Graphs in Representation Theory

这项跨学科的建议通过引入新的几何和组合结构，将代数、组合学和数论联系起来。这些领域处于现代数学研究的前沿，有望在从理论物理到计算机科学的各种应用中带来潜在的好处，以及在物流、经济和机器学习等广泛背景下的优化问题。通过引入特殊类型的图及其推广，如带状图和超图，到代数及其表示，我最近在几何、组合学和非交换代数之间建立了一种令人兴奋的联系。这项建议建立在这一新知识的基础上并加以扩展。更准确地说，通过新的想法，它将引入组合对象，即所谓的拟阵，以促进对最普遍的代数类别之一：野代数的研究。拟阵及其产生的超图是图的推广，在优化问题、图像聚类和人工智能中得到了应用。特别是非对易代数和表示理论，是通过向量空间上的线性变换集合的作用来研究对称性的。(有限)群是可逆的线性变换的集合。代数更具一般性，因为它们还模拟不可逆过程。代数可以分为两类：驯服的和狂野的。Tame代数通常具有良好的表示理论，到目前为止，表示理论的大部分工作都致力于对它的研究。相比之下，目前研究野代数及其表示理论的工具很少。同时，大多数自然生成的代数都是野性的。基于组合结构的新应用，包括超图和拟阵，该研究引入了几何表面模型形式的野代数的新工具。几何学研究的是点、线、圆等几何对象的形状和空间关系。在现代几何学中，这样的基本几何物体及其在空间中的排列将复杂的结构编码，其起源例如来自物理世界的模型，如弦理论，描述自然界基本力和所有形式物质的数学模型。数论中继整数之后最基本的物体是整数的分数，也被称为有理数。数论中一个重要的悬而未决的问题是绝对伽罗华群在Grothendieck引入的一组称为dessins d‘enfants的图上的作用的刻画，绝对伽罗瓦群是基于有理数的群。中心相关的公开问题是找到表征这一行为的不变量。这项研究的目的是通过应用所提出的研究中建立的代数和组合学的联系来产生新的此类不变量。

This intra-disciplinary proposal links algebra, combinatorics and number theory through the introduction of new geometric and combinatorial structures. These fields lie at the cutting edge of modern mathematics research and promise potential benefits in applications ranging from theoretical physics to computer science and optimization problems in a wide variety of contexts such as logistics, economics and machine learning.By introducing special classes of graphs and their generalizations, such as ribbon graphs and hypergraphs, to algebras and their representations, I recently established an exciting link between geometry, combinatorics and non-commutative algebra. This proposal builds and expands upon this new knowledge. More precisely, through novel ideas it will introduce combinatorial objects, the so-called matroids, to catalyse the study of one of the most ubiquitous classes of algebras: wild algebras. Matroids and the hypergraphs that give rise to them are generalizations of graphs that find applications in optimization problems, image clustering and artificial intelligence. Non-commutative algebra and representation theory in particular is the study of symmetries through the action of collections of linear transformations on vector spaces. A (finite) group is a collection of linear transformations that are invertible. Algebras are more general in that they also model non-invertible processes. Algebras can be divided into two classes: tame and wild. Tame algebras generally have a well-behaved representation theory and the majority of the work in representation theory to date has been devoted to their study. In contrast, there are currently very few tools available to study wild algebras and their representation theory. At the same time, most naturally occurring algebras are wild. The proposed research introduces new tools for wild algebras in the form of geometric surface models based on novel applications of combinatorial structures including hypergraphs and matroids. Geometry is concerned with the configurations and spatial relations of geometric objects such as points, lines and circles. In modern geometry, such basic geometric objects and their arrangements in space encode complicated structures whose origins arise, for example, from models of the physical world such as string theory, a mathematical model describing the fundamental forces in nature and all forms of matter.The most basic objects in number theory after integers are fractions of integers, also known as rational numbers. An important open question in number theory is the characterisation of the action of the absolute Galois group, a group based on the rational numbers, on a set of graphs introduced by Grothendieck, called dessins d'enfants. The central related open problem is to find invariants characterizing this action. This research aims to generate new such invariants through the application of the connections of algebra and combinatorics established in the proposed research.