Tropical Combinatorics of Graphs and Matroids

该项目由组合学计划和促进竞争研究的既定计划（EPSCoR）共同资助。组合学是研究离散结构的数学分支。本课题将探讨图和拟阵这两个组合学中的经典课题。图本质上与网络相同，由于社交网络的普及，这个概念现在已经广为人知。另一方面，矩阵是一种抽象了数学中线性无关概念的结构，例如平面上的三个给定点是否位于一条公共线上。组合学与代数几何（研究多项式方程的解）和热带几何（代数几何的组合形式）有许多重要的联系。热带几何的优势之一是代数几何中的一些难题可以简化为组合几何中的问题。这一理论有一个互补的优势：它为经典组合对象提供了一个新的视角，如图和拟阵，它们可以分别被视为热带曲线和热带线性空间，从而为这些领域开辟了新的技术和问题。这个项目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，并为佛蒙特州的农村高中提供外展活动。图可以看作是曲线在非阿基米德场上的热带化。然后将曲线的除数理论转化为芯片发射的经典研究。这使得对芯片发射的深入了解成为可能，例如图的黎曼-洛克定理。PI旨在进一步理解图的除数理论，与图方向的联系，组合表示理论和Tutte多项式。热带几何学关注的是平衡多面体复合体的研究，即使它们不是品种的热带化。这一观点近年来非常富有成果，因为所有拟矩阵，不仅是可实现的拟矩阵，都符合这一框架。PI将进一步研究拟阵的Hodge理论，以及与多面体理论更经典方面的热带联系，如缔合面体和其他广义复面体族。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。

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.