Jay Schweig

Projective dimension, graph domination parameters, and independence complex homology

We construct several pairwise-incomparable bounds on the projective dimensions of edge ideals. Our bounds use combinatorial properties of the associated graphs. In particular, we draw heavily from the topic of dominating sets. Through Hochster@?s Formula, we recover and strengthen existing results on the homological connectivity of graph independence complexes.

Bounding the projective dimension of a squarefree monomial ideal via domination in clutters

We introduce the concept of edgewise domination in clutters, and use it to provide an upper bound for the projective dimension of any squarefree monomial ideal. We then compare this bound to a bound given by Faltings. Finally, we study a family of clutters associated to graphs and compute domination parameters for certain classes of these clutters.

Catalan numbers, binary trees, and pointed pseudotriangulations

Abstract We study connections among structures in commutative algebra, combinatorics, and discrete geometry, introducing an array of numbers, called Borel’s triangle, that arises in counting objects in each area. By defining natural combinatorial bijections between the sets, we prove that Borel’s triangle counts the Betti numbers of certain Borel-fixed ideals, the number of binary trees on a fixed number of vertices with a fixed number of “marked” leaves or branching nodes, and the number of pointed pseudotriangulations of a certain class of planar point configurations.

Generalizing the Borel property

We introduce the notion of Q-Borel ideals: ideals which are closed under the Borel moves arising from a poset Q. We study decompositions and homological properties of these ideals, and offer evidence that they interpolate between Borel ideals and arbitrary monomial ideals.

Balanced Nontransitive Dice

Summary We study triples of labeled dice in which the relation “is a better die than” is nontransitive. Focusing on such triples with an additional symmetry we call balance, we prove that such triples of dice exist for all dice having at least three faces.We then examine the sums of the labels of such dice and use these results to construct an algorithm for verifying whether or not a triple of dice is balanced and nontransitive. We also consider generalizations to larger sets of dice and other related ideas.

A Convex-Ear Decomposition for Rank-Selected Subposets of Supersolvable Lattices

Let

A Survey of Stanley–Reisner Theory

L

A broad class of shellable lattices

be a supersolvable lattice with nonzero Mobius function. We show that the order complex of any rank-selected subposet of

Boij-Söderberg and Veronese decompositions

L

Bounds on the regularity and projective dimension of ideals associated to graphs

admits a convex-ear decomposition. This proves many new inequalities for the h-vectors of such complexes, and shows that their g-vectors are M-vectors.