Rigid structure in noncommutative, geometric and combinatorial problems

非交换、几何和组合问题中的刚性结构

• 批准号: EP/G007632/1
• 负责人: Iain Gordon
• 金额: $ 126.19万
• 依托单位: University of Edinburgh
• 依托单位国家: 英国
• 项目类别: Fellowship
• 财政年份: 2008
• 资助国家: 英国
• 起止时间: 2008 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FG007632%2F1
• 关键词: Rigid; structure; noncommutative; geometric; combinatorial

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.

该提案的研究领域是表示论，这是一个与计算科学、化学和物理学等其他科学有很强相互作用的纯数学领域。数学的基本思想是从自然发生的现象中提取最关键的属性，只留下需要研究和理解的情况的本质。我们都知道这个想法：很久以前，人们并不认为数字是抽象的数量，而是作为描述具体数量的方式。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的对象——我们可以对它们应用代数运算，例如加法和减法，我们可以比较它们等等——但它们并不总是我们可以在现实世界中可视化或指定的对象。 10亿这个数字是什么样的？ 1万亿？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本质上来自于我们像对待任何其他数字一样对待它们的能力，特别是那些我们确实有很好直觉的数字。抽象在数学中比比皆是。例如，对称性的研究被抽象地编码为群的概念。群是满足某些公理的元素的集合，这些公理显然满足对称性。然而，根据定义，群是抽象的，不需要表示为任何特定对象的对称性，而只需表示为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。给定群的一般抽象定义，人们就会想到简单的群，即可以构建每个群的构建块。长期以来，群理论家想要对简单群进行分类，大约二十五个他们提出了一个全面的列表。清单上的许多项目自群论诞生以来就已为人所知，但也有少数例外。这些是新的群，绝对是基础性的，因为它们是基本的构建块，但它们之前并没有作为某些著名数学对象的对称性被观察到。他们从哪里来？它们是某种东西的对称性吗？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）如何成为某些自然发生的对象的对称性。它结合了数学现实和抽象。因此，表示论是许多不同领域的研究人员感兴趣的强大工具。在纯数学中，它在研究抽象系统时非常重要，而且对于理解显示对称性的原始对象也非常有用。它也被用于更广泛的领域，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，或者在流体中帮助求解微分方程。在这个提案中，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，并且这也将引起表示理论家的内在兴趣。在此过程中，我将与英国和国外许多不同主题的研究人员进行互动。这项活动将为爱丁堡、英国及其他地区的数学带来好处。

项目成果

The Pfaffian-Grassmannian equivalence revisited

• DOI: 10.14231/ag-2015-015
• 发表时间: 2014-01
• 期刊: arXiv: Algebraic Geometry
• 作者: N. Addington;W. Donovan;E. Segal

New Trends in Noncommutative Algebra

非交换代数的新趋势

• DOI: 10.1090/conm/562/11131
• 发表时间: 2012
• 作者: Chlouveraki M

Contractions of 3-folds: deformations and invariants

三重收缩：变形和不变量

• DOI: 10.48550/arxiv.1511.01656
• 发表时间: 2015
• 作者: Donovan W

Moduli spaces of torsion sheaves on K3 surfaces and derived equivalences

K3 表面上扭力轮的模空间和导出的等价物

• DOI: 10.1112/jlms/jdw022
• 发表时间: 2016
• 期刊: Journal of the London Mathematical Society
• 作者: Addington N

Gaudin algebras, RSK and Calogero-Moser cells in Type A

A 型高丁代数、RSK 和 Calogero-Moser 细胞

• DOI: 10.1112/plms.12506
• 发表时间: 2023
• 期刊: Proceedings of the London Mathematical Society
• 影响因子: 1.8
• 作者: Brochier A