Title: Kronecker's Theory of Binary Bilinear Forms with Applications to Representations of Integers as Sums of Three Squares

Abstract: In 1883 Leopold Kronecker published a paper containing “a few explanatory remarks” to an earlier paper of his from 1866. His work loosely connected the theory of integral binary bilinear forms to the theory of integral binary quadratic forms. In this defense we shall discover the key statements within Kronecker's paper and offer insight into new, detailed arithmetic proofs. Further, I will present some additional results on the proper and complete class numbers for bilinear forms, before demonstrating their use in rigorously developing the connection between binary bilinear forms and binary quadratic forms. We conclude by giving an application of this material to the number of representations of an integer as a sum of three squares and show the resulting formula is equivalent to the well-known result due to Gauss.

Title: On Skew-Constacyclic Codes

Abstract: Cyclic codes are a well-known class of linear block codes with efficient decoding algorithms. In recent years they have been generalized to skew-constacyclic codes; such a generalization has previously been shown to be useful. After a brief introduction of skew-polynomial rings and their quotient modules, which we use to study skew-constacyclic codes algebraically, we motivate and define a notion of idempotent elements in these quotient modules. We are particularly concerned with the existence and uniqueness of idempotents that generate a given submodule; as such, we generalize relevant results from previous work on skew-constacyclic codes by Gao/Shen/Fu in 2013 and well-known results from the classical case.

Title:  New Perspectives of Quantum Analogues

Abstract:  In this talk we show the classical q-binomial can be expressed more compactly as a pair of statistics on a subset of 01-permutations via major index, an instance of the cyclic sieving phenomenon related to unitary spaces is also given. We then generalize this idea to q-Stirling numbers of the second kind using restricted growth words. The resulting expressions are polynomials in q and 1 + q. We extend this enumerative result via a decomposition of a new poset whose rank generating function is the q-Stirling number Sq[n,k] which we call the Stirling poset of the second kind. This poset supports an algebraic complex and a basis for integer homology is determined. This is another instance of Hersh, Shareshian and Stanton's homological version of the Stembridge q = -1 phenomenon. A parallel enumerative, poset theoretic and homological study for the q-Stirling numbers of the first kind is done beginning with de Médicis and Leroux's rook placement formulation. Time permitting, we will indicate a bijective argument à la Viennot showing the (q,t)-Stirling numbers of the first and second kind are orthogonal.

Title: A Linkage Constructions for Subspace Codes

Abstract: In this thesis defense, we will begin by giving an overview of random network coding and how subspace codes are used in this context. In this talk I will focus on the linkage construction, which builds a code by linking previously constructed codes. We will explore the properties of codes created by this construction. In particular, we will explore how to utilize the linkage construction to create partial spread codes. Finally we will look at cases in which linkage codes are efficiently decodable.

Title: Dissertation Defense

Abstract: We study a class of determinantal ideals called skew tableau ideals, which are generated by (t x t) minors in a subset of a symmetric matrix of indeterminates.  The initial ideals have been studied in the (2 x 2) case by Corso, Nagel, Petrovic and Yuen.  Using liaison techniques, we have extended their results to include the original determinantal ideals in the (2 x 2) case, and obtained some partial results in the (t x t) case.  A critical tool we use is an elementary biliaison, and producing these requires some technical determinantal calculations.  We have uncovered in error a previous determinantal lemma that was applied in several papers, and have used the straightening law for minors of a matrix to establish a new determinantal relation.  This new tool is quite versatile; it fixes the gaps in the previous papers and provides the main computational power in several of our own arguments.  This is joint work with Uwe Nagel.

Title: Combinatorics of the Descent Set Polynomial and the Diamond Product

Abstract: In this talk, we will first examine the descent set polynomial, which was defined by Chebikin, Ehrenborg, Pylyavskyy, and Readdy in terms of the descent set statistics of the symmetric group.  We will explain why large classes of cyclotomic polynomials are factors of the descent set polynomial, focusing on instances of the 2pth cyclotomic polynomial for a prime p.  Next, the diamond product of two Eulerian posets will be discussed, particularly the effect this product has on their cd-indices.  A combinatorial interpretation involving weighted lattice paths will be introduced to describe the outcome of applying the diamond product operator to two cd-monomials.

Title: Unimodality Questions in Ehrhart Theory

Abstract: An interesting open problem in Ehrhart theory is to classify the lattice polytopes having a unimodal h*-vector. Although various sufficient conditions have been found, necessary conditions remain a challenge. Some 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 talk, we will mainly examine the h*-vectors for two particular classes of polytopes, with special attention given to methods for proving unimodality.

Title:  Material tensors and pseydotensors of weakly-textured polycrystals with orientation measure defined on the orthogonal group

Abstract:  Material properties of polycrystalline aggregates should manifest the influence of crystallographic texture as defined by the orientation distribution function (ODF). A representation theorem on material tensors of weakly-textured polycrystals was established by Man and Huang (2012), by which a given material tensor can be expressed as a linear combination of an orthonormal set of irreducible basis tensors, with the components given explicitly in terms of texture coefficients and a number of undetermined material parameters. Man and Huang's theorem is based on the classical assumption in texture analysis that ODFs are defined on the rotation group SO(3), which strictly speaking makes it applicable only to polycrystals with (single) crystal symmetry defined by a proper point group. In the present study we consider ODFs defined on the orthogonal group O(3) and extend the representation theorem of Man and Huang to cover  pseudotensors and polycrystals with crystal symmetry defined by any improper point group. This extension  is important because many materials, including common metals such as aluminum, copper, iron, have their group of crystal symmetry being an improper point group. We present the restrictions on texture coefficients imposed by crystal symmetry for all the 21 improper point groups and we illustrate the extended representation theorem by its application to elasticity.

Title:  Homogeneous Gorenstein Ideals and Boij Söderberg Decompositions

Abstract:  This talk consists of two parts. Part one revolves around a construction for homogeneous Gorenstein ideals and properties of these ideals. Part two focuses on the behavior of the Boij-Söderberg decomposition of lex ideals. Gorenstein ideals are known for their nice duality properties. For codimension two and three, the structures of Gorenstein ideals have been established by Hilbert-Burch and Buchsbaum-Eisenbud, respectively. However, although some important results have been found about Gorenstein ideals of higher codimension,  there is no structure theorem proven for higher codimension cases. Kustin and Miller showed how to construct a Gorenstein ideals in local Gorenstein rings starting from smaller such ideals. We discuss a modification of their construction in the case of graded rings. In a Noetherian ring, for a given two homogeneous Gorenstein ideals, we construct another homogeneous Gorenstein ideal and so we describe the resulting ideal in terms of the initial homogeneous Gorenstein ideals. Gorenstein liaison theory plays a central role in this construction. For the second part, we talk about Boij-Söderberg theory which is a very recent theory. It arose from two conjectures given by Boij and Söderberg and their proof by Eisenbud and Schreyer.

It establishes a unique decomposition for Betti diagram of graded modules over polynomial rings. We focus on Betti diagrams of lex ideals which are the ideals having the largest Betti numbers among the ideals with the same Hilbert function. We describe Boij-Söderberg decomposition of a lex ideal in terms of Boij-Söderberg decomposition of some related lex ideals.

Title:  A Characterization of Serre Classes of Reflexive Modules Over a Complete Local Noetherian Ring

Abstract:  Serre classes of modules over a ring $R$ are important because they describe relationships between certain classes of modules and sets of ideals of $R$. In this talk we characterize the Serre classes of three different types of modules. First we characterize all Serre classes of noetherian modules over a commutative noetherian ring. By relating noetherian modules to artinian modules via Matlis duality, we characterize the Serre classes of artinian modules. When $R$ is complete local and noetherian, define $E$ as the injective envelope of the residue field of $R$. Then denote $M^\nu=Hom_R(M,E)$ as the dual of $M$. A module $M$ is reflexive if the natural evaluation map from $M$ to $M^{\nu\nu}$ is an isomorphism.  The main result provides a characterization of the Serre classes of reflexive modules over such a ring. This characterization depends on an ability to ``construct'' reflexive modules from noetherian modules and artinian modules. We find that Serre classes of reflexive modules over a complete local noetherian ring are in one-to-one correspondence with pairs of collections of prime ideals which are closed under specialization.