Combinatorics of Total Positivity: Amplituhedra and Braid Varieties

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。数学将这些复杂的行为抽象出来，并经常揭示隐藏的结构；抽象使人看到森林而不是树木。例如，物理学家Arkhani-Hamed和Trnka发现了一个叫做“振幅面体”的高维数学对象，它的几何形状应该控制着粒子的散射。然而，随着抽象的增加，直觉会减少；在云层中很容易看不见树木。代数组合学作为一门数学学科，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一种以更直观的方式表示一长串数字的工具。在振幅面体的情况下，组合学提供了一种将振幅面体分解成更小、更简单的部分的方法。它还提供了一种可视化每个部分的方法，即使这些部分不适合三维空间。正是通过这种组合，幅面体和粒子散射之间的推测关系是最明显的。在一个项目中，PI将与合作者Even-Zohar、Lakrec、Parisi、Tessler和Williams一起证明这种推测关系。总的来说，PI将寻求更好地理解幅面体的组合学和相关的数学对象，称为聚类变化。该项目将包括本科生和研究生。所提议的项目的更广泛的数学背景是总正性理论。经典地说，如果一个矩阵的所有子矩阵都是正的，那么它就是完全正的。Lusztig将总正性的概念推广到部分旗类，而Postnikov独立地定义了正的Grassmannian。总正的组合学是非常丰富的，这导致了famin和Zelevinsky对簇代数的定义。PI建议通过组合透镜研究总正性的两种推广。第一个项目涉及振幅面，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幅面平铺与散射振幅计算之间关系的猜想，以及关于m=2幅面平铺的各种猜想。第二个项目涉及编织品种的簇结构，它推广了正偏旗品种。PI将进一步发展这种簇结构的组合学，研究三维塑性图及其与编织的关系，并探索其应用。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。

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.