About Research Teaching Other
My research is in the field of algebraic and geometric combinatorics, and I am especially interested in posets, matroids, Borel ideals, and graph theory. Click here for my curriculum vitæ, and here for my Google Scholar profile.
Fix a field k. When Δ is a simplicial complex on n vertices with Stanley-Reisner ideal I_Δ, we define and study an invariant called the type defect of Δ. Except when Δ is a single simplex, the type defect of Δ, td(Δ), is the difference b_c(Δ)-c, where c is the codimension of Δ. We show that this invariant admits surprisingly nice properties. For example, it is well-behaved when one glues two complexes together along a face. Furthermore, Δ is Cohen-Macaulay if td(Δ) ≤0. On the other hand, if Δ is a simple graph (viewed as a one-dimensional complex), then td(Δ')≥0 for every induced subgraph Δ' of Δ if and only if Δ is chordal. Requiring connected induced subgraphs to have type defect zero allows us to define a class of graphs that we call treeish, and which we generalize to higher dimensions, proving sharp lower bounds for most Betti numbers of ideals with linear resolution, and classifying when equalities occur. As an application, we prove sharp lower bounds for Betti numbers of graded ideals (not necessarily monomial) with linear resolutions.
We give a complete classification of free and non-free multiplicities on the A_3 braid arrangement. Namely, we show that all free multiplicities on A_3 fall into two families that have been identified by Abe-Terao-Wakefield (2007) and Abe-Nuida-Numata (2009). The main tool is a new homological obstruction to freeness derived via a connection to multivariate spline theory.
Let M and N be two monomials of the same degree, and let I be the smallest Borel ideal containing M and N. We show that the toric ring of I is Koszul by constructing a quadratic Gröbner basis for the associated toric ideal. Our proofs use the construction of graphs corresponding to fibers of the toric map. As a consequence, we conclude that the Rees algebra is also Koszul.
We introduce a new class of lattices, the modernistic lattices, and their duals, the comodernistic lattices. We show that every modernistic or comodernistic lattice has shellable order complex. We go on to exhibit a large number of examples of (co)modernistic lattices. We show comodernism for two main families of lattices that were not previously known to be shellable: the order congruence lattices of finite posets, and a weighted generalization of the k-equal partition lattices. We also exhibit many examples of (co)modernistic lattices that were already known to be shellable. To start with, the definition of modernistic is a common weakening of the definitions of semimodular and supersolvable. We thus obtain a unified proof that lattices in these classes are shellable. Subgroup lattices of solvable groups form another family of comodernistic lattices that were already proven to be shellable. We show not only that subgroup lattices of solvable groups are comodernistic, but that solvability of a group is equivalent to the comodernistic property on its subgroup lattice. Indeed, the definition of comodernistic requires, on every interval, a lattice-theoretic analogue of the composition series in a solvable group. Thus, the relation between comodernistic lattices and solvable groups resembles, in several respects, that between supersolvable lattices and supersolvable groups.
We study triples of labeled dice in which the relation ''is a better die than'' is non-transitive. Focusing on such triples with an additional symmetry we call ''balanced,'' we prove that such triples of n-sided dice exist for all n > 2. We then examine the sums of the labels of such dice, and use these results to construct an O(n^2) algorithm for verifying whether or not a triple of n-sided dice is balanced and non-transitive. Finally, we consider generalizations to larger sets of dice.
Boij-Söderberg theory has had a dramatic impact on commutative algebra. We determine explicit Boij-Söderberg coefficients for ideals with linear resolutions and illustrate how these arise from the usual Eliahou-Kervaire computations for Borel ideals. In addition, we explore a new numerical decomposition for resolutions based on a row-by-row approach; here, the coefficients of the Betti diagrams are not necessarily positive. Finally, we demonstrate how the Boij-Söderberg decomposition of an arbitrary homogeneous ideal with a pure resolution changes when multiplying the ideal by a homogeneous polynomial.
We characterize the lcm lattices that support a monomial ideal with a pure resolution. Given such a lattice, we provide a construction that yields a monomial ideal with that lcm lattice and whose minimal free resolution is pure.
We study the relationship between the projective dimension of a squarefree monomial ideal and the domination parameters of the associated graph or clutter. In particular, we show that the projective dimensions of graphs with perfect dominating sets can be calculated combinatorially. We also generalize the well-known graph domination parameter τ to clutters, obtaining bounds on the projective dimension analogous to those for graphs. Through Hochster's Formula, our bounds on projective dimension also give rise to bounds on the homologies of the associated Stanley-Reisner complexes.
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.
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.
We survey the Stanley-Reisner correspondence in combinatorial commutative algebra, describing fundamental applications involving Alexander duality, associated primes, f- and h-vectors, and Betti numbers of monomial ideals.
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.
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.
In this paper we give new upper bounds on the regularity of edge ideals whose resolutions are k-steps linear; surprisingly, the bounds are logarithmic in the number of variables. We also give various bounds for the projective dimension of such ideals, generalizing other recent results. By Alexander duality, our results also apply to unmixed square-free monomial ideals of codimension two. We also discuss and connect these results to more classical topics in commutative algebra.
White has conjectured that the toric ideal of a matroid is generated by quadric binomials corresponding to symmetric basis exchanges. We prove a stronger version of this conjecture for lattice path polymatroids by constructing a monomial order under which these sets of quadrics form Gröbner bases. We then introduce a larger class of polymatroids for which an analogous theorem holds. Finally, we obtain the same result for lattice path matroids as a corollary.
We prove a theorem allowing us to find convex-ear decompositions for rank- selected subposets of posets that are unions of Boolean sublattices in a coherent fashion. We then apply this theorem to geometric lattices and face posets of shellable complexes, obtaining new inequalities for their h-vectors. Finally, we use the latter decomposition to give a new interpretation to inequalities satisfied by the flag h- vectors of face posets of Cohen-Macaulay complexes.
We use the notion of Borel generators to give alternative methods for computing standard invariants, such as associated primes, Hilbert series, and Betti numbers, of Borel ideals. Because there are generally few Borel generators relative to ordinary generators, this enables one to do manual computations much more easily. Moreover, this perspective allows us to find new connections to combinatorics involving Catalan numbers and their generalizations. We conclude with a surprising result relating the Betti numbers of certain principal Borel ideals to the number of pointed pseudo-triangulations of particular planar point sets.
Stanley has conjectured that the h-vector of a matroid complex is a pure M- vector. We prove a strengthening of this conjecture for lattice path matroids by constructing a corresponding family of discrete polymatroids.
Let L be a supersolvable lattice with nonzero Möbius function. We show that the order complex of any rank-selected subposet of L 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.
We investigate properties of rim-finite subsets of the plane (those which have topological bases whose elements have finite boundaries), which are also arc-free.
This is my PhD thesis, so most of the results here are contained in the papers above. By reading the results in this thesis, however, you get the additional benefit of several typos and errors.