Rees Products of Posets and Inequalities

In this dissertation we will look at properties of two different posets from different perspectives. The first poset is the Rees product of the face lattice of the n-cube with the chain. Specifically we study the Möbius function of this poset. Our proof techniques include straightforward enumeration and a bijection between a set of labeled augmented skew diagrams and barred signed permutations which label the maximal chains of this poset. Because the Rees product of this poset is Cohen-Macaulay, we find a basis for the top homology group and a representation of the top homology group over the symmetric group both indexed by the set of labeled augmented skew diagrams. We also show that the Möbius function of the Rees product of a graded poset with the t-ary tree and the Rees product of its dual with the t-ary tree coincide. We discuss labelings for Rees and Segre products in general, particularly the Rees product of the face lattice of a polytope with the chain. We also look at cases where the Möbius function of a poset is equal to the permanent of a matrix and we consider local h-vectors for the barycentric subdivision of the n-cube. In each section we state open conjectures. The second poset in this dissertation is the Dowling lattice. In particular we look at the k = 1 case, that is, the partition lattice. We study inequalities on the flag vector of the partition lattice via a weighted boustrophedon transform and determine a more generalized version for the Dowling lattice. We generalize a determinantal formula of Niven and conclude with conjectures and avenues of study

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

Introductory mathematics and statistics through sports: supplementary activities and writing projects

Introductory Mathematics and Statistics through Sports uses sport as a tool to help students get to grips with mathematics and statistics, placing great emphasis on communication, application, and internalization of mathematics

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

A Professional Development Framework for the Flipped Classroom Model: Design and Implementation of a Literacy and Math Integrated Professional Development Initiative

Experience”(ADISRE), a component of a five-year (2018–2023) National Institutes of Health (NIH) grant led by principal investigator Karla-Sue Marriott, has focused on developing close reading and critical thinking skills for cohorts of seven freshmen at Savannah State University (SSU), an HBCU. Marriott, who I met through a Governor’s Teaching Fellows Program, currently serves as interim chair of the Chemistry and Forensic Science Department at SSU. I work at another local university—Georgia Southern University—in the College of Education and was invited to collaborate on this NIH grant by presenting workshops at several points throughout the year to students and monitoring their progres