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Inequalities for the h and flag hvectors of geometric lattices
 DISC. AND COMP. GEOM
, 2003
"... We prove that the order complex of a geometric lattice has a convex ear decomposition. As a consequence, if ∆(L) is the order complex of a rank (r+1) geometric lattice L, then the for all i ≤ r/2 the hvector of ∆(L) satisfies, hi−1 ≤ hi and hi ≤ hr−i. We also obtain several inequalities for the f ..."
Abstract

Cited by 6 (2 self)
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We prove that the order complex of a geometric lattice has a convex ear decomposition. As a consequence, if ∆(L) is the order complex of a rank (r+1) geometric lattice L, then the for all i ≤ r/2 the hvector of ∆(L) satisfies, hi−1 ≤ hi and hi ≤ hr−i. We also obtain several inequalities for the flag hvector of ∆(L) by analyzing the weak Bruhat order of the symmetric group. As an application, we obtain a zonotopal cdanalogue of the DowlingWilson characterization of geometric lattices which minimize Whitney numbers of the second kind. In addition, we are able to give a combinatorial flag hvector proof of hi−1 ≤ hi when i ≤ 2 5 (r + 7 2).
COMPLEMENTS AND HIGHER RESONANCE VARIETIES OF HYPERPLANE ARRANGEMENTS
"... Abstract. Hyperplane arrangements form the geometric counterpart of combinatorial objects such as matroids. The shape of the sequence of Betti numbers of the complement of a hyperplane arrangement is of particular interest in combinatorics, where they are known, up to a sign, as Whitney numbers of t ..."
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Abstract. Hyperplane arrangements form the geometric counterpart of combinatorial objects such as matroids. The shape of the sequence of Betti numbers of the complement of a hyperplane arrangement is of particular interest in combinatorics, where they are known, up to a sign, as Whitney numbers of the first kind, and appear as the coefficients of chromatic, or characteristic, polynomials. We show that certain combinations, some nonlinear, of these Betti numbers satisfy Schur positivity. At the same time, we study the higher degree resonance varieties of the arrangement. We draw some consequences, using homological algebra results and vector bundles techniques, of the fact that all resonance varieties are determinantal. 1.
Research Statement
"... My research is in the area of algebraic combinatorics. I find the most captivating aspect of combinatorics is the approachability and elegance of its questions. In combinatorics one finds both open problems which are accessible to undergraduate students, and simply stated problems which prove to hav ..."
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My research is in the area of algebraic combinatorics. I find the most captivating aspect of combinatorics is the approachability and elegance of its questions. In combinatorics one finds both open problems which are accessible to undergraduate students, and simply stated problems which prove to have deep and sophisticated solutions. In addition, combinatorial problems are integral to many other disciplines; for example my research deals with enumerative questions related to geometry and algebra. Suppose there are n points in the plane. Three natural questions one might ask are: * How many lines do the points determine? * How many triangular shaped regions are formed by the determined lines? * What is the maximum number of lines through a point? All of the information about how an arrangement of points and lines breaks up space is encoded by the flag f and flag hvectors, which are combinatorial invariants of a partially ordered set associated to the arrangement. While these vectors are easily defined, they have eluded complete characterization. For example it in not known how to calculate the number of lines which can be determined by n points. My work establishes inequalities which the flag f and h vectors satisfy.