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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.

数学用最简单和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。真实的世界往往比我们可以用数学建模的世界更复杂，但这并不意味着研究底层的数学不重要。以球体的数学概念为例：我们永远不可能建造出一个完美的球体（每一个真实的生命模型都会有微观上的缺陷），但理解球体可以帮助我们理解地球。毕竟，地球是一个旋转的球体（数学对称），因此给了我们黑夜和白天-可以说是最基本的对称。数学每天都围绕着我们：秋千和圆，著名的艺术和建筑，机器人在工厂车间的布局，化合物的对称性，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”来研究“代数对象”。让我们先来看看方法。“拓扑学”是一个数学领域，它研究一个物体的基本“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，这个洞永远不会改变。类似地，数字8具有两个孔。一个物体中的“洞”可以用拓扑学中的“同调”概念来描述，这可能是拓扑学所能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。开始时，我将不采用一个具体的对象，如一个圆，而是采用一个代数对象，并建立一个在某种意义上“类似”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关于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。第一种类型的代数对象，我将研究在这个建议是“阿廷群”。在数学中，物体的“对称性”是一种保持物体“不变”的变换：真实的生活中的例子是洗牌一叠纸，旋转一个环形交叉路口，或者编织你的头发。一个对象的对称性被打包成一个代数对象，称为“群”。Artin群是一个广泛的群族，最简单的例子是辫子群-人们可以认为辫子群是所有使用特定数量的股编织头发的方法的数学体现。Artin群的数学定义非常简单，但非常神秘：即使经过50多年的研究，许多基本问题仍然没有答案。我的目标是建立具体的拓扑对象来研究阿廷群，并回答其中的一些基本问题。第二种类型的代数对象，我将研究在这个建议是“图代数”。考虑地面上的2n个洞，n只鼹鼠必须从一个洞里跳出来，然后到达一个自由的洞。如果我们考虑的路径这些摩尔创建我们得到一个图属于一些图代数我将研究：允许摩尔交叉对方的路径给我们的“布劳尔代数”，如果我们不允许他们交叉我们得到的“坦珀利-李伯代数”。图代数与物理学有很强的联系：用粒子代替我们的摩尔，给出了物理学家的“连接图”。我的目标是分别研究这些代数的同调，并调整一个研究同调的一般框架，以包括这些代数的同调，使用拓扑学的方法研究这些代数对象将导致有趣的纯数学，并增加我们对我们生活的世界背后的纯现象的集体理解。广义编织和图代数与数学和科学的其他领域的联系也意味着这个项目将产生连锁效应，使整个科学研究受益。