Totally Positive Spaces and Cluster Algebras

组合学研究的是离散结构，如排列和图，而拓扑学则研究在连续变形下保持不变的几何形状的性质。该项目的目标是将拓扑学和组合技术的组合应用于数学和物理的各个领域中出现的问题。例如，最近组合学和散射幅度之间的联系产生了指导高能粒子物理实验的新方法。其他应用出现在通过边界测量确定材料的内部属性，以及在电阻抗断层成像和其他类型的医学成像中。与底层拓扑空间相关的令人惊讶的自然组合结构允许人们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人的项目。这个空间的拓扑最近已经确定；然而，几个相关空间的拓扑仍然没有完全了解。例如，散射振幅的物理学与振幅四面体密切相关，振幅四面体是正格拉斯曼数的线性投影。其中一个项目涉及了解其拓扑结构和其三角剖分的组合学。另一个例子是平面Ising网络空间，它被证明与正正交Grassmanian重合。这种方法在伊辛模型的普适性和共形不变性问题上有很好的应用前景。由此产生的平面伊辛网络空间的胞元分解表明，与平面电网络空间有着惊人的直接联系。这类空间的基本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。该奖项反映了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。

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.