Combinatorics of Total Positivity: Amplituhedra and Braid Varieties

真实的世界问题的答案，如确定粒子加速器中粒子的行为，往往是相当复杂的。数学抽象了这些复杂的行为，并经常揭示隐藏的结构;抽象使人们看到森林而不是树木。例如，物理学家Arkhani-Hamed和Trnka发现了一个被称为“振幅面体”的高维数学对象，其几何形状应该控制粒子散射。然而，随着抽象的增加，直觉减少了;很容易在云中失去树木。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。在幅面体的情况下，组合学提供了一种将幅面体分解成更小、更简单的块的方法。它还提供了一种可视化每一块的方法，即使这些块不适合三维。正是通过这种组合，振幅面体和粒子散射之间的几何关系是最明显的。在一个项目中，PI将与合作者Even-Zohar、Lakrec、Parisi、Tessler和威廉姆斯一起努力证明这种推测关系。一般来说，PI将寻求更好地理解振幅面体和相关数学对象的组合，称为集群品种。PI将涉及本科生和研究生在这项研究中。更广泛的数学背景下提出的项目是理论的总积极性。经典地，如果所有子式都是正的，则矩阵是完全正的。Lusztig将完全正性的概念扩展到部分旗簇，而Postnikov独立定义了正格拉斯曼。全正性的组合学非常丰富，导致了Fomin和Zelevinsky对簇代数的定义。PI建议通过组合透镜研究总积极性的两种概括。第一个项目涉及振幅面体，它推广了正格拉斯曼并出现在粒子物理学中。PI将致力于解决m=4振幅面体的平铺与散射振幅计算之间的关系的问题，以及m=2振幅面体的平铺的各种问题。第二个项目是关于辫子簇的簇结构，它推广了正偏旗簇。PI将进一步发展这种集群结构的组合学，研究3D plabic图及其与编织的关系，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.