Towards a unified approach to functor calculus

走向函子微积分的统一方法

• 批准号: RGPIN-2017-04114
• 负责人: Bauer, Kristine
• 金额: $ 1.02万
• 依托单位: University of Calgary
• 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=675450
• 关键词: Towards; unified; approach; functor; calculus

Topology is the part of mathematics which provides a careful analysis of shapes, or topological spaces. Some of these spaces can be visually inspected because we can draw them or model them. But others have so many dimensions that they can hardly be imagined. The job of an algebraic topologist is to find ways to imagine the unimaginable by assigning mathematical values to topological spaces which appropriately represent the space. This assignment is accomplished using homotopy functors, a method of replacing something which is difficult to visualize, analyze or classify by something which is simpler to imagine or even compute. Unfortunately, many of the most valuable homotopy functors - those that are the most descriptive - are themselves very difficult to compute. In order to make this task easier, there is a kind of calculus for homotopy functors, which was originally pioneered by Goodwillie and which is known as Goodwillie calculus. The idea of Goodwillie calculus is to approximate valuable but complicated homotopy functors by simpler functors which are easier to compute. Some of the first applications of Goodwillie calculus were to famous functors like K-theory, which is important in many branches of mathematics including number theory, algebra, algebraic geometry, topology and analysis.******My main research goal is to make it easier to use Goodwillie calculus. In particular, one of my research goals is to formalize the relationship between Goodwillie's calculus and the calculus of Newton and Leibniz that we teach to first-year university students. These are not the same, but they have many of the same properties. Since mathematician have understood calculus of functions very well for several hundred years, if I can make this relationship precise then this will make it easier to use Goodwillie's calculus, too. Together with my colleagues Johnson, Osborne, Riehl and Tebbe, we have already found the precise relationship we want in a special case using the theory of differential categories.******Topology is used to model problems of all kinds. The positions of robots in an automated warehouse is modelled by a topological space, as is spacetime. In a world of data collected with an ever-increasing number of parameters, topology gives us one way of understanding the data as a whole, modelling the data, and using this to draw conclusions and make predictions from the information we have gained. For each application of topology in the real world, there are homotopy functors lurking which can simplify this information and make it easier to understand. Indeed, this is already being done: by using a simple homotopy functor to analyze scans of hepatic lesions, Carlsson et al were able to classify these lesions into a small number of disease types. As applications of topology become more prevalent, the tools we use will become more sophisticated. My work will simplify complicated homotopy functors when these tools are needed.

拓扑学是数学的一部分，它提供了对形状或拓扑空间的仔细分析。其中一些空间可以通过视觉检查，因为我们可以画出它们或为它们建模。但另一些则有很多难以想象的维度。代数拓扑学家的工作是通过赋予拓扑空间适当的数学值来找到想象不可想象的空间的方法。这个赋值是用同伦函子完成的，这是一种用更容易想象甚至计算的东西代替难以想象、分析或分类的东西的方法。不幸的是，许多最有价值的同伦函子——那些最具描述性的——本身是很难计算的。为了简化这个任务，有一种关于同伦函子的微积分，它最初是由古德威利首创的，被称为古德威利微积分。古德威利微积分的思想是用更容易计算的更简单的函子来近似有价值但复杂的同伦函子。古德威利微积分最早的一些应用是在著名的函子上，比如k理论，它在数学的许多分支中都很重要，包括数论、代数、代数几何、拓扑和分析。******我的主要研究目标是使古德威利微积分更容易使用。特别是，我的研究目标之一是形式化古德威利微积分与牛顿和莱布尼茨微积分之间的关系，我们教一年级的学生。它们不一样，但它们有很多相同的性质。由于数学家对函数演算的理解已经有几百年了，如果我能使这种关系精确，那么这也会使古德威利的演算更容易使用。与我的同事Johnson， Osborne， Riehl和Tebbe一起，我们已经使用微分范畴理论在一个特殊情况下找到了我们想要的精确关系。******拓扑学用于对各种问题建模。机器人在自动化仓库中的位置是由拓扑空间建模的，就像时空一样。在一个参数数量不断增加的数据收集世界中，拓扑为我们提供了一种整体理解数据、对数据建模并使用它从我们获得的信息中得出结论和做出预测的方法。对于拓扑在现实世界中的每一个应用，都潜伏着同伦函子，可以简化这些信息，使其更容易理解。事实上，这已经在做了：通过使用一个简单的同伦泛子来分析肝脏病变的扫描，Carlsson等人能够将这些病变分为少数疾病类型。随着拓扑学的应用越来越普遍，我们使用的工具也将变得越来越复杂。当需要这些工具时，我的工作将简化复杂的同伦函子。