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Kazhdan-Lusztig Theory of Matroids

Kazhdan-Lusztig 拟阵理论

基本信息

批准号：
1954050
负责人：
Nicholas Proudfoot
金额：
$ 20万
依托单位：
University of Oregon Eugene
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-07-01 至 2024-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1954050&HistoricalAwards=false
关键词：
Kazhdan Lusztig Theory Matroids

项目摘要

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.

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。拟阵理论是这一概念的抽象。虽然线性代数提供了许多拟阵的丰富例子，但可以证明几乎所有拟阵的例子都不能通过线性代数实现。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，该项目还将为研究生提供研究培训的机会。这些定理中的第一个是道林和威尔逊的头重脚轻猜想，它指出，给定一个拟阵沿着的自然数k是最多一半的秩，那么单位的秩k的数量小于或等于单位的corank k的数量。第二种是拟阵的Kazhdan-Lusztig多项式的系数都是非负的。这两种说法都证明了可实现拟阵在过去的五年中应用霍奇理论的交叉上同调群的某一代数簇的实现。这个项目的目标是表明，即使不可能为一般拟阵构造类似的代数簇，人们仍然可以定义它的交上同调群的类似物，并证明它们具有足够好的性质来暗示这两个结果。这个奖项反映了美国国家科学基金会的法定使命，并被认为是值得通过利用基金会的知识价值和更广泛的影响审查进行评估来支持的的搜索.