Tropical Combinatorics of Graphs and Matroids

图和拟阵的热带组合

• 批准号: 2246967
• 负责人: Spencer Backman
• 金额: $ 20万
• 依托单位: University of Vermont & State Agricultural College
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-09-01 至 2026-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2246967&HistoricalAwards=false
• 关键词: Tropical; Combinatorics; Graphs; Matroids

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.

该项目由Combinatorics计划和启发竞争性研究的既定计划共同资助（EPSCOR）。组合学是与离散结构研究有关的数学子场。 该项目将研究图形和矩形，这是组合学中的两个经典主题。 图与网络基本相同，该概念由于社交网络的普及而现在广为人知。 另一方面，矩阵是一种在数学中提取线性独立概念的结构，例如平面中的三个给定点是否位于公共线上。 Combinatorics接受了代数几何形状的许多重要联系，这是对多项式方程的解决方案的研究，以及热带几何形状，这是代数几何形状的组合版本。热带几何形状的优势之一是代数几何学中的一些困难问题可以简化为组合学中的问题。 该理论具有互补的强度：它为图形和矩形等经典组合对象提供了新的观点，可以将其视为热带曲线和热带线性空间，从而将这些领域开放到新技术和问题上。 该项目将从热带几何形状的角度研究图形和成曲线。 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外展活动。图可以看作是非架构场上曲线的热带化。 然后，曲线的除数理论转化为芯片传火的经典研究。 这可以深入了解芯片射击，例如用于图的Riemann-Roch定理。 PI旨在进一步了解图形，图取向的连接，组合表示理论和Tutte多项式。 热带几何形状与平衡多面体复合物的研究有关，即使它们不作为品种的热带化。 近年来，这种观点一直非常富有成果，因为所有矩阵，不仅是可实现的曲目，都适合该框架。 PI将进一步调查曲霉的杂物理论，以及与多层理论的更经典方面的热带联系，例如Associahedra和其他广义定义性的家族。该奖项反映了NSF的法定任务，并已通过评估该基金会的智力功能和广泛的影响来审查Criteria。