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I am interested in finite combinatoric structures, especially difference sets in abelian groups. The set {0,1,3,9} is a difference set in the additive group Z13, since the differences between the elements of this set give exactly one copy of the nonzero elements of Z13. Difference sets with classical parameters can be constructed using finite fields and correspond to finite Desarguesian geometries. There are many difficult but natural open questions in this area. I have been investigating induced substructures in subgroups, when the group contains a difference set with classical parameters.
Difference sets correspond to sequences which have special correlation properties, and these correlation properties make them useful in signal transmission, for example.Other interests, motivated by colleagues' interests here in Galway include problems in linear algebra, specifically the study of spaces of non-nilpotent matrices of maximal dimension over finite fields. Another exciting theme on which my colleagues work is the application of linear algebra to problems in homotopy theory, specifically concerning the design of algorithms which describe the shape of curves.
My Ph.D. thesis was on difference sets with classical parameters in abelian groups, which was written in UCD, Dublin under the supervision of Professor Rod Gow in 2007.On substructures of abelian difference sets with classical parameters. Journal of Combinatorial Designs 16 (2008), 182-190.
A preprint on difference sets with parameters of 3-dimensional projective geometries is here on the arxiv.:
Together with Dr Catherine Paolucci from the School of Education, we were responsible for the successful implementation and direction of the BA in Mathematics and Education between 2009 and 2013. Our first cohort of 12 teachers graduated in 2012 and the second cohort of 24 graduated in 2013. This programme attracts very talented students and they undergo a very thorough training in Mathematics and Applied Mathematics together with a teaching qualification and innovative courses in Mathematics Pedagogy and Education. Further details are on the course webpage or may be obtained from Dr Máire Ní Ríordáin (Education) or Dr Aisling McCluskey (Mathematics).