DOCTORAL DEFENSE

Speaker:  Benjamin Reese

Title:  Geometry of pipe dream complexes

DOCTORAL DEFENSE

Speaker:  Benjamin Reese

Title:  Geometry of pipe dream complexes

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.