Categorical Invariants of Matroids

拟阵是编码线性依赖概念的组合抽象的对象。 这个名字来源于矩阵这个词：给定一个矩阵A，有一个相关联的拟阵，它记录了A的哪些列的子集是线性相关的。 不是所有的拟阵都是以这种方式产生的，但是那些确实形成了一大类拟阵的拟阵，它们为一般的拟阵提供了良好的直觉。 在过去的十年里，拟阵的工作出现了爆炸式的增长，这在很大程度上要归功于June Huh的工作，他获得了2022年的菲尔兹奖。 Huh使用代数几何的技术来解决多年来一直开放的拟阵问题，包括1976年的Welsh对数猜想，以及1975年的Dowling和Wilson头重脚轻的猜想。 这个项目的各个方向背后的共同主题是将涉及拟阵的数值不变量的各种定理、结构和构造升级为考虑拟阵对称性的更丰富的结构。该项目的更广泛影响包括指导研究生和组织讲习班。第一个项目将丰富我们的理解的对数和真实的根在拟阵不变量的上下文中的等价工作。 在特殊情况下，它允许问新的问题的配置空间的点在平面上的上同调，并关于欧拉多项式。 第二个项目揭示了神秘的概念，价值重铸它在范畴方面，并提供了一种方法来计算大类的拟阵不变量等变。第三个项目引入了拟阵的局部h-多项式的概念，并有可能对拟阵的可实现性和图的可着色性的经典问题作出新的进展。 最后一个项目涉及图的小定理的分类，与应用程序的配置空间的同源graphs.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。

A matroid is an object that encodes a combinatorial abstraction of the notion of linear dependence. The name derives from the word matrix: given a matrix A, there is an associated matroid that records the information of which subsets of the columns of A are linearly dependent. Not all matroids arise in this manner, but those that do form a large class of matroids that provide good intuition for matroids in general. There has been an explosion of work on matroids in the past ten years, thanks largely to the work of June Huh, which was recognized with a 2022 Fields Medal. Huh has used techniques from algebraic geometry to resolve conjectures about matroids that have been open for many years, including Welsh's log concavity conjecture from 1976, and Dowling and Wilson's top heavy conjecture from 1975. The common theme behind the various directions of this project is to upgrade various theorems, conjectures, and constructions involving numerical invariants of matroids to richer structures that take symmetries of matroids into account. Broader impacts of the project include mentoring of graduate students and workshop organization. The first project will enrich our understanding of log concavity and real rootedness in the context of matroid invariants by working equivariantly. In special cases, it allows to ask new questions about the cohomology of the configuration space of points in the plane, and about Eulerian polynomials. The second project sheds light on the mysterious notion of valuativity by recasting it in categorical terms, and provides a method for compute large classes of matroid invariants equivariantly. The third project introduces the notion of the local h-polynomial of a matroid, and has the potential to make new inroads into classical questions about matroid realizability and graph colorability. The last project involves a categorification of the graph minor theorem, with applications to the homology of configuration spaces of graphs.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.