ALGEBRAIC PROPERTIES OF EDGE IDEALS

Given a simple graph G, the corresponding edge ideal IG is the ideal generated by the edges of G. In 2007, Ha and Van Tuyl demonstrated an inductive procedure to construct the minimal free resolution of certain classes of edge ideals. We will provide a simplified proof of this inductive method for the class of trees. Furthermore, we will provide a comprehensive description of the finely graded Betti numbers occurring in the minimal free resolution of the edge ideal of a tree. For specific subclasses of trees, we will generate more precise information including explicit formulas for the projective dimensions of the quotient rings of the edge ideals. In the second half of this thesis, we will consider the class of simple bipartite graphs known as Ferrers graphs. In particular, we will study a class of monomial ideals that arise as initial ideals of the defining ideals of the toric rings associated to Ferrers graphs. The toric rings were studied by Corso and Nagel in 2007, and by studying the initial ideals of the defining ideals of the toric rings we are able to show that in certain cases the toric rings of Ferrers graphs are level

Path ideals of rooted trees and their graded Betti numbers

Let $\Gamma$ be a rooted tree and let $t$ be a positive integer. We study algebraic invariants and properties of the path ideal generated by monomial corresponding to paths of length $(t-1)$ in $\Gamma$. In particular, we give a recursive formula to compute the graded Betti numbers, a general bound for the regularity, an explicit computation of the linear strand, and we characterize when this path ideal has a linear resolution.Comment: 18 page

Fibonacci numbers and resolutions of domino ideals

This paper considers a class of monomial ideals, called domino ideals, whose generating sets correspond to the sets of domino tilings of a $2\times n$ tableau. The multi-graded Betti numbers are shown to be in one-to-one correspondence with equivalence classes of sets of tilings. It is well-known that the number of domino tilings of a $2\times n$ tableau is given by a Fibonacci number. Using the bijection, this relationship is further expanded to show the relationship between the Fibonacci numbers and the graded Betti numbers of the corresponding domino ideal

Finite Sum Representations of Elements in R and R2

In February 2017, a number theoretic problem was posed in Mathematics Magazine by Souvik Dey, a master’s student in India. The problem asked whether it was possible to represent a real number by a finite sum of elements in an open subset of the real numbers that contained one positive and one negative number. This paper not only provides a solutionto the original problem, but proves an analogous statement for elements of R2

Finite Sum Representations of Elements in R and R2

In February 2017, a number theoretic problem was posed in Mathematics Magazine by Souvik Dey, a master’s student in India. The problem asked whether it was possible to represent a real number by a finite sum of elements in an open subset of the real numbers that contained one positive and one negative number. This paper not only provides a solutionto the original problem, but proves an analogous statement for elements of R2

A constructive approach to minimal free resolutions of path ideals of trees

For a rooted tree $\Gamma$, we consider path ideals of $\Gamma$, which are ideals that are generated by all directed paths of a fixed length in $\Gamma$. In this paper, we provide a combinatorial description of the minimal free resolution of these path ideals. In particular, we provide a class of subforests of $\Gamma$ that are in one-to-one correspondence with the multi-graded Betti numbers of the path ideal as well as providing a method for determining the projective dimension and the Castelnuovo-Mumford regularity of a given path ideal