A lecturer at the Cork Institute of Technology. A recent cv, (February 2019) may be found here.

**Current and Recent Teaching Interests**

– Mechanical Engineering (MATH7016)

– Civil & Environmental Engineering (MATH7019, MATH7021)

– Biomedical Engineering & Sustainable Engineering (MATH6015, MATH6040)

– Industrial Measurement Control (MATH6037, MATH6038)

– Computing (MATH6055, MATH6000, STAT6000)

– Professional Diploma in Mathematics for Teachers (MB5003, MB5014, MB5021)

– Mathematical Studies in University College Cork (MS2001, MS2002, MS3011)

**Publications**

*Diaconis–Shahshahani Upper Bound Lemma for Finite Quantum Groups,*Journal of Fourier Analysis and Applications, doi: 10.1007/s00041-019-09670-4 (earlier preprint available here)

**Research Interests**

– Random Walks on Finite Quantum Groups *In my current research, the results and ideas contained in my MSc thesis *The Cut-off Phenomenon in Finite Groups* are greatly extended to the case of finite quantum groups. Given a group, an algebra of complex-valued functions may be defined on the group and this forms a commutative C*-algebra. This C*-algebra inherits a natural structure from the group axioms. There exist non-commutative C*-algebras that have this natural structure and although there is no longer an underlying space it is natural to call such an algebra (of functions on) a non-commutative or quantum group. In this setting the C*-algebra of functions on a finite group is called a commutative or classical quantum group.*

*The notion of a random walk on a finite group is well studied and this idea can be suitably extended or quantised to the case of quantum groups. Most of the techniques that utilise the underlying space used in the analysis of classical random walks are no longer useful for the analysis of quantum random walks as there is no longer an underlying space to exploit, but many techniques that use just the algebra of functions are.*

*One such technique that can possibly be adapted from the classical case to the quantum setting is that of Diaconis-Fourier Theory. This quantum Diaconis-Fourier Theory would be used to produce qualitative bounds on how long it takes a quantum random walk to get random. It is the aim of this work to apply this theory to quantum random walks.*

** **– Random Walks on Finite Groups *How many shu**ﬄ**es are needed to mix up a deck of cards? This question may be **answered in the language of a random walk on the symmetric group, . This generalises neatly to the study of random walks on ﬁnite groups — themselves a special class of Markov chains. Ergodic random walks exhibit nice limiting behaviour, and both the quantitative and qualitative aspects of the convergence to this limiting behaviour is examined. A particular qualitative behaviour — the cut-o**ﬀ** phenomenon — occurs in many examples. For random walks exhibiting this behaviour, after a period of time, convergence to the limiting behaviour is abrupt.*

**Education**

- PhD, Mathematics, 2017, with Dr. Stephen Wills (UCC). The research was in Quantum Groups and the thesis title was
*Random Walks on Finite Quantum Groups — Diaconis-Shahshahani Theory for Quantum Groups*. - MSc by Research, Mathematics, 2010, with Dr. Stephen Wills (UCC). The research was in Random Walks on Finite Groups and the thesis title was
*The Cut-Off Phenomenon in Random Walks on Finite Groups*. - BSc, Joint Honours Maths & Physics, 2008, 2:1 awarded broken into maths (1:1) & physics (2:2).

**Other **

– I am a keen user of Math.StackExchange, MathOverflow and MathEducators.StackExchange.

** **

## 4 comments

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January 25, 2011 at 10:09 pm

jim kellyhttp://www.projectmaths.com/index.php/news-comments/

Check out the clarifications on algebra

January 25, 2011 at 10:13 pm

J.P. McCarthyJim,

These clarifications refer to Paper 1. I was wondering about Hypothesis Testing (i.e. Paper 2) – it is suspended from the syllabus. The normal distribution is still there as far as I can see but the person I was helping had never seen the tables for the normal distribution.

Regards,

J.P.

May 25, 2013 at 9:24 pm

Harold GreenHi J.P. Thank you for discovering my blog http://www.throughharoldslens.com. It gave me the opportunity to discover, explore and Follow your blog. Hope we both enjoy our journeys. Best, Harold

July 5, 2013 at 10:55 am

toorajHi, I saw your math page it was very useful , specially the c*-algebras &…(13). thanks for you.