#
Padraig Ó Catháin

Research

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Areas of interest

This page contains a brief overview of the type of problems that interest me.
Definitions of all of the objects mentioned here may be found in the preprints and slides given

here.

#### Design Theory

I am interested in all aspects of the theory of combinatorial designs, but particularly in the application of algebraic techniques to problems in design theory. It is possible to associate to any given design a group of automorphisms, which is generally considered as a permutation group on the points of the design. Typical problems are then the classification of all designs for which the automorphism group has some given property (e.g. primitivity, multiple transitivity etc.).
My research has primarily focussed on Hadamard matrices, their related 2- and 3-designs, and algebraic objects associated with these. Foremost among these algebraic objects are difference sets and relative difference sets. My Masters thesis contains a detailed study and computation enumeration of the regular subgroups of automorphism groups of the Hadamard matrices of order less than 28. My PhD thesis considers a similar classification problem for Hadamard matrices having a doubly transitive automorphism group.

#### Group theory

Many interesting problems in group theory arise from consideration of problems in design theory. I am particularly interested in finite permutation groups, both from a theoretical and a computational perspective. Theoretical topics that interest me include classification results for permutation groups of small rank, the structure theory of primitive permutation groups and the character and representation theory of permutation groups. Computational topics that interest me include algorithms for computing regular subgroups of permutation groups and algorithms for computing the full automorphism group of a given combinatorial structure.

#### Other interests

More broadly, I have interests ranging across algebra, combinatorics and related areas. These include, but are not limited to, coding theory, cryptography, elementary number theory, graph theory, the classification of finite simple groups, etc.