Coxeter Groups, Scattering Diagrams, and Shards

Coxeter 组、散点图和碎片

• 批准号: 2054489
• 负责人: Nathan Reading
• 金额: $ 27万
• 依托单位: North Carolina State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-08-15 至 2024-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2054489&HistoricalAwards=false
• 关键词: Coxeter; Groups; Scattering; Diagrams; Shards

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.

这个项目的目标是更好地理解某些被称为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。Coxeter群是高度对称对象的对称性的集合(通常称为“群”)，例如柏拉图固体(四面体、立方体、八面体、十二面体和二十面体)和更高维的类似物。在过去的一百年左右的时间里，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获得和有用的钥匙。这个项目探索了Coxeter群在Artin群(包括描述辫子链数学的“辫子群”)、散射图(弦论中“镜像对称”的关键工具)和簇代数(一种较新的理论，也是其他数学理论的关键)中的应用。该奖项还将资助从事这一项目的研究生。该项目的一部分涉及星系团散布图的组合及其theta函数。我们的目标是将theta函数连接到突变扇(编码矩阵突变的分段线性几何的扇形)，并在曲面/奥比诺尔德的情况下构造簇散射扇，并证明在这种情况下theta函数是手镯的基础。项目的第二部分是扩展(从有限类型到仿射类型)不相交的划分格(出现在Artin群理论中)和广义结合面体/散射扇之间的深层联系。与这一目标密切相关的是构造实现经典仿射类型的不相交划分格的平面图的项目。该项目的第三部分(在其最雄心勃勃的提法中)是将有限Coxeter群上的弱序推广到完全一般Coxeter群的无限格(而不是半格)，然后进行完全一般化的寒武纪星团散射图的Fan构造。在有限类型中，弱序可以用碎片和有限半分配格基本定理(FTFSDL)来理解。在无限的情况下，计划是用碎片和无限版本的FTFSD构建一个网格。这个奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。