Theory and applications of tangent categories
切范畴的理论与应用
基本信息
- 批准号:RGPIN-2019-04081
- 负责人:
- 金额:$ 1.24万
- 依托单位:
- 依托单位国家:加拿大
- 项目类别:Discovery Grants Program - Individual
- 财政年份:2021
- 资助国家:加拿大
- 起止时间:2021-01-01 至 2022-12-31
- 项目状态:已结题
- 来源:
- 关键词:
项目摘要
Calculus is the most useful and practical area of mathematics. It deals with how to calculate with changing quantities, and is thus a fundamental tool in sciences, economics, and engineering. However, while it is extremely important in applied contexts, its study is but one part of a much wider world of mathematics, which includes many other topics, such as combinatorics (counting), topology (the study of abstract shapes) and computer science. On the other hand, in some of these other areas of mathematics, researchers have started to develop ideas that share many formal similarities with calculus. These have gone by various names: polynomial functors (in combinatorics), the functor calculus (in topology) and differential linear logic (in computer science). As these ideas have been developed, it has not been clear how they are related to each other, or if they truly share a direct similarity with ordinary calculus. The research proposed for this grant offers a way to resolve this problem, by allowing one to view all these various notions as all aspects of one common idea, the notion of a tangent category. By providing a common framework for these different ideas, tangent categories provide an essential language to translate ideas between different areas of mathematics. Moreoever, one can develop much of differential geometry within the abstract setting of a tangent category, allowing the sophisticated ideas of differential geometry to be transferred into the study of polynomial functors, functor calculus, and differential linear logic. This work thus allows us an entirely new way to advance our understanding of these important ideas.
微积分是数学中最有用和最实用的领域。它涉及如何计算变化的量,因此是科学,经济学和工程学的基本工具。然而,尽管它在应用背景中非常重要,但它的研究只是更广泛的数学世界的一部分,其中包括许多其他主题,如组合数学(计数),拓扑学(抽象形状的研究)和计算机科学。另一方面,在其他一些数学领域,研究人员已经开始发展与微积分有许多形式上相似之处的思想,这些思想有各种各样的名字:多项式函子(组合学中),函子微积分(拓扑学中)和微分线性逻辑(计算机科学中)。 随着这些思想的发展,人们还不清楚它们是如何相互联系的,或者它们是否真的与普通微积分有直接的相似之处。 这项研究提供了一种解决这个问题的方法,通过允许一个人将所有这些不同的概念视为一个共同概念的所有方面,切线范畴的概念。通过为这些不同的概念提供一个共同的框架,切线范畴提供了一种必要的语言来翻译不同数学领域之间的思想。人们可以在正切范畴的抽象设置中发展许多微分几何,允许微分几何的复杂思想转移到多项式函子,函子演算,和微分线性逻辑。这项工作因此使我们能够以一种全新的方式来推进我们对这些重要思想的理解。
项目成果
期刊论文数量(0)
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会议论文数量(0)
专利数量(0)
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Cruttwell, Geoffrey其他文献
Cruttwell, Geoffrey的其他文献
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{{ truncateString('Cruttwell, Geoffrey', 18)}}的其他基金
Theory and applications of tangent categories
切范畴的理论与应用
- 批准号:
RGPIN-2019-04081 - 财政年份:2022
- 资助金额:
$ 1.24万 - 项目类别:
Discovery Grants Program - Individual
Theory and applications of tangent categories
切范畴的理论与应用
- 批准号:
RGPIN-2019-04081 - 财政年份:2020
- 资助金额:
$ 1.24万 - 项目类别:
Discovery Grants Program - Individual
Theory and applications of tangent categories
切范畴的理论与应用
- 批准号:
RGPIN-2019-04081 - 财政年份:2019
- 资助金额:
$ 1.24万 - 项目类别:
Discovery Grants Program - Individual
Abstract tangent functors
抽象正切函子
- 批准号:
435766-2013 - 财政年份:2018
- 资助金额:
$ 1.24万 - 项目类别:
Discovery Grants Program - Individual
Abstract tangent functors
抽象正切函子
- 批准号:
435766-2013 - 财政年份:2017
- 资助金额:
$ 1.24万 - 项目类别:
Discovery Grants Program - Individual
Abstract tangent functors
抽象正切函子
- 批准号:
435766-2013 - 财政年份:2015
- 资助金额:
$ 1.24万 - 项目类别:
Discovery Grants Program - Individual
Abstract tangent functors
抽象正切函子
- 批准号:
435766-2013 - 财政年份:2014
- 资助金额:
$ 1.24万 - 项目类别:
Discovery Grants Program - Individual
Abstract tangent functors
抽象正切函子
- 批准号:
435766-2013 - 财政年份:2013
- 资助金额:
$ 1.24万 - 项目类别:
Discovery Grants Program - Individual
Doctor's Level Studies in Category Theory
范畴论博士级研究
- 批准号:
317865-2005 - 财政年份:2007
- 资助金额:
$ 1.24万 - 项目类别:
Postgraduate Scholarships - Doctoral
Doctor's Level Studies in Category Theory
范畴论博士级研究
- 批准号:
317865-2005 - 财政年份:2006
- 资助金额:
$ 1.24万 - 项目类别:
Postgraduate Scholarships - Doctoral
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