Tropical Combinatorics of Graphs and Matroids

该项目由组合学计划和已建立的激励竞争研究计划(EPSCoR)共同资助。组合学是数学的一个子领域，研究离散结构。这个项目将研究图和拟阵，这是组合学中的两个经典主题。图表本质上等同于网络，由于社交网络的流行，这个概念现在已经广为人知。另一方面，拟阵是一种结构，它抽象了数学中的线性独立概念，例如平面上的三个给定点是否位于一条公共直线上。组合数学与代数几何和热带几何有许多重要的联系，前者研究多项式方程的解，后者是代数几何的组合版本。热带几何的优点之一是代数几何中的一些困难问题可以归结为组合学中的问题。这个理论有一个互补的优势：它为经典的组合对象，如图和拟阵提供了一个新的视角，这些组合对象分别可以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学的角度研究图形和拟阵。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开展外联活动。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烧成的经典研究。这使得人们对切屑烧成有了更深入的了解，比如图的Riemann-Roch定理。PI的目的是进一步了解图的除数理论、与图定向的联系、组合表示理论和图特多项式。热带几何学涉及平衡多面体复合体的研究，即使它们不是作为变种的热带化出现的。近年来，这一观点非常有成效，因为所有拟阵动物，不仅是可实现的，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，如缔合面和其他广义置换面体家族。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。

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.