Artin groups and diagram algebras via topology

通过拓扑的 Artin 群和图代数

基本信息

批准号：
EP/V043323/1
负责人：
Rachael Boyd
金额：
$ 41.06万
依托单位：
University of Cambridge
依托单位国家：
英国
项目类别：
Fellowship
财政年份：
2022
资助国家：
英国
起止时间：
2022 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FV043323%2F1
关键词：
Artin groups diagram algebras via

项目摘要

Mathematics describes in the simplest and purest terms the objects around us, and the world we live in. The real world is often more complex that we can model mathematically, but that does not mean that studying the underlying mathematics is not important. Take the mathematical concept of the sphere: we can never build a perfect sphere (every real life model will have microscopic imperfections), but understanding spheres can help us understand the earth. After all, the earth is a sphere which spins (a mathematical symmetry) and thus gives us night and day - arguably the most fundamental symmetry. Mathematics surrounds us every day: swings and roundabouts, famous art and architecture, the layout of robots on factory floors, symmetries in chemical compounds, and hair braids are all rich examples of mathematical phenomena.In this project I will study 'algebraic objects' using 'topological methods'. Let's look at the methods first. "Topology" is a field of mathematics which studies the underlying 'shape' of an object, no matter how much you bend, squash, or stretch it. For example, a circle has one hole, and no matter how much you squash it or stretch it, the hole is never changing. Similarly, a figure 8 has two holes. 'Holes' in an object are described by the topological notion of "homology", which is potentially the most successful tool topology has to offer, and an example of something mathematicians call a "topological invariant". It can, for example, tell us that a circle and a figure 8 are inherently 'different' shapes. Rather than taking a concrete object like a circle to begin with, I will take an algebraic object, and build a concrete object which 'resembles' the algebraic object in some sense. Computing topological invariants of this concrete object can then provide information about the original algebraic object.So let's return to our algebraic objects, of which there are 2 types. The first type of algebraic objects I will study in this proposal are "Artin groups". In mathematics, a "symmetry" of an object is a transformation that leaves the object 'unchanged': real life examples are shuffling a stack of papers, rotating a roundabout, or braiding your hair. The symmetries of an object are packaged up into an algebraic object called a "group". Artin groups are a broad family of groups with the simplest example being the braid group - one can think of the braid group as the mathematical embodiment of all the ways to braid your hair using only a specific number of strands. Artin groups are very simple to define mathematically, yet very mysterious: even after over 50 years of work, many fundamental questions remain unanswered. I aim to build concrete topological objects to study Artin groups, and answer some of these fundamental questions. The second type of algebraic objects I will study in this proposal are "diagram algebras". Consider 2n holes in the ground, and n moles which have to pop out of one hole and make their way to a free hole. If we consider the paths these moles create we get a diagram belonging to some of the diagram algebras I will study: allowing the moles to cross each others' paths gives us the "Brauer algebra", and if we don't allow them to cross we get the "Temperley-Lieb algebra". Diagram algebras have strong connections with physics: replacing our moles with particles, gives a physicists "connection diagram". I aim to study the homology of these algebras individually, and also to adapt a general framework for studying homology to include the homology of these algebras.Studying these algebraic objects using methods from topology will result in interesting pure mathematics, and add to our collective understanding of the pure phenomena underlying the world we live in. The links that generalised braiding and diagram algebras have to other areas of mathematics and science also means that this project will have a knock-on effect that will benefit research across the sciences.

数学用最简单、最纯粹的术语描述了我们周围的物体以及我们生活的世界。现实世界往往更加复杂，我们可以用数学建模，但这并不意味着研究底层数学不重要。以球体的数学概念为例：我们永远无法构建一个完美的球体（每个现实生活中的模型都会存在微观缺陷），但了解球体可以帮助我们了解地球。毕竟，地球是一个旋转的球体（数学对称性），从而给我们带来了黑夜和白天——可以说是最基本的对称性。数学每天都围绕着我们：秋千和环形交叉口、著名的艺术和建筑、工厂车间的机器人布局、化合物的对称性和发辫都是数学现象的丰富例子。在这个项目中，我将使用“拓扑方法”。我们先来看看方法。 “拓扑学”是一个数学领域，它研究物体的潜在“形状”，无论你如何弯曲、挤压或拉伸它。例如，一个圆有一个洞，无论你如何挤压或拉伸它，洞都不会改变。同样，8 字形也有两个孔。物体中的“洞”是用“同调”的拓扑概念来描述的，这可能是拓扑提供的最成功的工具，也是数学家称之为“拓扑不变量”的一个例子。例如，它可以告诉我们圆形和数字 8 本质上是“不同”的形状。我不会一开始就采用一个像圆这样的具体对象，而是采用一个代数对象，并构建一个在某种意义上“类似于”代数对象的具体对象。计算这个具体对象的拓扑不变量可以提供有关原始代数对象的信息。所以让我们回到我们的代数对象，其中有两种类型。我将在本提案中研究的第一类代数对象是“Artin 群”。在数学中，物体的“对称性”是一种使物体保持“不变”的变换：现实生活中的例子是拖放一叠纸、旋转环形交叉路口或编辫子。对象的对称性被打包成称为“群”的代数对象。 Artin 群是一个广泛的群族，最简单的例子是辫子群 - 人们可以将辫子群视为仅使用特定数量的股线来编织头发的所有方法的数学体现。 Artin 群的数学定义非常简单，但却非常神秘：即使经过 50 多年的研究，许多基本问题仍然没有得到解答。我的目标是构建具体的拓扑对象来研究 Artin 群，并回答其中一些基本问题。我将在本提案中研究的第二种代数对象是“图代数”。考虑地上有 2n 个洞，以及 n 只鼹鼠，它们必须从一个洞中弹出并进入一个空闲的洞。如果我们考虑这些鼹鼠创建的路径，我们会得到一个属于我将研究的一些图代数的图：允许鼹鼠相互交叉的路径给我们“布劳尔代数”，如果我们不允许它们交叉我们得到“Temperley-Lieb 代数”。图代数与物理学有着密切的联系：用粒子代替摩尔，给出了物理学家的“连接图”。我的目标是单独研究这些代数的同源性，并调整研究同源性的通用框架以包括这些代数的同源性。使用拓扑方法研究这些代数对象将产生有趣的纯数学，并增加我们的集体理解广义编织和图代数与数学和科学其他领域的联系也意味着该项目将产生连锁反应，使整个领域的研究受益科学。