Cluster-tilting theory

流形流形是一种数学空间，它在局部上类似于通常的固定维数的坐标空间，称为欧几里德空间。地球本身就是一个二维流形的例子：地图利用了这样一个事实：如果你不离开一个给定的点太远，地球的表现就很像一个平面。利用这种局部平面结构的小地图可以覆盖整个地球，尽管整个世界的精确地图必须是一个地球仪。一个一维流形，比如一个圆，局部看起来像一条线，也有更高维度的流形。流形在数学中扮演着重要的角色，因为有很多例子，因为它们可以用欧几里得空间的性质来研究。对称一个物体的对称是使它看起来和原来完全一样的变换，比如一个正方形旋转四分之一圈。两种对称总是可以组成第三种对称。这与其他性质一起，使任何物体的对称集成为群的结构。正方形只有8种对称，而圆有无限多的对称，形成一维流形。流形和群结构是相容的，这样的对象被称为李群。李群通常以某些对称群的形式出现，在物理学中也被称为规范群。李群在19世纪70年代，Sophus Lie通过对李群进行线性化来研究它们，从而得到被称为李代数的更简单的空间。李代数可以使用线性代数中更初级的数学概念来研究，线性代数是对向量空间（如欧几里得空间）及其线性变换的研究。这和其他的发展导致了李论这一现代学科的出现，李代数是通过它们的表示理论来研究的：李代数的每个元素都是由向量空间的变换来表示的。这种转换可以通过矩阵更具体地表示，矩阵是矩形的数字数组，只要选择了基（本质上是坐标的选择）。量子群1985年，德林菲尔德和金博引入了经典李代数的变形版本，或量子群，这导致了李代数研究方式的一场革命。特别是，Kashiwara和Lusztig在1990年引入了正则基，它具有美丽的性质。它同时产生了（某种）李代数的所有表示的基。正则基在其他领域也有应用，如仿射Hecke代数的表示理论。该项目试图明确地描述规范基，导致了许多有趣的数学，包括Fomin和Zelevinsky的聚类代数，引入模型的乘法性质。为了理解簇代数，引入了簇范畴和簇倾斜理论。这些对象是用颤振的表示来定义的：颤振是一个有方向边的图，一个表示是每个顶点的向量空间和每个边的变换。除了提供对规范基础的洞察外，簇倾斜理论已迅速成为研究颤振表示的有力工具。规范基的这个模型仍然不完整，所提议的项目将有助于通过量子簇代数获得更好的图像。该项目还将从颤振表示的角度发展集群倾斜理论，通过描述其组合特性并将其推广到几个方向的更广泛的背景下。该项目还将发展与统计力学中出现的图代数的更强联系。

ManifoldsA manifold is a mathematical space which is locally like the usual coordinate space of some fixed dimension, known as Euclidean space. The earth itself is an example of a two-dimensional manifold: maps exploit the fact that, if you don't go too far from a given point, the earth behaves pretty much like a plane. The earth could be covered entirely by small maps making use of this local planar structure, although an accurate map of the entire world must be a globe. A one-dimensional manifold, such as a circle, looks locally like a line, and there are manifolds of higher dimension as well. Manifolds play an important role in mathematics because there are so many examples and because they can be studied using the properties of Euclidean space. SymmetriesThe symmetries of an object are the transformations which leave it looking exactly as it was before, such as a rotation of a square through a quarter of a revolution. Two symmetries can always be composed to give a third. This, together with other properties, gives the set of symmetries of any object the structure of a group. While there are only 8 symmetries of a square, the circle has infinitely many symmetries, forming a one-dimensional manifold. The manifold and group structures are compatible, and such an object is known as a Lie group. Lie groups often arise as certain symmetry groups known as gauge groups in physics as well.Lie groupsIn the 1870s, Sophus Lie studied Lie groups by linearizing them, to obtain simpler spaces known as Lie algebras. Lie algebras can be studied using more elementary mathematical notions from linear algebra, which is the study of vector spaces (such as Euclidean spaces) and their linear transformations. This and other developments led to the modern subject of Lie theory in which Lie algebras are studied via their representation theory: each element of the Lie algebra is represented by a transformation of a vector space. Such transformations can be represented more concretely via matrices, which are rectangular arrays of numbers, provided a basis is chosen (essentially a choice of coordinates).Quantum groupsIn 1985, deformed versions of the classical Lie algebras, or quantum groups, were introduced by Drinfeld and Jimbo, and this led to a revolution in the way that Lie algebras were studied. In particular, Kashiwara and Lusztig introduced the canonical basis in 1990, with beautiful properties. It simultaneously gives rise to bases for all representations for the Lie algebra (of a certain kind). The canonical basis has applications to other fields as well, such as the representation theory of affine Hecke algebras.The projectAttempts to describe the canonical basis explicitly have led to a lot of interesting mathematics, including the cluster algebras of Fomin and Zelevinsky, introduced to model its multiplicative properties. Cluster categories and cluster-tilting theory were introduced in order to understand cluster algebras. These objects were defined using representations of quivers: a quiver is a graph with oriented edges and a representation is a vector space for each vertex and a transformation for each edge. As well as giving insight into the canonical basis, cluster-tilting theory has rapidly become a powerful tool for the study of representations of quivers.This model of the canonical basis is still incomplete and work on the proposed project will help towards a better picture via quantum cluster algebras. The project will also work on developing cluster-tilting theory from the perspective of representations of quivers by describing its combinatorial properties and generalising it to a wider context in several directions. The project will also develop stronger connections to the diagram algebras arising in statistical mechanics.