权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

Applications of Higher Dimensional Algebra to Stable Homotopy Theory

高维代数在稳定同伦理论中的应用

基本信息

批准号：
EP/K007343/1
负责人：
Michael Nicholas Gurski
金额：
$ 31.24万
依托单位：
University of Sheffield
依托单位国家：
英国
项目类别：
Fellowship
财政年份：
2013
资助国家：
英国
起止时间：
2013 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FK007343%2F1
关键词：
Applications Higher Dimensional Algebra Stable

项目摘要

Category theory is the abstract study of relationships between mathematical objects. These relationships can be simple, such as one number being larger or smaller than another, or they can be more complicated and many-layered, such as spatial relationships between two points on a surface. Higher-dimensional category theory is then the study of similar kinds of relationships, but we now allow the existence of relationships between relationships, and so on. One example might describe all the ways to walk from one point to another. If the space between these two points is an empty field, say, then any two different paths are essentially the same, as we could devise a series of paths that gradually changed from the first given path to the second. On the other hand, if there is a pond between these two points, then there are at least two essentially different paths, one for each way around the pond. The higher dimensional category theory in this proposal is in the subfield of higher dimensional algebra which is the study of objects with many different kinds of algebraic operations on them, and the interactions between those; a simple example would just be the real numbers with the operations plus and times, and one example interaction between these is a(b+c)=ab+ac.The other field of mathematics in this proposal is topology, which is the study of shapes. In topology, aspects like distance or smoothness do not matter, so a circle is the same as a square and a doughnut is the same as a coffee cup. One of the most profitable ways of understanding shapes is by assigning them algebraic invariants. For instance, there is an invariant called the fundamental group which counts certain kinds of holes in a shape; computing the fundamental group of both a circle and a square will give the same answer, as they both have a single hole in the middle. On the other hand, computing the fundamental group of a filled-in square will give a different answer, precisely because the hole has now been filled.The research in this proposal is about using higher dimensional algebra to describe shapes. As an example, imagine two flexible tubes. We can combine these in a variety of ways: we can just set them next to each to get a pair of tubes, we could glue the end of one tube to the other and get a single very long tube, or we could even glue both ends of the first tube to the corresponding ends of the second tube to get a circular tube (a shape called a torus). Gluing surfaces together is one kind of algebraic operation, and setting surfaces next to each other is another, and these two operations behave in ways that are governed by laws appearing in higher dimesional algebra. If we additionally take into account symmetries of the shapes involved (for our example, tubes can be rotated or the ends can be swapped), then both the topological and algebraic descriptions become more complicated. These constructions have been studied using topology, but this research is aimed at utilising the tools of higher dimensional algebra in order to shed new light on old problems.

范畴论是对数学对象之间关系的抽象研究。这些关系可以是简单的，例如一个数字比另一个数字大或小，也可以是更复杂和多层的，例如曲面上两点之间的空间关系。高维范畴论研究的是类似的关系，但我们现在允许关系之间存在关系，等等。一个例子可以描述从一个点走到另一个点的所有方式。如果这两个点之间的空间是一个空白区域，那么任何两条不同的路径本质上都是相同的，因为我们可以设计一系列从第一条给定路径到第二条给定路径逐渐变化的路径。另一方面，如果在这两点之间有一个池塘，那么至少有两条本质上不同的路径，沿着池塘的每一条路径。本提案中的高维范畴论属于高维代数的子领域，高维代数研究的是具有许多不同代数运算的对象，以及它们之间的相互作用；一个简单的例子就是实数的加号和乘号，它们之间的一个相互作用的例子是A (b+c)=ab+ac。这个提议中的另一个数学领域是拓扑学，也就是对形状的研究。在拓扑学中，距离或平滑度等方面并不重要，因此圆形与正方形相同，甜甜圈与咖啡杯相同。理解形状最有效的方法之一是赋予它们代数不变量。例如，有一个叫做基本群的不变量，它计算形状中某些类型的洞；计算圆形和正方形的基本群会得到相同的答案，因为它们中间都有一个洞。另一方面，计算一个已填满的正方形的基本群会得到一个不同的答案，正是因为这个洞现在已经填满了。本提案的研究是关于使用高维代数来描述形状。举个例子，想象两个柔性管。我们可以用各种方法把它们组合起来：我们可以把它们放在一起得到一对管子，我们可以把一个管子的一端粘在另一个管子上得到一个很长的管子，或者我们甚至可以把第一根管子的两端粘在第二根管子的相应末端上得到一个圆形的管子（一种叫做环面的形状）。将曲面粘合在一起是一种代数操作，将曲面相邻放置是另一种代数操作，这两种操作的行为方式受到高维代数中出现的定律的支配。如果我们额外考虑到所涉及形状的对称性（例如，管子可以旋转或两端可以交换），那么拓扑和代数描述都会变得更加复杂。这些结构已经用拓扑学进行了研究，但这项研究的目的是利用高维代数的工具来揭示老问题的新亮点。