Categorical Invariants of Matroids

矩阵是一个对象，它编码了线性相关概念的组合抽象。这个名字来源于矩阵这个词：给定一个矩阵a，有一个相关的矩阵，它记录了a的列的子集是线性相关的信息。并不是所有的拟阵都以这种方式出现，但那些以这种方式出现的拟阵形成了一个大的类，为一般的拟阵提供了很好的直觉。在过去的十年里，对类人机器人的研究出现了爆炸式的增长，这主要归功于June Huh的工作，他在2022年获得了菲尔兹奖。Huh使用了代数几何的技术来解决已经开放多年的关于拟阵的猜想，包括1976年的Welsh的对数凹性猜想，以及1975年的Dowling和Wilson的顶重猜想。本项目各个方向背后的共同主题是将涉及拟阵数值不变量的各种定理、猜想和构造升级为考虑拟阵对称性的更丰富的结构。该项目更广泛的影响包括指导研究生和组织研讨会。第一个项目将通过等量工作丰富我们对矩阵不变量背景下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位形空间的上同调和关于欧拉多项式的新问题。第二个项目通过在分类术语中重新定义它，揭示了价值性的神秘概念，并提供了一种等效计算大类矩阵不变量的方法。第三个项目引入了矩阵的局部h多项式的概念，并有可能对矩阵可实现性和图可着色性等经典问题做出新的突破。最后一个项目涉及图小定理的分类，并应用于图的组态空间的同调。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.