FRG: Collaborative Research: Dimers in Combinatorics and Physics

统计力学是在小尺度上对物质进行的数学研究。它的主要目标是分析相变：例如，液体到固体的转变，即物质的物理性质突然改变。二聚体模型最初被认为是一种简化的二维物质模型，可以在其中研究相变。然而，最近的工作将该模型与从组合学到弦论的许多其他数学领域联系在一起，在这些领域中，提出了作为对小尺度时空的基本描述的膜二聚体。PI建议联合研究数学和物理中的一些相互关联的主题，每个主题都有二聚体模型作为其基本的组合结构。该项目将导致为私人投资机构及其研究生和博士后研究员组织讲习班和定期会议，继续推动个人投资机构努力让年轻的数学家和物理学家参与这些主题。PI将通过他们对年轻学者的指导，在会议和研讨会上的研究演讲，在同行评议的期刊上发表的论文，以及关于这些主题的精选书籍，为数学界做出贡献。二聚体模型研究圆盘或黎曼曲面上平面二部图G上的所有二聚体或完美匹配的集合。尽管定义很简单，但关于二聚体模型以及在几何、代数和物理中的应用，还有许多悬而未决的问题。通过二聚体的生成函数满足Plucker关系这一事实，圆盘上的二聚体模型与Grassman模型之间存在着根本的联系。这一事实导致了Grassman齐次坐标环上的簇代数结构，以及正Grassman环的美丽组合。这个项目将探索上述对象的无数推广，并将显著提高我们对以下方面的理解：非平面图上的二聚体模型；环面和其他表面上的二聚体模型的极限行为；环面上的二聚体与弦理论中的膜片之间的联系；KP方程和可实现为孤子图的二部图的孤子解；凸多边形片与相应的带Kasteleyn权的二部平面图之间的关系；二聚体模型与振幅四面体三角之间的联系，其体积计算散射幅度；以及更高维的二聚体模型、彩色颤动和广义的星团突变概念，激发了超对称量子场论中的对偶性激发的新对象。这一奖项反映了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。

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.