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# J.P. McCarthy Profile

## Current and Recent Teaching Interests

- Civil Engineering (MATH7019, MATH7021)
- Mechanical Engineering (MATH7016)
- Biomedical & Building Energy Systems (MATH6015, MATH6040)
- Industrial Measurement Control (MATH6037, MATH6038)
- Computing (MATH6000, STAT6000)
- Professional Diploma in Mathematics for Teachers (MB5003, MB5014, MB5021)
- Mathematical Studies in University College Cork (MS2001, MS2002, MS3011)

## 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 shuﬄes 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, S_52. 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-oﬀ 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 (Mathematics, 2009 - 2010): Masters in Mathematics by research 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 (2004 - 2008; 2:1 awarded broken into maths (1:1) & physics (2:2)).

Other

- I have a webpage at jpmccarthymaths.com.
- I am a keen user of Math.StackExchange, MathOverflow and MathEducators.StackExchange.