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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.

簇簇是某些参数的几何空间;这些参数的某些函数形成所谓的簇代数。这些发现在整个数学和数学物理的应用。本项目从两个方面研究簇代数的几何。第一个是应用代数几何的工具来研究簇。这将有望使我们能够理解在粒子物理应用中常见的簇簇簇上计算的积分的结构。第二种意义是用反射群的几何形状来描述簇的多样性。这些对称性是由镜子的反射形成的。这样的连接已经为有限反射群所熟知，但是典型的簇应该与无限反射群的性质有关，这些仍然是非常不好理解的。更确切地说，第一个项目是计算簇上同调的混合Hodge结构。这是我和托马斯林的合作。在以前的工作中，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，他们打算研究这些计算，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描述集群球迷和集群复杂的无限Coxeter集团。Nathan阅读和PI以前对有限类型的簇变体（对应于有限Coxeter群）做了这件事，并且在一般情况下部分成功。进一步发展的障碍是他们需要一个格，可以在其中嵌入一个无限的考克斯特群;在有限的情况下，考克斯特群本身就是这样一个格。目前的项目，联合内森阅读和休托马斯，建议解决这个问题，使用的想法，从表示理论的箭袋。他们还打算找到这些晶格的组合模型，并希望最终找到应用集群variety.This奖项反映了NSF的法定使命，并已被认为是值得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的支持。