FRG: Collaborative Research: Dimers in Combinatorics and Physics

统计力学是对小尺度物质的数学研究。 它的主要目标是分析相变：例如液体到固体的转变，其中物质的物理性质突然改变。 二聚体模型最初被认为是二维物质的简化模型，其中可以研究相变。然而，最近的工作已经将这个模型与许多其他数学领域联系起来，从组合数学到弦论，在弦论中，“膜二聚体”被提出作为小尺度时空的基本描述。 PI建议共同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础组合结构。这个项目将导致组织讲习班和定期会议的PI和他们的研究生和博士后研究员，继续PI的努力，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讲座，在同行评议的期刊上发表的论文以及关于这些主题的书籍，为数学界做出贡献。二聚体模型研究圆盘或黎曼曲面上的平面二分图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、代数和物理学中的应用，还有许多悬而未决的问题。圆盘上的二聚体模型和格拉斯曼模型之间存在着一种基本的联系，即二聚体的生成函数满足Plucker关系。这一事实导致了格拉斯曼的齐次坐标环上的簇代数结构，以及正格拉斯曼的美丽组合。 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问题的理解：非平面图上的二聚体模型;二聚体模型在环面和其他表面上的极限行为;环面上的二聚体与弦理论中的膜镶嵌之间的联系; KP方程的孤子解和可实现为孤子图的二分图;凸多边形平铺和相应的Kasteleyn权重的二分平面图之间的关系;二聚体模型和幅面体的三角剖分之间的联系，其体积计算散射幅度的对象;以及更高维度的二聚体模型、彩色颤动和集群突变的广义概念，该奖项反映了NSF的法定使命，并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。

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.