Mathematical Sciences Department
PhD Dissertation Proposal
Guillermo Nunez Ponasso, Mathematical Sciences PhD Candidate
Friday, February 17, 2023
8:00 am
SH 203 and Zoom Meeting ID: 942 7502 9251
Title: Combinatorics of Complex Maximal Determinant Matrices
Abstract: We consider the question of determining the maximal value of the determinant of a matrix with entries taken from a subset of the complex unit circle. In 1893 Jacques Hadamard gave an absolute determinantal upper bound that holds for all complex matrices with bounded entries. In his honour, matrices that achieve this bound are called Hadamard matrices. These matrices have a surprising amount of mathematical depth, as well as a broad range of applications in fields such as statistics, coding theory, knot theory, or quantum information theory.
If the subset of entries that we consider is the entire unit circle, then an Hadamard matrix exists at every order. However when this subset is more restricted, it is not always possible to find Hadamard matrices, and the calculation of the maximal determinant may require large amounts of computation and the use of arithmetic obstructions to prune the search space.
Some of the results we will present include new lower bounds for the determinant of matrices over the third roots of unity using generalised Paley matrices, an overview of our results on maximal determinants on certain matrix algebras, and a discussion on the solvability of Grammian equations.
Committee Members:
John Bamberg, University of Western Australia
Ada Chan, York University
Padraig Ó Catháin, Dublin City University & WPI
William J. Martin, WPI
Gabor Sárközy, WPI
Adam Wagner, WPI