FRG: Collaborative Research: Dimers in Combinatorics and Physics

FRG：合作研究：组合学和物理学中的二聚体

基本信息

批准号：
1940932
负责人：
Richard Kenyon
金额：
$ 26.49万
依托单位：
Yale University
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：
2019-07-01 至 2023-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1940932&HistoricalAwards=false
关键词：
FRG Collaborative Research Dimers Combinatorics

项目摘要

Statistical mechanics is the mathematical study of matter at small scales. Its primary goals are to analyze phase transitions: for example liquid-to-solid transitions where the physical properties of a substance change abruptly. The dimer model was originally conceived as a simplified model of two-dimensional matter in which phase transitions can be studied. Recent work, however, has linked the model to many other areas of mathematics, from combinatorics to string theory, where ''brane dimers'' are proposed as fundamental descriptions of spacetime at small scales. The PIs propose to jointly investigate a number of interrelated topics in mathematics and physics, each of which has the dimer model as its underlying combinatorial structure. This project will lead to the organization of workshops and regular meetings of the PIs and their graduate students and postdoctoral fellows, continuing the PIs' efforts to get young mathematicians and physicists involved in these topics. The PIs will contribute to the mathematical community through their mentorship of young scholars, research talks in conferences and workshops, papers published in peer-reviewed journals, and books on a selection of these topics.The dimer model studies the set of all dimers, or perfect matchings, on a planar bipartite graph G on a disk or Riemann surface. Despite the simple definition, there are many open problems about the dimer model, as well as applications to geometry, algebra, and physics. There is a fundamental connection between the dimer model on the disk and the Grassmannian, via the fact that generating functions of dimers satisfy Plucker relations. This fact leads to the cluster algebra structure on the homogeneous coordinate ring of the Grassmannian, and the beautiful combinatorics of the positive Grassmannian. This project will explore a myriad of generalizations of the objects mentioned above, and will significantly improve our understanding of: the dimer model on non-planar graphs; limiting behaviors of the dimer model on a torus and other surfaces; the connection between dimers on a torus and brane tilings in string theory; soliton solutions to the KP equation and the bipartite graphs realizable as soliton graphs; the relationship between convex polygon tilings and the corresponding bipartite planar graphs with Kasteleyn weightings; the connection between the dimer model and triangulations of the amplituhedron, an object whose volume computes scattering amplitudes; and higher-dimensional dimer models, colored quivers and a generalized notion of cluster mutation, exciting new objects motivated by dualities in supersymmetric quantum field theory.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.

统计力学是小尺度上物质的数学研究。它的主要目标是分析相变：例如，物质的物理特性突然变化。该二聚体模型最初被认为是二维物质的简化模型，可以研究相变。然而，最近的工作将模型与数学的许多其他领域联系起来，从组合学到弦理论，其中“ Brane Dimers”被提出为在小尺度上时空的基本描述。 PI提议共同研究数学和物理学中的许多相互关联的主题，每个主题都有二聚体模型作为其基础组合结构。该项目将导致组织的研讨会和PIS的定期会议及其研究生和博士后研究员，继续PIS努力使年轻的数学家和物理学家参与这些主题。 PI将通过对年轻学者的指导，会议和讲习班的研究演讲，在同行评审的期刊上发表的论文以及有关这些主题的选择的书籍为数学社区做出贡献。二聚体模型研究所有二聚体的集合或完美的匹配，在平面bipartite Graph g on Planar Biptite Graper g on Planar Biptite g on Disk con a Disk con a Disk cons corme timank或riemann surface。尽管有一个简单的定义，但有关二聚体模型以及对几何，代数和物理的应用仍存在许多开放问题。磁盘上的二聚体模型与司法二聚体模型之间存在基本联系，这是因为产生二聚体的功能满足了拔出者的关系。这一事实导致了格拉斯曼尼亚人均匀坐标环的集群代数结构，以及阳性格拉斯曼尼亚的美丽组合。该项目将探讨上述对象的无数概括，并将大大提高我们对：非平面图上的二聚体模型；在圆环和其他表面上限制二聚体模型的行为；弦理论中的二聚体与蓝砖之间的连接； KP方程的孤子解决方案和双方图形可作为孤子图实现的图形；凸多边形瓷砖与具有Kasteleyn权重的相应的两部分平面图之间的关系；二聚体模型和Amplituhedron的三角形之间的连接，该对象的体积计算散射幅度；以及高维二聚体模型，彩色颤动和集群突变的普遍概念，令人兴奋的新物体是由超对称量子场理论中的二元性促进的。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子和更广泛影响的审查标准来通过评估来通过评估来支持的。