Tropical Combinatorics of Graphs and Matroids

该项目由组合学计划和刺激竞争研究的既定计划（EPSCoR）共同资助。组合数学是研究离散结构的数学分支。 本计画将探讨图与拟阵，这是组合数学中的两个经典主题。 图本质上与网络相同，由于社交网络的流行，网络是现在广为人知的概念。 另一方面，拟阵是一种结构，它抽象了数学中线性独立的概念，例如平面上的三个给定点是否位于公共线上。 组合数学承认许多重要的连接与代数几何，这是研究解决方案的多项式方程，和热带几何，这是一个组合版本的代数几何。热带几何的优点之一是，代数几何中的一些困难问题可以简化为组合数学中的问题。 这个理论有一个互补的优势：它提供了一个新的视角，经典的组合对象，如图和拟阵，可以被视为热带曲线和热带线性空间，分别，从而打开这些领域的新技术和问题。 本专题将从热带几何的角度研究图和拟阵。 该项目还将为研究生提供研究机会和支持，并向佛蒙特州的农村高中提供推广活动。一个图可以看作是一条曲线在非阿基米德域上的热带化。 然后，曲线的除数理论转化为切屑燃烧的经典研究。 这使得对芯片发射的深入了解成为可能，例如图形的黎曼-罗奇定理。 PI旨在进一步理解图的除数理论，与图方向的连接，组合表示理论和Tutte多项式。 热带几何学研究的是平衡的多面体复合体，即使它们不是作为变体的热带化而出现的。 这种观点在近年来已经非常富有成效，因为所有拟阵，而不仅仅是可实现的拟阵，都适合这个框架。 PI将进一步研究霍奇理论的拟阵以及热带连接到多面体理论的更经典的方面，如associahedra和其他家庭的广义置换hedra.This奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。

This project is jointly funded by the Combinatorics Program and the Established Program to Stimulate Competitive Research (EPSCoR). Combinatorics is a subfield of mathematics concerned with the study of discrete structures. This project will investigate graphs and matroids, which are two classical topics in combinatorics. A graph is essentially the same as a network, a concept which is now widely known due to the popularity of social networks. On the other hand, a matroid is a structure which abstracts the notion of linear independence in mathematics, e.g. whether or not three given points in a plane lie on a common line. Combinatorics admits many important connections with algebraic geometry, which is the study of solutions to polynomial equations, and tropical geometry which is a combinatorial version of algebraic geometry. One of the strengths of tropical geometry is that some difficult questions in algebraic geometry can be reduced to problems in combinatorics. There is a complementary strength of this theory: it provides a new perspective on classical combinatorial objects such as graphs and matroids, which can be viewed as tropical curves and tropical linear spaces, respectively, thus opening these fields up to new techniques and questions. This project will investigate graphs and matroids from the perspective of tropical geometry. The project will also provide research opportunities and support for graduate students, as well as outreach activities to rural high schools in Vermont. A graph can be viewed as the tropicalization of a curve over a non-Archimedean field. Divisor theory for curves then translates to the classical study of chip-firing. This has allowed for deep insights into chip-firing such as the Riemann-Roch theorem for graphs. The PI aims to further understand divisor theory for graphs, connections to graph orientations, combinatorial representation theory, and the Tutte polynomial. Tropical geometry is concerned with the study of balanced polyhedral complexes even when they do not arise as the tropicalizations of varieties. This perspective has been very fruitful in recent years as all matroids, not only the realizable ones, fit into this framework. The PI will further investigate Hodge theory for matroids as well as tropical connections to more classical aspects of polytope theory such as associahedra and other families of generalized permutahedra.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.