Geometry of Cluster Algebras

簇代数的几何

基本信息

批准号：
1855135
负责人：
David Speyer
金额：
$ 24万
依托单位：
Regents of the University of Michigan - Ann Arbor
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-08-01 至 2024-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1855135&HistoricalAwards=false
关键词：
Geometry Cluster Algebras

项目摘要

Cluster varieties are certain geometric spaces of parameters; certain functions of those parameters form so-called cluster algebras. These have found applications throughout mathematics and mathematical physics. This project proposes to study the geometry of cluster algebras in two senses. The first is to apply tools from algebraic geometry to the study of cluster varieties. This will hopefully enable us to understand the structure of integrals computed on cluster varieties, which are common in applications to particle physics. The second sense is to describe cluster varieties using the geometry of reflection groups. These are the symmetries formed by reflections over collections of mirrors. Such connections are already well known for finite reflection groups, but the typical cluster variety should be related to the properties of infinite reflection groups, and these are still very poorly understood.More precisely, the first project is to compute the mixed Hodge structure on the cohomology of cluster varieties. This is joint work with Thomas Lam. In previous work, the PI and his collaborator have discovered a "curious Lefschetz symmetry" in these mixed Hodge structures. They intend to build a spectral sequence to help compute these mixed Hodge structures, and they intend to study connections between these computations, rational Catalan theory and character varieties. The second project has as its ultimate goal describing cluster fans and cluster complexes in terms of infinite Coxeter groups. Nathan Reading and the PI previously did this for cluster varieties of finite type, which corresponded to finite Coxeter groups, and partially succeeded in doing so in general. The obstacle to further progress was that they needed a lattice into which they could embed an infinite Coxeter group; in the finite case, the Coxeter group itself is such a lattice. The current project, joint with Nathan Reading and Hugh Thomas, proposes to solve this problem by using ideas from the representation theory of quivers. They also intend to find combinatorial models of these lattices, and hope to eventually find applications to cluster varieties.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.

簇簇是参数的某些几何空间；这些参数的某些函数形成所谓的簇代数。这些在数学和数学物理中都有应用。该项目提出从两种意义上研究簇代数的几何。第一个是将代数几何的工具应用于簇簇的研究。这有望使我们能够理解在簇簇上计算的积分结构，这在粒子物理学的应用中很常见。第二种意义是利用反射群的几何形状来描述簇的多样性。这些是由镜子集合的反射形成的对称性。对于有限反射群，这种连接已经众所周知，但是典型的簇变化应该与无限反射群的属性有关，而这些仍然知之甚少。更准确地说，第一个项目是计算簇簇的上同调。这是与 Thomas Lam 的合作作品。在之前的工作中，PI 和他的合作者在这些混合霍奇结构中发现了“奇怪的莱夫谢茨对称性”。他们打算建立一个谱序列来帮助计算这些混合霍奇结构，并打算研究这些计算、理性加泰罗尼亚理论和字符品种之间的联系。第二个项目的最终目标是用无限 Coxeter 群来描述集群风扇和集群复合体。 Nathan Reading 和 PI 之前曾对有限类型的簇变种（对应于有限 Coxeter 群）进行过此操作，并且总体上取得了部分成功。进一步进展的障碍是他们需要一个可以嵌入无限考克塞特群的格子。在有限情况下，Coxeter 群本身就是这样一个格子。当前的项目与 Nathan Reading 和 Hugh Thomas 合作，提出利用箭袋表示理论的思想来解决这个问题。他们还打算找到这些晶格的组合模型，并希望最终找到聚类品种的应用。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。