Tropical Combinatorics of Graphs and Matroids

点击翻译按钮获取中文摘要

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.