Cluster algebras and applications

用字母a、B和c表示三个数的乘积，有两种不同的括号表示方法：（ab）c和a（bc）。结合律说这两个乘积相等。对于四个数字，有五种不同的括号法。将结合律应用于三个数五次表明它们都相等：从（（ab）c）d开始，我们得到（a（bc））d、a（（bc）d）、a（B（cd））和（ab）（cd）;再应用一次结合律，我们又得到（（ab）c）d。这五个可能的括号可以被视为五边形的顶点，其边对应于结合规则的应用。取更长的积，我们得到更高维的多面体。例如，对于5个数的乘积，我们得到一个有14个顶点的三维多面体。一般来说，所得到的形状被称为协面体（associahedron），或Stasheff多面体（Stasheff polytope）-多面体是多边形、多面体或它们的高维形式的总称。簇代数是由Fomin和Zelevinsky在2001年发现的。每个有限型的簇代数都有一个相关联的多面体。特别是，有集群代数，产生上述Stasheff多面体。因此，任意有限型的簇代数可以被看作是结合律的形式推广，其形式类似于Stasheff多面体。这个美丽的理论最初是为了描述李理论中的各种代数而发展的（量化包络代数），这是与矩阵，但自成立以来，它有许多应用在不同的领域，例如物理学中的热力学贝特定理和几何对象Teichmueller空间的描述。我们研究的主要目的是理解簇代数。建议的研究助理，斯科特，已经表明，他们描述了格拉斯曼，这是一个空间的所有可能的嵌入空间（固定维度）在另一个。使用某些称为Postnikov图的平面图，我们计划确定哪些簇代数描述格拉斯曼以及其中包含的某些关键子变种，称为Gelfand-Serganova变种。我们已经发现，正多边形的菱形平铺是一个丰富的自然集群代数的来源，并打算利用这一点来实现上述目标。因此，我们希望能够更好地理解格拉斯曼，特别是，找到很好的因式分解的元素使用集群代数的方法。有这么多的格拉斯曼集群代数，我们猜想，他们将给予洞察集群代数的一般理论。每个簇代数都有一组与之相关的图，我们想知道这是如何划分所有图的集合的;特别是我们想知道树是如何融入这幅图的，树是没有环的图。簇代数的乘法规则也可以变形，产生一族新的代数。我们计划使用的描述格拉斯曼作为集群代数，以了解他们的变形。我们还计划解决集群代数理论中的问题（例如某些正性问题），将它们与其他数学领域联系起来，例如李代数的特征理论，以及应用数学中的狄利克雷到诺依曼问题。这可能会导致我们的工作令人兴奋的应用。

There are two different ways of bracketing the product of three numbers, represented by the letters a,b and c: (ab)c and a(bc). The rule of associativity says that these two products are equal. For four numbers, there are five different ways of bracketing. Applying the associativity rule for three numbers five times shows that they are all equal: starting from ((ab)c)d, we obtain (a(bc))d, a((bc)d), a(b(cd)) and (ab)(cd); applying the rule once more we obtain again ((ab)c)d. These five possible bracketings can be regarded as the vertices of a pentagon with the edges corresponding to applications of the associativity rule. Taking longer products, we obtain higher-dimensional polyhedra. For example, for a product of five numbers, we obtain a 3-dimensional polyhedron with 14 vertices. In general, the shape obtained is known as the associahedron, or Stasheff polytope - a polytope is the generic term for a polygon, polyhedron, or higher dimensional version of one of these.Cluster algebras were discovered in 2001 by Fomin and Zelevinsky. Each cluster algebra of finite type has an associated polytope. In particular, there are cluster algebras which give rise to the Stasheff polytopes mentioned above. So an arbitrary cluster algebra of finite type can be regarded as giving a formal generalisation of the associativity rule, in the form of a polytope similar to the Stasheff polytope.This beautiful theory was initially developed in order to describe various algebras in Lie theory (quantised enveloping algebras) which are associated with matrices, but since inception, it has had many applications in different areas, such as the Thermodynamic Bethe Ansatz in physics and the description of geometric objects known as Teichmueller spaces.The main aim of our research is to understand cluster algebras. The proposed research assistant, Scott, has shown that they describe the Grassmannian, which is the space of all possible embeddings of one space (of fixed dimension) in another. Using certain planar diagrams known as Postnikov diagrams, we plan to determine which cluster algebras describe the Grassmannian as well as certain key subvarieties contained in it, known as Gelfand-Serganova varieties. We have discovered that tilings of regular polygons by rhombuses are a rich source of natural cluster algebras and intend to exploit this to achieve the above aim. As a result, we hope to be able to understand the Grassmannians better and, in particular, to find nice factorisations of their elements using the cluster algebra approach.There are so many Grassmannian cluster algebras, that we conjecture that they will give insight into the general theory of cluster algebras. Each cluster algebra has a set of graphs associated to it, and we would like to understand how this divides up the collection of all graphs; in particular we would like to know how trees, which are graphs with no loops, fit into this picture.The multiplication rule for a cluster algebra can also be deformed to produce a whole family of new algebras. We plan to use the description of Grassmannians as cluster algebras in order to understand their deformations. We also plan to solve problems in the theory of cluster algebras (such as certain positivity questions), by relating them to other areas of mathematics, such as the character theory of Lie algebras and, from applied mathematics, Dirichlet-to-Neumann problems. This is likely to lead to exciting applications of our work.