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Totally Positive Spaces and Cluster Algebras

完全正空间和簇代数

基本信息

批准号：
1954121
负责人：
Pavel Galashin
金额：
$ 19.78万
依托单位：
University of California-Los Angeles
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-03-01 至 2024-02-29
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1954121&HistoricalAwards=false
关键词：
Totally Positive Spaces Cluster Algebras

项目摘要

Combinatorics is the study of discrete structures such as permutations and graphs, while topology deals with properties of geometric shapes that stay invariant under continuous deformations. The goal of the project is to apply a mix of topological and combinatorial techniques to questions arising in various areas of mathematics and physics. For example, recent connections between combinatorics and scattering amplitudes give rise to new methods to guide high-energy particle physics experiments. Other applications arise in determining interior properties of materials from boundary measurements, as well as in electrical impedance tomography and other types of medical imaging. The surprisingly natural combinatorial structures associated with the underlying topological spaces allow one to find unexpected connections between seemingly unrelated areas. The award provides research training of graduate studetns.This work comprises several projects that directly involve the positive Grassmannian. The topology of this space has been recently determined; however, the topology of several related spaces remains not fully understood. For example, the physics of scattering amplitudes is intimately related to the amplituhedron, which is a linear projection of the positive Grassmannian. One of the projects involves understanding its topological structure and the combinatorics of its triangulations. Another example is the space of planar Ising networks, which was shown to coincide with the positive orthogonal Grassmannian. This approach has promising applications to the questions of universality and conformal invariance of the Ising model. The resulting cell decomposition of the space of planar Ising networks suggests a surprising direct connection with the space of planar electrical networks. The underlying algebraic structure of such spaces is described by cluster algebras; one of the projects involves studying integrability properties of dynamical systems arising from cluster algebras.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.

组合学是对排列和图等离散结构的研究，而拓扑学则研究在连续变形下保持不变的几何形状的属性。该项目的目标是将拓扑和组合技术结合应用来解决数学和物理各个领域中出现的问题。例如，最近组合学和散射振幅之间的联系产生了指导高能粒子物理实验的新方法。其他应用包括通过边界测量确定材料的内部特性，以及电阻抗断层扫描和其他类型的医学成像。与底层拓扑空间相关的令人惊讶的自然组合结构使人们能够在看似不相关的区域之间找到意想不到的联系。该奖项为研究生提供研究培训。这项工作包括几个直接涉及积极格拉斯曼主义的项目。该空间的拓扑最近已确定；然而，几个相关空间的拓扑结构仍未完全被理解。例如，散射振幅的物理原理与振幅面体密切相关，振幅面体是正格拉斯曼函数的线性投影。其中一个项目涉及了解其拓扑结构及其三角剖分的组合。另一个例子是平面伊辛网络的空间，它被证明与正正交格拉斯曼一致。这种方法对于解决伊辛模型的普适性和共形不变性问题具有广阔的应用前景。由此产生的平面伊辛网络空间的细胞分解表明与平面电气网络空间存在令人惊讶的直接联系。这些空间的底层代数结构由簇代数描述；其中一个项目涉及研究由簇代数产生的动力系统的可积性质。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。