Categorical Invariants of Matroids

拟阵是对线性依赖概念的组合抽象进行编码的对象。这个名字来源于单词Matrix：给定一个矩阵A，有一个相关的拟阵，它记录了A的哪些列子集是线性相关的信息。并不是所有的拟阵都是以这种方式产生的，但那些确实形成了一大类拟阵的拟阵，通常为拟阵提供了良好的直觉。在过去的十年里，关于拟阵的研究工作出现了爆炸性的增长，这在很大程度上要归功于琼·胡的工作，他被授予了2022年菲尔兹奖。Hh使用代数几何的技术来解决多年来关于拟阵的猜想，包括1976年的威尔士对数凹性猜想，以及1975年的道林和威尔逊的顶重猜想。这个项目不同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、猜想和构造升级到考虑拟阵的对称性的更丰富的结构。该项目的更广泛影响包括对研究生的指导和研讨会的组织。第一个项目将通过等价地工作来丰富我们对拟阵不变量上下文中的对数凹性和实根性的理解。在特殊情况下，它允许提出关于平面上点的配置空间的上同调以及关于欧拉多项式的新问题。第二个项目通过用范畴术语重新塑造值的神秘概念来阐明它，并提供了一种等量计算大类拟阵不变量的方法。第三个项目引入了拟阵的局部h-多项式的概念，并有可能在关于拟阵的可实现性和图的可染性的经典问题上取得新的进展。最后一个项目涉及图的次要定理的分类，并应用于图的配置空间的同调。这一奖项反映了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。

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.