Kazhdan-Lusztig Theory of Matroids

线性相关的概念是线性代数的基础，而线性代数又是所有数学分支的核心。拟阵理论就是这个概念的抽象。虽然线性代数提供了许多丰富的拟阵例子，但我们可以证明，几乎所有的拟阵例子都不能通过线性代数来实现。这个项目的主要目的是证明关于可实现拟阵的两个主要定理实际上对所有拟阵都成立。此外，该项目将为研究生提供研究培训机会。这些定理中的第一个是Dowling和Wilson的头重猜想（Top-Heavy Conjecture），它指出，给定一个矩阵和一个自然数k最多是秩的一半，那么秩k的平面数小于或等于秩k的平面数。第二个定理说，矩阵的Kazhdan- Lusztig多项式的系数都是非负的。在过去的五年中，通过将Hodge理论应用于与实现相关的某代数变量的交上同调群，证明了这两种说法。这个项目的目标是表明，即使不可能构造一个一般矩阵的类似代数变量，人们仍然可以定义它的交上同群的类似物，并证明它们有足够好的性质来暗示这两个结果。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。

The notion of linear dependence is fundamental to linear algebra, which in turn lies at the core of all branches of mathematics. The theory of matroids is an abstraction of this notion. While linear algebra provides many rich examples of matroids, one can prove that almost all examples of matroids are not realizable via linear algebra. The main purpose of this project to show that two major theorems about realizable matroids in fact hold for all matroids. In addition the project will provide research training opportunities for graduate students.The first of these theorems is the Top-Heavy Conjecture of Dowling and Wilson, which states that, given a matroid along with a natural number k that is at most half the rank, then the number of flats of rank k is less than or equal to the number of flats of corank k. The second says that the coefficients of the Kazhdan--Lusztig polynomial of a matroid are all non-negative. Both of these statements were proved for realizable matroids in the past five years by applying Hodge theory to the intersection cohomology groups of a certain algebraic variety associated with the realization. The goal of this project is to show that, even though it is impossible to construct an analogous algebraic variety for a general matroid, one can still define analogues of its intersection cohomology groups and prove that they have enough nice properties to imply these two results.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.