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RESEARCH STATEMENT
"... Broadly speaking, my research interests run the breadth of combinatorics, from weighted graph packing to preference aggregation for the Major League Baseball draft, with stops at the combinatorial calculation of algebraic invariants (such as Stanley depth), and the relationships between graph expans ..."
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Broadly speaking, my research interests run the breadth of combinatorics, from weighted graph packing to preference aggregation for the Major League Baseball draft, with stops at the combinatorial calculation of algebraic invariants (such as Stanley depth), and the relationships between graph expansion and selfish traffic routing. By inclination, and as a result of my training in the interdisciplinary Algorithms, Combinatorics, and Optimization program1, my research interests span a large number of combinatorial topics and involve a variety of different collaborators. Going forward, in addition to continuing to expand the breadth of my research and collaborations, I intend to continue work on three areas in which I have had particular research success; combinatorial Stanley depth of monomial ideals, dimensionlike measures for posets, and spectral graph theory and applications. In this section I provide a less technical overview of my efforts in these areas while providing a more detailed summary in the following sections. Since it was introduced, one of the fundamental problems in the study of Stanley depth has been to develop a finite time (not necessarily efficient, just finite) means of calculating the Stanley depth of a given module. Recently, a group of commutative algebraists has made progress on this problem by providing a correspondence between the Stanley depth of monomial ideals and a class
WHEN WILL THE STANLEY DEPTH INCREASE?
, 2013
"... Let I ⊂ S = K, [x1,...,xn] be an ideal generated by squarefree monomials of degree ≥ d. If the number of degree d minimal generating monomials is μd(I) ≤ min ( ( n) ∑n−d ( 2j−1) , d+1 j=1), then the Stanley depth j sdepthS (I) ≥ d +1. ..."
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Let I ⊂ S = K, [x1,...,xn] be an ideal generated by squarefree monomials of degree ≥ d. If the number of degree d minimal generating monomials is μd(I) ≤ min ( ( n) ∑n−d ( 2j−1) , d+1 j=1), then the Stanley depth j sdepthS (I) ≥ d +1.