Artin groups and diagram algebras via topology

数学用最简单、最纯粹的术语描述我们周围的物体和我们生活的世界。现实世界往往比我们可以用数学建模的复杂得多，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：我们永远不可能造出一个完美的球体（每个现实生活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），因此我们有白天和黑夜——可以说是最基本的对称。数学每天都在我们身边：秋千和环岛，著名的艺术和建筑，工厂里机器人的布局，化合物的对称性，以及辫子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”。让我们先看看方法。“拓扑学”是一个数学领域，研究对象的潜在“形状”，无论你如何弯曲、挤压或拉伸它。例如，一个圆有一个洞，不管你怎么挤压或拉伸它，这个洞永远不会改变。同样，数字8有两个孔。物体中的“洞”由拓扑概念“同调”来描述，这可能是拓扑所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，它可以告诉我们一个圆和数字8本质上是“不同”的形状。我将从一个像圆这样的具体对象开始，而不是从一个代数对象开始，并构建一个在某种意义上“类似”代数对象的具体对象。计算该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。让我们回到代数对象，代数对象有两种类型。在这个proposal中我要研究的第一类代数对象是“Artin groups”。在数学中，物体的“对称”是一种变换，使物体保持“不变”：现实生活中的例子是洗一叠纸，旋转一个环形交叉路口，或者编你的头发。一个对象的对称性被打包成一个称为“群”的代数对象。Artin组是一个广泛的组族，最简单的例子是辫子组——你可以把辫子组想象成用特定数量的头发编成辫子的所有方法的数学体现。Artin群在数学上的定义非常简单，但也非常神秘：即使经过50多年的研究，许多基本问题仍未得到解答。我的目标是建立具体的拓扑对象来研究Artin群，并回答其中的一些基本问题。我将在这个提案中研究的第二种代数对象是“图代数”。假设地上有2n个洞，有n只鼹鼠要从一个洞里钻出来，然后爬到另一个自由的洞里。如果我们考虑这些摩尔所创造的路径，我们得到一个属于我将要学习的一些图代数的图：允许摩尔穿过彼此的路径给我们“布劳尔代数”，如果我们不允许它们穿过我们得到“坦波利-利布代数”。图解代数与物理学有很强的联系：用粒子代替摩尔，给物理学家一个“连接图”。我的目的是单独研究这些代数的同调，并适应一个研究同调的一般框架来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产生有趣的纯数学，并增加我们对我们所生活的世界背后的纯粹现象的集体理解。广义编织代数和图解代数与其他数学和科学领域的联系也意味着这个项目将产生连锁效应，这将有利于整个科学领域的研究。

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.