Mathematical Sciences Department
PhD Dissertation Defense
Guillermo Nunez Ponasso, PhD Candidate
Tuesday, August 1, 2023
7:00 am - 9:00 am
Zoom Meeting ID: 973 0946 3924
https://wpi.zoom.us/j/97309463924
Title: Combinatorics of Complex Maximal Determinant Matrices
Abstract: Hadamard's determinant inequality gives an upper bound
for the determinant of a complex matrix whose entries are taken from
the complex unit disk. Furthermore, a matrix meets this bound with
equality if and only if all its entries have absolute value equal to 1, and
its rows are pairwise orthogonal. Such a matrix is called an Hadamard
matrix.
Hadamard matrices exist at all orders n>0. However, if we restrict
ourselves to matrices with entries in some proper subset of the unit
circle, such as +/- 1, then Hadamard matrices with these entries may
not always exist. For example, +/- 1 Hadamard matrices can only exist
at orders n=1,2, or n a multiple of 4.
The study of Hadamard matrices is not only interesting by its intrinsic
beauty, but also because of the wide range of applications that these
matrices enjoy. For example, Hadamard matrices play an important
role in quantum information theory, coding theory, signal processing,
and the statistical theory of design of experiments. They have also
been applied in operator theory, and harmonic analysis.
In this talk we will present our results on maximal determinants for
matrices with entries in the complex m-th roots of unity. Some of the
results we present include non-existence results for designs and
Hadamard matrices using quadratic forms and algebraic number
theory, a study of the asymptotic existence of BH(12p,p) matrices, and
an account of results of maximal determinant matrices over the third
and fourth roots of unity. Additionally, we discuss an application of
finite geometry to privacy in database communications.
Dissertation Committee:
Padraig Ó Catháin, Dublin City University (Advisor)
William J. Martin, WPI
Gabor Sárközy, WPI
Adam Wagner, WPI
John Bamberg, University of Western Australia
Ada Chan, York University