括熟悉的计数、有理数和实数，还包括逐点变化的数值量。空间上的温度，例如物体表面的温度地球，以及通过空间分布的量，例如通过地球大气层分布的某些气体的量。函数、测量和分布的数学概念以及各种其他相关概念被设计来在数学上表示此类变量。空间有各种略有不同的数学概念，例如拓扑空间、流形和簇的概念，每种概念都有自己的数学分支，例如拓扑、微分几何和代数几何。相应地，有许多与不同空间概念相关的略有不同的函数和分布概念，并且这些概念最终以意想不到的方式应用，例如描述实验可能结果的空间中的可能性分布，如所提出的研究采用范畴论的方法（数学结构的一般研究）来识别和理解所有这些细微不同的空间和数量概念所共有的共同结构特征，并开发一个系统范畴论框架、公理论和结果，促进对生成和表征一般变量和分布量的结构的新的、统一的理解，这将使实践者更容易理解数学各个分支的概念和方法。此外，这些结构研究使数学见解能够更容易地转移到应用领域，例如概率规划的设计。语言，并且它们支持为此目的对这些想法进行有效的变化和适应。