Totally Positive Spaces and Cluster Algebras

组合学研究的是像排列和图这样的离散结构，而拓扑学研究的是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和组合技术的混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。其他应用还包括通过边界测量来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允许人们在看似不相关的区域之间发现意想不到的联系。该奖项为研究生提供研究培训。这项工作包括几个项目，直接涉及积极的格拉斯曼。这个空间的拓扑结构是最近确定的；然而，一些相关空间的拓扑结构仍未被完全理解。例如，散射振幅的物理学与振幅面体密切相关，它是正格拉斯曼的线性投影。其中一个项目涉及了解它的拓扑结构和三角形的组合。另一个例子是平面Ising网络空间，它被证明与正正交的Grassmannian重合。这种方法在解决伊辛模型的普适性和保角不变性问题上具有很好的应用前景。由此产生的平面伊辛网络空间的细胞分解表明，它与平面电网络空间有着惊人的直接联系。这种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。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.