Factorization theorems and canonical representations for generating functions of special sums

Dissertation Defense
Wednesday, July 6, 2022 - 3:00pm for 1 hour (actually 50 minutes)
Hybrid - Skiles 006 and Zoom
Maxie Dion Schmidt – Georgia Tech – maxieds@gmail.comhttps://mschmidt34.math.gatech.edu/
Maxie Schmidt
ABSTRACT: This manuscript explores many convolution (restricted summation) type sequences via certain types of matrix based factorizations that can be used to express their generating functions. These results are a main focus of the author's publications from 2017-2021. The last primary (non-appendix) section of the thesis explores the topic of how to best rigorously define a so-termed "canonically best" matrix based factorization for a given class of convolution sum sequences. The notion of a canonical factorization for the generating function of such sequences needs to match the qualitative properties we find in the factorization theorems for Lambert series generating functions (LGFs). The expected qualitatively most expressive expansion we find in the LGF case results naturally from algebraic constructions of the underlying LGF series type. We propose a precise quantitative requirement to generalize this notion in terms of optimal cross-correlation statistics for certain sequences that define the matrix based factorizations of the generating function expansions we study. We finally pose a few conjectures on the types of matrix factorizations we expect to find when we are able to attain the maximal (respectively minimal) correlation statistic for a given sum type. COMMITTEE:
  • Dr. Josephine Yu, Georgia Tech
  • Dr. Matthew Baker, Georgia Tech
  • Dr. Rafael de la Llave, Georgia Tech
  • Dr. Jayadev Athreya, University of Washington
  • Dr. Bruce Berndt, University of Illinois at Urbana-Champaign