Combinatorics of Total Positivity: Amplituhedra and Braid Varieties

对现实世界问题的答案，例如确定粒子加速器中粒子的行为，通常是相当复杂的。数学抽象了这些复杂的行为，往往揭示了隐藏的结构；抽象让人看到的是森林，而不是树木。例如，物理学家Arkhani-Hamed和Trnka发现了一个高维的数学物体，称为“振幅多面体”，它的几何形状应该控制粒子的散射。然而，随着抽象的增加，直觉会减少；人们很容易看不到云中的树。作为一门数学学科，代数组合学是一种以更具体的方式表示抽象数学的工具--类似于条形图或散点图是以更直观的方式表示一长串数字的工具。在振幅多面体的情况下，组合数学提供了一种将振幅多面体分解成更小、更简单的片段的方法。它还提供了一种可视化每一件作品的方法，即使这些作品不适合三维。正是通过这种组合，振幅四面体与粒子散射之间的猜想关系最为明显。在一个项目中，PI将致力于证明这种猜测的关系，甚至与合作者-佐哈尔，拉克雷克，帕里西，特斯勒和威廉姆斯。总体而言，PI将寻求更好地理解振幅体和相关数学对象的组合，这些对象称为簇变种。PI将包括本科生和研究生参与这项研究。被提议的项目的更广泛的数学背景是完全积极的理论。经典地说，如果一个矩阵的所有子项都是正的，那么它是完全正的。Lusztig将完全正性的概念推广到部分旗帜种类，而波斯尼科夫独立地定义了正格拉斯曼。全正性的组合学非常丰富，导致了Fomin和Zlevinsky对簇代数的定义。PI建议通过组合透镜研究全正性的两个推广。第一个项目涉及振幅，它推广了正格拉斯曼定律，出现在粒子物理学中。PI将致力于解决关于m=4幅多面体的平铺与散射幅度的计算之间的关系的猜想，以及关于m=2幅多面体的平铺的各种猜想。第二个项目涉及编织品种上的集群结构，它概括了积极的部分旗帜品种。PI将进一步开发这种集群结构的组合学，调查3D塑料图形及其与编织的关系，并探索应用。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。

The answers to real world problems, such as determining the behavior of particles in particle accelerators, are often quite complicated. Mathematics abstracts these complicated behaviors, and often reveals hidden structures; abstraction allows one to see the forest rather than the trees. For example, physicists Arkhani-Hamed and Trnka uncovered a high-dimensional mathematical object called the "amplituhedron" whose geometry should govern particle scattering. However, as abstraction increases, intuition decreases; it is easy to lose sight of the trees among the clouds. Algebraic combinatorics, as a mathematical discipline, is a tool to represent abstract mathematics in a more concrete way--similar to how a bar graph or scatter plot is a tool to represent a long list of numbers in a more intuitive way. In the case of the amplituhedron, combinatorics provides a way to break the amplituhedron up into smaller, simpler pieces. It also provides a way to visualize each piece, even though the pieces do not fit in three dimensions. It is through this combinatorics that the conjectural relationship between the amplituhedron and particle scattering is most apparent. In one project the PI will work to prove this conjectural relationship with collaborators Even-Zohar, Lakrec, Parisi, Tessler, and Williams. In general, the PI will seek to better understand the combinatorics of amplituhedra and related mathematical objects called cluster varieties. The PI will involve both undergraduate and graduate students in thisd research.The broader mathematical context for the proposed projects is the theory of total positivity. Classically, a matrix is totally positive if all minors are positive. Lusztig extended the notion of total positivity to partial flag varieties, while Postnikov independently defined the positive Grassmannian. The combinatorics of total positivity is incredibly rich, leading to the definition of cluster algebras by Fomin and Zelevinsky. The PI proposes to study two generalizations of total positivity through a combinatorial lens. The first project concerns amplituhedra, which generalize the positive Grassmannian and arise in particle physics. The PI will work to resolve conjectures on the relationship between tilings of m=4 amplituhedra and the computation of scattering amplitudes, as well as various conjectures on tilings of m=2 amplituhedra. The second project concerns cluster structures on braid varieties, which generalize positive partial flag varieties. The PI will further develop the combinatorics of this cluster structure, investigating 3D plabic graphs and their relationship to weaves, and explore applications.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.