Current research
Mathematics Education and Data Analysis:
Working with Patrick Browne of Limerick Institute of Technology on a SATLE-funded mathematics education/data analysis project to lead second-level teachers in implementing an online system to assist with students' transition from second- to third-level study.
Previous research Computer Science Education and Machine Learning:
Led a summer research group with second- and third-level students as a case study in using game implementation as a method
of teaching introductory computer science (programming, data analysis, web services and machine learning).
Combinatorics:
Coauthored a survey paper on the Hadamard Maximal Determinant Problem. In 1893, Jacques Hadamard posed the problem of investigating the
maximal determinant of matrix with entries 1 or -1. Our survey paper provided proofs of the results obtained thus far, in standardised
language, and made an improvement on one of the bounds contained within.
Compressed Sensing:
As part of the Theoretical and Computational Optimization Research Group in the University of Birmingham, I worked on compressed sensing -- branch of signal processing. This area has applications in signal recovery and denoising, image reconstruction, statistical estimation, and financial engineering, amongst others. More specifically, I worked on the construction and manipulation of deterministic and structured compressed sensing matrices and algorithms for efficient recovery of compressed data.
Computational Homology:
My doctoral thesis focused on developing some theory and software for efficiently computing the homology of Euclidean data sets in up to four dimensions. This software was an extension of work done by Graham Ellis as part of HAP, a homological algebra library for use with the GAP computer algebra system. This extension exploited certain mathematical advantages of using permutahedrally-based structures, rather than traditional simplicial or cubical ones, to speed up and expand the scope of certain implementations of homological image computations. Further details are available here or see my thesis defence slides or this blitz presentation for some toy examples.
Publications
Harnessing the Webwork platform for student engagement in mathematics and analysing its data, with Patrick Browne, published in the TUS Compendium of Active Learning and Assessment for Student Engagement, 2022.
A Survey of the Hadamard Maximal Determinant Problem with Patrick Browne, Ronan Egan, and Padraig Ó Cathá published in the Electronic Journal of Combinatorics, 2021.
Sparsification of Matrices and Compressed Sensing with Padraig Ó Catháin and Yunbin Zhao, published in the Irish Mathematical Society Bulletin, 2018.
Computational homotopy of finite regular CW-spaces with Graham Ellis, published in the Journal of Homotopy and Related Structures, 2013.
Computational homology of cubical and permutahedral complexes, PhD thesis supervised by Graham Ellis, awarded Feb 2013.
And for nostalgia...
What is the Riemann Hypothesis? final year project, supervised by Götz Pfeiffer, 2008.
Software and Data
Sparsification in Compressed Sensing: A small library of the data used to generate graphs and tables in my paper with Ó Catháin and Zhao, along with other relevant simulations and the Matlab algorithms used are available here .
HAP_Permutahedral - The GAP package for homological analysis created as part of my doctoral thesis, has been incorporated into Graham Ellis' HAP, but if you would like a copy of the original code, documentation, and tutorials, please let me know.