Kazhdan-Lusztig Theory of Matroids

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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.