DOCTORAL DEFENSE

Speaker:  Benjamin Reese

Title:  Geometry of pipe dream complexes

Title:  GLOBAL WELL-POSEDNESS FOR THE DERIVATIVE NONLINEAR SCHR¨ODINGER EQUATION THROUGH INVERSE SCATTERING

Abstract:  We study the Cauchy problem of the derivative nonlinear Schro¨dinger equation in one space dimension. Using the method of inverse scattering, we prove global well-posedness of the derivative nonlinear Schro¨dinger equation for initial conditions in a dense and open subset of weighted Sobolev space that can support bright solitons.

Title: Artin's Conjecture for additive forms.

Abstract: We prove a special case of a conjecture of Emil Artin; namely, we prove that if K is an unramified extension of Q_p where p is an odd prime, and if f is an additive form of degree d and dimension d^2+1, then f is K-isotropic.

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.