My research interests are generally in the area of algebra, especially linear algebra and its interactions with group theory, combinatorics, and field theory. I have a particular interest in linear and affine spaces of matrices that have special properties relating to rank. I am also interested in the ordinary and projective representation theory of finite groups and in group rings. My PhD thesis, completed in 2000 at the University of Alberta, was concerned with the classification of irreducible projective representations of finite groups over fields, using the group ring of a *generic central extension*.

I am also interested in mathematics education research at university

level.

I am currently supervising the research of two PhD students at the University of

Galway, Badriah Safarji and Dana Saleh.

My former PhD students are Kirsten Pfeiffer (PhD 2011), James McTigue (PhD 2015), Olga O’Mahony (PhD 2018), Hieu Ha Van (PhD 2019), and Cian O’Brien (PhD 2020).

##### Selected Publications and Preprints

- C. O’Brien and R. Quinlan, Alternating sign matrices of finite multiplicative order,
*Linear Algebra and its Applications*, Vol 651, pages 332–358 (2022) - R. Quinlan, M. Shau and F. Szechtman, Linear Diophantine equations in several variables,
*Linear Algebra and its Applications*, Vol 640, pages 67–90 (2022) - C. O’Brien, K. Jennings and R. Quinlan, Alternating signed bipartite graphs and difference-1 colourings,
*Linear Algebra and its Applications*, Vol 604, pages 370–398 (2020) - O. O’Mahony and R. Quinlan, Exponent-critical primitive graphs and the Kronecker product,
*Electronic Journal of Graph Theory and its Applications*, Vol 7, no. 2, 329–347, 2019. - Hieu Ha Van and Rachel Quinlan, Almost-nonsingular entry pattern matrices,
*Linear Algebra and its Applications*, Vol 578, 334–355, 2019. - J. Cruickshank, R. Quinlan and F. Szechtman, Hermitian and skew hermitian forms over local rings,
*Linear Algebra and its Applications*, Vol 551, 147–161, 2018. - O. O’Mahony and R.Quinlan, ,
*Linear Algebra and its Applications*, Vol 525, 1–19, 2017. - HIeu Ha Van and Rachel Quinlan, On the maximum rank of completions of entry pattern matrices
*, Linear Algebra and its Applications*, Vol 542, 66–83, 2018. - K. Pfeiffer and R. Quinlan, Proof-evaluation as a step towards proof authorship, in Beyond Lecture, Mathematical Association of America, 2016.
- J. McTigue and R. Quinlan, Partial matrices of constant rank,
*Linear Algebra and its Applications*, 2014. - J. Cruickshank, A. Herman, R. Quinlan, F. Szechtman, Unitary groups over local rings,
*Journal of Algebra and its Applications*, Vol. 13, Issue 02, 2014. - J. McTigue and R. Quinlan, Partial matrices whose completions all have the same rank,
*Linear Algebra and its Applications*, Vol 438, no. 1, pages 348–360 (2013). - K. Pfeiffer and R. Quinlan, A proof evaluation exercise in an elementary linear algebra course, Proceedings of the Fifth National Conference on Research in Mathematics Education, St Patrick’s College Drumcondra, 2013.
- J. McTigue and R. Quinlan, Partial matrices whose completions have ranks bounded below,
*Linear Algebra and its Applications*, Vol 435, no. 9, pages 2259-2271 (2011). - R. Quinlan, Spaces of matrices without non-zero eigenvalues in their field of definition, and a question of Szechtman,
*Linear Algebra and its Applications*, Vol. 434, no. 6, pages 1580-1587 (2011) - R. Quinlan, Acts of knowing in university mathematics curricula, Proceedings of the Fourth National Conference on Research in Mathematics Education, St Patrick’s College Drumcondra, 2011.
- R. Quinlan, ‘Guessing at our mental models’ or ‘committed knowing acts’ – what are our students doing?,
*Bulletin of the Irish Mathematical Society*, no. 66 (2010) - R. Quinlan, What makes mathematics attractive at university?, Proceedings of the Third National Conference on Research in Mathematics Education, St Patrick’s College Drumcondra, 2009.
- R. Gow and R. Quinlan, Galois extensions and subspaces of alternating bilinear forms with special rank properties,
*Linear Algebra and its Applications*, Vol. 430, Issues 8-9, 2009. - R. Gow and R. Quinlan, Galois theory and linear algebra,
*Linear Algebra and its Applications*, Vol. 430, Issue 7, 2009. - R. Gow and R. Quinlan, On the vanishing of subspaces of alternating bilinear forms,
*Linear and Multilinear Algebra*, Vol. 54, no. 6, 2006 - R. Gow and R. Quinlan, Covering groups of rank 1 of elementary abelian groups,
*Communications in Algebra*, Vol 34, no. 4, 2006. abstract - R. Gow and R. Quinlan, On central difference sets in certain non-abelian 2-groups,
*J. Combin. Theory Ser. A*, Vol 113, no. 4, 2006. - R. Quinlan, Real elements and real-valued characters of covering groups of elementary abelian 2-groups,
*Journal of Algebra*, Vol 275, no. 1, 2004. abstract - R. Quinlan, Twisted group rings of metacyclic groups,
*Representation Theory*, Vol. 7, 2003 - R. Quinlan, Generic central extensions and projective representations of finite groups,
*Representation Theory*, Vol. 5, 2001. - R. Quinlan, Irreducible Projective Representations of Finite Groups, PhD thesis, University of Alberta, 2000.