Combinatorial Representation Theory: Discovering the Interfaces of Algebra with Geometry and Topology

组合表示理论：发现代数与几何和拓扑的接口

基本信息

批准号：
EP/W007509/1
负责人：
Karin Baur
金额：
$ 325.55万
依托单位：
University of Leeds
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2022
资助国家：
英国
起止时间：
2022 至无数据
项目状态：
未结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FW007509%2F1
关键词：
Combinatorial Representation Theory Discovering Interfaces

项目摘要

A fundamental, and often successful, way of studying an abstract mathematical object is to consider methods of representing it in another, more concrete object. This is a powerful idea, and recent progress in algebraic representation theory and related areas has given rise to strong opportunities for the transformation of other fields. In particular, geometric and combinatorial phenomena initially specific to representation theory have emerged in many other fields, leading to effective new techniques and applications. Our team is at the forefront of these developments. The PI and the five CoIs have contributed to major advances in the past decade, with their expertise ranging from algebra, geometry, and topology to mathematical physics. This provides new ways to link algebra and geometry & topology. Examples include the categorification of the Grassmannian cluster structure, the McKay correspondence for reflection groups, the lifting of Lie-theoretic techniques to 2-dimensional category theory, with applications to topological physics, and the derivation of decomposition matrices of Brauer algebras from generalised Lie geometry. In all cases, the medium for interpolating between the theories is an emergent geometrical property which is not well understood. For the advancement of research, there is a strong need for explaining these phenomena and placing them in an encompassing novel paradigm. Our proposal hence seeks to understand and investigate relations between very different areas, and so to push on from there in a more systematic framework. This aim would benefit from a broad, holistic view of representation theory, embracing Lie theory, algebraic geometry, low dimensional topology and mathematical physics. Our team in Leeds is uniquely qualified to pursue this programme. Together with specialist collaboration of many mathematicians at our international partner institutions, we will address the current challenges, provide solutions to open questions and develop applications by establishing bridging to other fields. We are in a position to embrace the perspectives of both pure and application-driven mathematics, and with the potential, in the long term, for serving the needs of physical sciences, life sciences and engineering. This unification of perspectives requires a programme-level research structure and algebra is the right core platform for such an ambitious venture. Thus our proposal will push forward the mathematical state-of-the-art and will shape the future directions in the areas we touch upon.

研究抽象数学对象的一种基本且通常成功的方法是考虑在另一个更具体的对象中表示它的方法。这是一个强有力的想法，代数表示论及相关领域的最新进展为其他领域的转型带来了强大的机会。特别是，最初特定于表示论的几何和组合现象已经出现在许多其他领域，从而产生了有效的新技术和应用。我们的团队处于这些发展的最前沿。 PI 和五个 CoI 在过去十年中为重大进步做出了贡献，他们的专业知识涵盖代数、几何、拓扑到数学物理。这提供了连接代数和几何与拓扑的新方法。例子包括格拉斯曼簇结构的分类、反射群的麦凯对应、将李理论技术提升到二维范畴论以及在拓扑物理中的应用，以及从广义李几何推导布劳尔代数的分解矩阵。在所有情况下，理论之间插值的媒介是一种新兴的几何性质，但尚未得到很好的理解。为了研究的进步，强烈需要解释这些现象并将它们置于一个包容性的新颖范式中。因此，我们的建议旨在理解和研究不同领域之间的关系，并以此为基础在一个更系统的框架中推进。这一目标将受益于对表示论的广泛、整体的看法，包括李理论、代数几何、低维拓扑和数学物理学。我们在利兹的团队具有独特的资质来开展该项目。我们将与国际合作机构的许多数学家的专业合作一起，应对当前的挑战，为开放性问题提供解决方案，并通过建立与其他领域的桥梁来开发应用程序。我们能够接受纯数学和应用驱动数学的观点，并且从长远来看，有潜力满足物理科学、生命科学和工程的需求。这种观点的统一需要一个程序级的研究结构，而代数是这样一个雄心勃勃的事业的正确核心平台。因此，我们的建议将推动最先进的数学发展，并将塑造我们所涉及领域的未来方向。