Discrete Mathematics and Combinatorics Commons^™

Deletion-Induced Triangulations, Clifford T. Taylor

Theses and Dissertations--Mathematics

Let d > 0 be a fixed integer and let A ⊆ ℝ^d be a collection of n ≥ d + 2 points which we lift into ℝ^d+1. Further let k be an integer satisfying 0 ≤ k ≤ n-(d+2) and assign to each k-subset of the points of A a (regular) triangulation obtained by deleting the specified k-subset and projecting down the lower hull of the convex hull of the resulting lifting. Next, for each triangulation we form the characteristic vector defined by Gelfand, Kapranov, and Zelevinsky by assigning to each …

Polyhedral Problems In Combinatorial Convex Geometry, Liam Solus

Theses and Dissertations--Mathematics

In this dissertation, we exhibit two instances of polyhedra in combinatorial convex geometry. The first instance arises in the context of Ehrhart theory, and the polyhedra are the central objects of study. The second instance arises in algebraic statistics, and the polyhedra act as a conduit through which we study a nonpolyhedral problem.

In the first case, we examine combinatorial and algebraic properties of the Ehrhart h*-polynomial of the r-stable (n,k)-hypersimplices. These are a family of polytopes which form a nested chain of subpolytopes within the (n,k)-hypersimplex. We show that a well-studied unimodular triangulation of the (n,k)-hypersimplex restricts to a …

Unimodality Questions In Ehrhart Theory, Robert Davis

Theses and Dissertations--Mathematics

An interesting open problem in Ehrhart theory is to classify those lattice polytopes having a unimodal h^*-vector.Although various sufficient conditions have been found, necessary conditions remain a challenge. Highly-structured polytopes, such as the polytope of real doubly-stochastic matrices, have been proven to possess unimodal h^*-vectors, but the same is unknown even for small variations of it.

In this dissertation, we mainly consider two particular classes of polytopes: reflexive simplices and the polytope of symmetric real doubly-stochastic matrices. For the first class, we discuss an operation that preserves reflexivity, integral closure, and unimodality of the h^{* …}

Boij-Söderberg Decompositions, Cellular Resolutions, And Polytopes, Stephen Sturgeon

Theses and Dissertations--Mathematics

Boij-Söderberg theory shows that the Betti table of a graded module can be written as a linear combination of pure diagrams with integer coefficients. In chapter 2 using Ferrers hypergraphs and simplicial polytopes, we provide interpretations of these coefficients for ideals with a d-linear resolution, their quotient rings, and for Gorenstein rings whose resolution has essentially at most two linear strands. We also establish a structural result on the decomposition in the case of quasi-Gorenstein modules. These results are published in the Journal of Algebra, see [25].

In chapter 3 we provide some further results about Boij-Söderberg decompositions. We …