Rigid structure in noncommutative, geometric and combinatorial problems

该提案的拟议研究是在表示论中，这是一个纯数学领域，与其他科学（包括计算科学，化学和物理学）有很强的相互作用。数学的一个基本思想是从自然发生的现象中提取最关键的性质，只留下待研究和理解的情况的本质。我们都知道这个想法：很久以前，人们不认为数字是抽象的量，而是描述具体量的方式。一只羊，两只羊，三只羊。抽象地思考数字是一个了不起的进步：它们是我们可以做事情的对象-我们可以对它们进行代数运算，例如加减，我们可以比较它们，等等-但它们并不总是我们可以在真实的世界中可视化或指定的对象。10亿这个数字是什么样子的？一万亿？我们知道它们是很大的数字，因为我们可以将它们与其他数字进行比较，但是有多大呢？我们对这些数字的感觉主要来自于我们把它们当作任何其他数字来对待的能力，特别是那些我们确实有很好的直觉的数字。数学中有大量的抽象。例如，对对称性的研究是抽象地编码在群的概念中的。群是满足某些公理的元素的集合，这些公理显然适用于对称性。然而，一个群根据定义是抽象的，并且不需要被表示为任何特定对象的对称，而仅仅是满足给定公理列表的对象。公理化方法是一种非常强大的方法，在这种情况下，它允许人们证明许多一般定理，所有这些定理都可以应用于任何群。给出了群的一般抽象定义，人们就会想到简单群，每个群都可以从这些简单群中构建。很长一段时间以来，群论学家都想对简单的群进行分类，大约在25岁时，他们提出了一个全面的列表。自从群论诞生以来，这个列表上的许多项目都是已知的，但也有少数例外。这些是新的群，绝对是基本的，因为它们是基本的积木，但它们以前没有被观察到作为一些著名的数学对象的对称性。他们是从哪里来的？它们是什么东西的对称吗？这就是表示论的用武之地：它研究一个群（或其他抽象数学结构）如何成为某些自然存在的对象的对称性。它结合了数学实在性和抽象性，因此表现论是一个强有力的工具，受到许多不同领域研究者的关注。在纯数学中，它在研究抽象系统时很重要，但对于理解显示对称性的原始对象也非常有用。它也被用于更远的领域，例如在化学中帮助研究分子的对称性，在物理学中研究空间的性质，或在流体中帮助求解微分方程。在这个建议中，我打算使用非交换代数表示论所产生的刚性结构来构建数学工具，然后将其应用于解决许多不同领域的问题，这也将是表示论理论家的内在兴趣。在此过程中，我将与来自英国和国外许多不同主题的研究人员进行互动。这项活动将有利于数学在爱丁堡，英国和超越。

The proposed research of this proposal is in representation theory, a field of pure mathematics with strong interactions with other sciences including computing science, chemistry and physics. A basic idea of mathematics is to distill the most crucial properties from naturally occurring phenomena, leaving simply the essence of the situation to be studied and understood. We all know this idea already: long ago people did not think of numbers as abstract quantities, but as way of describing specific quantities. One sheep, two sheep, three sheep. It was a remarkable step forward to think of numbers abstractly: they are objects to which we can do things - we can apply algebraic operations to them, such as adding and subtracting, we can compare them, etc - but they are not always objects that we can visualise or specify in the real world. What does the number 1 billion look like? 1 trillion? We know that they are big numbers because we can compare them to other numbers, but how big? Our feeling for such numbers comes essentially from our ability to treat them just like any other number, and in particular ones for which we do have a very good intuition. Abstraction abounds in mathematics. For instance, the study of symmetry is encoded abstractly in the notion of a group. Groups are collections of elements which satisfy certain axioms, axioms which obviously hold for symmetries. However, a group is abstract by definition and need not be presented as symmetries of any particular object, but only as an object satisfying the given list of axioms. The axiomatic approach is a very powerful method which, in this instance, allows one to prove many general theorems, all of which can be applied to any group. Given the general abstract definition of a group, one is lead to think about simple groups, the building blocks from which every group can be built. For a long time group theorists wanted to classify simple groups, and around twenty five they came up with a comprehensive list. Many items on the list had been known since the birth of group theory, but there were a small number of exceptions. These were new groups, absolutely fundamental since they were basic building blocks, but they had not been observed earlier as symmetries of some well-known mathematical object. Where did they come from? Were they symmetries of something? This is where representation theory comes in: it studies how a group (or other abstract mathematical structures) can be the symmetry of some naturally occurring object. It weds mathematical reality and abstraction.Representation theory is thus a powerful tool that is of interest to researchers in many different fields. Within pure mathematics it is important when studying abstract systems, but it is also a very useful for understanding the orginal objects which display the symmetry. It is also used further afield, for instance in chemistry to help to study the symmetry of molecules, in physics when studying the nature of space, or in fluids to help to solve differential equations. In this proposal I intend to build mathematical tools using the rigid structure arising from the representation theory of noncommutative algebras which can then be applied to solve problems in a number of different fields and which will also be of intrinsic interest to representation theorists. In doing this, I will interact with researchers from many different topics, both in the UK and abroad. This activity will have benefits for mathematics in Edinburgh, the UK and beyond.