Coxeter Groups, Scattering Diagrams, and Shards

该项目的目标是更好地理解某些称为Coxeter群的数学对象，并使用Coxeter群在各种其他数学问题上取得进展。 考克斯特群是高度对称对象的对称性的集合（通常称为“群”），例如柏拉图立体（四面体，立方体，八面体，十二面体和二十面体）和高维类似物。 近百年来数学的一个重要主题是Coxeter群和相关的几何对象为解开许多其他数学理论提供了一把非常容易理解和有用的钥匙。 这个项目探索了Coxeter群在Artin群（包括“辫子群”，它描述了编织线的数学），散射图（弦理论中“镜像对称”的关键工具）和簇代数（一个较新的理论，也是其他数学理论的关键）中的应用。该奖项还将资助研究生从事这个项目。该项目的一部分涉及集群散射图及其theta函数的组合学。 我们的目标是连接theta函数的突变风扇（编码矩阵突变的分段线性几何的风扇），并在表面/orbifolds的情况下构建集群散射风扇，并表明theta函数是手镯的基础，在这种情况下。 该项目的第二部分是扩展（从有限型到仿射型）之间的非交叉分区格（出现在理论的阿廷集团）和广义associahedra/散射风扇的深层连接。 与这个目标密切相关的是构建实现经典仿射类型的非交叉划分格的平面图的项目。 该项目的第三部分是（在其最雄心勃勃的制定），以扩大弱顺序有限Coxeter组的无限格（而不是半格）的完全一般Coxeter组，然后进行寒武纪风扇建设集群散射图在完全的一般性。 在有限型中，弱序可以用分片和有限半分配格基本定理（FTFSDL）来理解。 在无限的情况下，计划是用碎片和FTFSDL的无限版本构建一个网格。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The goal of this project is to better understand certain mathematical objects called Coxeter groups, and use Coxeter groups to make progress on a variety of other mathematical questions. Coxeter groups are collections (usually called "groups") of symmetries of highly symmetric objects, like for example the Platonic solids (tetrahedron, cube, octahedron, dodecahedron, and icosahedron) and higher-dimensional analogues. An important theme of mathematics in the last hundred years or so is that Coxeter groups and related geometric objects provide a very accessible and useful key to unlock many other mathematical theories. This project explores the applications of Coxeter groups to Artin groups (including the "braid group", which describes the mathematics of braided strands), scattering diagrams (a key tool for "mirror symmetry" in string theory), and cluster algebras (a newer theory that also functions as a key to other mathematical theories). The award will also fund graduate students working on this project.One part of the project concerns the combinatorics of cluster scattering diagrams and their theta functions. The goal is to connect theta functions to the mutation fan (a fan that encodes the piecewise-linear geometry of matrix mutation) and also to construct the cluster scattering fan in the surfaces/orbifolds case and show that the theta functions are the bracelets basis in that case. A second part of the project is to extend (from finite type to affine type) the deep connections between the noncrossing partition lattices (which appear in the theory of Artin groups) and generalized associahedra/scattering fans. Closely related to this goal is the project of constructing planar diagrams that realize noncrossing partition lattices of classical affine types. The third part of the project is (in its most ambitious formulation) to extend the weak order on a finite Coxeter group to an infinite lattice (not semilattice) for completely general Coxeter groups and then carry out the Cambrian fan construction of cluster scattering diagrams in complete generality. In finite type, the weak order can be understood in terms of shards and the Fundamental Theorem of Finite Semidistributive Lattices (FTFSDL). In the infinite case, the plan is to construct a lattice from shards and an infinite version of FTFSDL.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.