Artin groups and diagram algebras via topology

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，但这并不意味着学习基本的数学知识不重要。以球体的数学概念为例：我们永远不可能建立一个完美的球体(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，地球是一个自转的球体(一种数学对称性)，因此给了我们白天和黑夜--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，工厂里的机器人布局，化合物中的对称性，以及发辫，都是丰富的数学现象的例子。在这个项目中，我将使用‘拓扑学方法’来研究‘代数对象’。让我们先来看看这些方法。“拓扑学”是一个数学领域，它研究一个物体的基本‘形状’，无论你将它弯曲、挤压或拉伸多少。例如，一个圆有一个洞，无论你如何挤压它或拉伸它，洞永远不会改变。同样，图8有两个洞。物体中的“洞”是由“同调”的拓扑学概念来描述的，这可能是拓扑学所提供的最成功的工具，也是数学家称之为“拓扑不变性”的一个例子。例如，它可以告诉我们，圆和图形8的形状本质上是“不同的”。我不会一开始就拿一个像圆这样的具体对象，而是拿一个代数对象，然后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。因此，让我们回到代数对象上，其中有两种类型。在这项建议中，我要研究的第一类代数对象是“Artin群”。在数学中，对象的“对称性”是一种使对象保持“不变”的变换：现实生活中的例子是拖着一叠纸，旋转一个环形交叉路口，或编发。一个对象的对称性被包装成一个称为“群”的代数对象。Artin组是一个广泛的组家族，最简单的例子是辫子组-人们可以将辫子组视为仅使用特定数量的辫子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：即使在50多年的工作之后，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Artin群，并回答这些基本问题中的一些。在这项建议中，我将研究的第二类代数对象是“图代数”。考虑地面上的2n个洞，以及n个鼹鼠，它们必须从一个洞中弹出，然后到达一个自由的洞。如果我们考虑这些鼹鼠创造的路径，我们会得到一个属于我将要研究的某些图代数的图：允许鼹鼠相互交叉的路径给我们提供了“Brauer代数”，如果我们不允许它们交叉，我们就得到了“Temperley-Lieb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分子，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调，并调整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之间的联系也意味着，这个项目将产生连锁反应，使各科学领域的研究受益。

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.