# Dr Matthew Lettington

Senior Lecturer

Director of Admissions

- lettingtonmc@cardiff.ac.uk
- +44 (0)29 2087 5670
- 2.17, Abacws, Senghennydd Road, Cathays, Cardiff, CF24 4AG

- Media commentator
- Available for postgraduate supervision

## Overview

Following the award of my PhD (Number Theory) in 2010, I joined Cardiff University as a Lecturer, obtaining a tenured position as an SBP Fellow in May 2012.

My main research interests lie on the interface of Number Theory, Combinatorics and Linear Algebra. In particular I am interested in generating discrete sets (possibly infinite) with combinatorial and number theoretic properties, via elements of superalgebras under matrix multiplication.

In addition to my research I am also interested in mathematics education. Recently I appeared on The One Show, with the ground-breaking story "It Doesn't Add Up" concerning errors in GCSE revision guides. This story also appeared on the front page of The Times (24/01/2017) and in a number of other newspapers and media outlets, leading to radio and TV interviews. Since then I have acted as a consultant to the secondary education sector regarding the new GCSE and A level programmes.

On Friday 15 November 2019 I appeared on the Radio 4 Today Programme live to read out a riddle that I had composed encoding the history of Cardiff Univeristy in the style of the epigram of Diophantus of Alexandria circa 300 AD. More information is available here BBC Radio 4 - Today - Today Puzzle #612

### Administrative duties

Undergraduate Project Co-Ordinator (2012-2020)

School of Mathematics Newsletter Editor (2013-Present)

Year 3 Tutor (2015_Present)

Director for Mathematics in Media Communication (2020-Present)

## Biography

### Honours and awards

SBP Fellowship awarded May 2012

### Professional memberships

Memeber of the London Mathematical Society (LMS)

### Academic positions

- June 2006 - Aug 2010: Mathematical Consultant, JSC Management und-Technologieberatung AG, Eltville, Germany
- Oct. 2001 - April 2010: Lecturer in Mathematics, Gloucestershire College, Cheltenham.

### Speaking engagements

- August 2008. A dynamic approach to product sales forecasting. SMiCR, Cardiff.
March 2012. Recent progress in the distribution of lattice points near to convex surfaces and polytopes. Analysis Seminar Programme, Cardiff.

May 2013. Recent progress in Bernoulli relations, their implications for the Ramanujan polynomials and the special values of the Riemann zeta function, Wales Mathematics Colloquium, Gregynog.

- February 2014. On rational convergents to the cosine of rational multiples of Pi, Analysis Seminar Programme, Cardiff.
- May 2014. Higher-Dimensional Lucas and Fibonacci Sequences, Wales Mathematics Colloquium, Gregynog.
- March 2015, On Higher Dimensional Interlacing Fibonacci Sequences, Continued Fractions and Chebyshev Polynomials (Abstract), LMS, WIMCS – Bath Analysis Day, Swansea University.
- July 2015, On Higher Dimensional Interlacing Fibonacci Sequences, Continued Fractions and Chebyshev Polynomials (Abstract), 29th Journées Arithmétiques JA 2015, Debrecen University, Hungary.
- Friday, September 4 2015, 3.00pm, Chauvenet Hall, Room 143, Invited Talk, Applied Mathematics and Statistics Colloquium, Colorado School of Mines, On Higher Dimensional Interlacing Fibonacci Sequences. (Abstract).
- May 2016. On Block Representations and Spectral Properties of Semimagic Square Matrices, Wales Mathematics Colloquium, Gregynog.
- October 14 2016, Modern Aspects of Analysis and Scientific Computing (A joint Cardiff University — KU Leuven meeting), Plenary talk entitled
*On polynomial expressions for higher-dimensional Fibonacci sequences, Faulhaber sums, and some theorems of Fermat*. - January 24 2017, Television interview with BBC Wales Today and Radio interview live on Good Morning Wales regarding the ground-breaking story screened by BBCs The One Show, and appearing on the front page of The Times newspaper, entitled “It Doesn’t Add Up.
- March 21 2017, Tinted Lens – Chapter Arts Centre, Cardiff, Invited Talk entitled
*Concrete Mathematics in relation to the Theodore Melfi film The Hidden Figures*. - July 3 2017, On Binomial Polynomials Mimicking Riemann's Zeta Function, 30th Journées Arithmétiques JA 2017, Caen University, France.
- May 2018. On Sum-and-Distance Systems, Reversible Square Matrices and Divisor Functions, Wales Mathematics Colloquium, Gregynog.
- June 2018. On Sum-and-Distance Systems, Reversible Square Matrices and Divisor Functions, Bohemian Matrices and Applications, June 20-22, 2018, Manchester University. http://www.maths.manchester.ac.uk/~higham/conferences/bohemian/slides_lettington.pdf
- October 2018. On Sum-and-Distance Systems, Reversible Square Matrices and Divisor Functions, Invited Colloquium Talk, The Open University, Milton Keynes, UK, http://mathematics.open.ac.uk/events/sum-and-distance-systems-reversible-square-matrices-and-divisor-functions
- November 2018. On Sum-and-Distance Systems, Reversible Square Matrices and Divisor Functions, Invited Seminar Talk, University of Essex, Colchester, UK,https://www.essex.ac.uk/events/2018/11/15/department-of-mathematical-sciences-seminar
- July 2019. On Reversible Square Matrices and Divisor Functions, Invited Minisymposium Talk (MS A6-2-2 9) on Bohemian Matrices, organised by Nick higham and Rob Corless, SIAM 2019, Valencia, Spain.
- January 2020, Invited Colloquium Talk, University of South Wales, Divisor Functions and Sum systems, https://maths.research.southwales.ac.uk/upcoming-seminars/matthew-lettington-combinatorics/
- February 2020, Invited Pure Maths Seminar Talk, Reading University, Divisor Functions and the Number of Sum Systems https://janivirtanen.wordpress.com/pure-mathematics-seminar-at-reading/
- Friday 23rd July 2021, Bohemain Matrices Minisymposium invited talk entitled
*Recent Progress in the Rational Factorisation of Integer Matrices*, SIAM AN21, Spokane, Washington SIAM AN21 Minisymposium on Bohemian Matrices and Applications – Nick Higham (nhigham.com) See also Conference Program Online (siam.org)

### ESGI 80 Collaboration

April 2011. Loading on the electricity grid from distributed energy storage systems.

### Educational and media outreach

- In 2015 I advised and participated in an episode for the Discovery channel series entitled Man V Expert, featuring the celebrity Alexis Conran, and the Oxford Number Theorist Professor Minhyong Kim.
- In late 2016 I put together a team of PhD students and staff from Cardiff School of Mathematics to investigate errors in GCSE Maths Book on behalf of the BBCs
*The One Show*. The results made the front page of The Times Newspaper and were screened on the BBCs The One Show on January 24 2017. They story was also covered in many other national newspapers including The Herald, The Independent, The Daily Mirror and The Daily Express. On the same day I appeared on both Television (BBC Wales Today) and Radio (Good Morning Wales) to discuss our findings. - In December 2018, in conjunction with Admiral and Brand Content, I wrote the mathematics underpinning the hugely successful Admiral Bauble Calculator. This asks for details of your Christmas tree and preferences, from which it calculates the number of baubles and length of fairly lights required. The Admiral Bauble Calculator is available here Bauble Calculator - Admiral
- On Friday 15 November 2019 I appeared on the Radio 4 Today Programme live to read out a riddle that I had composed encoding the history of Cardiff Univeristy in the style of the epigram of Diophantus of Alexandria circa 300 AD. The riddle reads as follows:
`Cardiff University’, the wonder behold,

In November 2018, the riddle tells how old:

Infancy lasted the first full nine years,

One-ninth of life later, a building appears.

After three-fifths more, I merged my abode;

And another one-fifteenth, full powers bestowed.

Twenty-one years on and I'm now flying high,

At the heart of Wales capital, how old am I?

The solution can be found at https://www.bbc.co.uk/programmes/articles/26hdyfBgB2PSWs3kMBp0kJ2/today-puzzle-612

### Committees and reviewing

Year Three Tutor

Reviewer for Zentralblatt

Referee for: Journal of Mathematical Analysis and its Applications; Moscow journal of combinatorial number theory; American Mathematical Monthly; Miskolc Mathematical Notes.

## Publications

### 2022

- Higham, N. J. and Lettington, M. C. 2022. Optimizing and factorizing the Wilson matrix. American Mathematical Monthly 129(5), pp. 454-465. (10.1080/00029890.2022.2038006)

### 2021

- Higham, N. J., Lettington, M. C. and Schmidt, K. M. 2021. Integer matrix factorisations, superalgebras and the quadratic form obstruction. Linear Algebra and its Applications 622, pp. 250-267. (10.1016/j.laa.2021.03.028)

### 2020

- Lettington, M. C. and Schmidt, K. M. 2020. Divisor functions and the number of sum systems. Integers 20, article number: A61.
- Hill, S. L., Huxley, M. N., Lettington, M. C. and Schmidt, K. M. 2020. Some properties and applications of non-trivial divisor functions. Ramanujan Journal 51 (10.1007/s11139-018-0093-9)
- Coffey, M. W. and Lettington, M. C. 2020. Binomial polynomials mimicking Riemann's zeta function. Integral Transforms and Special Functions 31(11), pp. 856-872. (10.1080/10652469.2020.1755672)

### 2019

- Huxley, M. N., Lettington, M. C. and Schmidt, K. M. 2019. On the structure of additive systems of integers. Periodica Mathematica Hungarica 78, pp. 178-199. (10.1007/s10998-018-00275-w)

### 2018

- Hill, S., Lettington, M. C. and Schmidt, K. M. 2018. Block representations and spectral properties of constant sum matrices. Electronic Journal of Linear Algebra 34, pp. 170-190. (10.13001/1081-3810.3530)

### 2017

- Lettington, M. C., Schmidt, K. M. and Hill, S. 2017. On superalgebras of matrices with symmetry properties. Linear and Multilinear Algebra 66(8), pp. 1538-1563. (10.1080/03081087.2017.1363153)
- Coffey, M. W., Hindmarsh, J. L., Lettington, M. C. and Pryce, J. D. 2017. On higher-dimensional Fibonacci numbers, Chebyshev polynomials and sequences of vector convergents. Journal de Theorie des Nombres de Bordeaux 29(2), pp. 369-423. (10.5802/jtnb.985)
- Coffey, M. W. and Lettington, M. C. 2017. Binomial polynomials mimicking Riemann's Zeta Function. [Online]. arXiv. Available at: https://arxiv.org/abs/1703.09251

### 2016

- Lettington, M. and Coffey, M. W. 2016. On Fibonacci polynomial expressions for sums of m-th powers, their implications for Faulhaber's formula and some theorems of fermat. [Online]. arXiv. Available at: https://arxiv.org/abs/1510.05402

### 2015

- Coffey, M. W. and Lettington, M. C. 2015. Mellin transforms with only critical zeros: Legendre functions. Journal of Number Theory 148, pp. 507-536. (10.1016/j.jnt.2014.07.021)

### 2014

- Brunnock, R., Lettington, M. C. and Schmidt, K. M. 2014. On square roots and norms of matrices with symmetry properties. Linear Algebra and its Applications 459, pp. 175-207. (10.1016/j.laa.2014.06.054)

### 2013

- Lettington, M. C. 2013. A trio of Bernoulli relations, their implications for the Ramanujan polynomials and the special values of the Riemann zeta function. Acta Arithmetica 158(1), pp. 1-31. (10.4064/aa158-1-1)

### 2011

- Lettington, M. C. 2011. Fleck's congruence, associated magic squares and a zeta identity. Functiones et Approximatio, Commentarii Mathematici 45(2), pp. 165-205. (10.7169/facm/1323705813)

### 2010

- Lettington, M. C. 2010. Integer points close to convex hypersurfaces. Acta Arithmetica 141(1), pp. 73-101. (10.4064/aa141-1-4)

### 2009

- Lettington, M. C. 2009. Integer points close to convex surfaces. Acta Arithmetica 138(1), pp. 1-23. (10.4064/aa138-1-1)

### 2008

- Lettington, M. C. 2008. Topics related to the theory of numbers: integer points close to convex hypersurfaces, associated magic squares and a zeta identity. PhD Thesis, Cardiff University.

## Teaching

My current teaching duties include:

- MA4011 Analytic Number Theory
- MA2011 Introduction to Number Theory I

Previously I have taught:

- MA3004 Combinatorics
- MA3011 Introduction to Number Theory II
- MA0216 Elementary Number Theory II
- MA2002 Matrix Algebra
- MA0111 Elementary Number Theory I
- MA1000 Calculus
- MA0104 Modelling (with Difference and Differential Equations)
- MA0003 Preliminary Mathematics

My main research interests lie on the interface of Number Theory, Combinatorics and Linear Algebra. In particular I am interested in generating discrete sets (possibly infinite) with combinatorial and number theoretic properties, via elements of superalgebras under matrix multiplication.

More generally I am interested in:

- Combinatorial and number theoretic properties of matrices with symmetry properties under matrix multiplication.
- Congruence properties of rational sequences generated by orthogonal polynomial systems.
- Integer points close to convex hypersurfaces and polytopes.
- Identities for the Zeta function at integer values.

MathSciNet Overview: https://mathscinet.ams.org/mathscinet/search/author.html?mrauthid=877964

ORCID Number: https://orcid.org/0000-0001-9327-143X

## Supervision

## Project Descriptions

The student will learn about ideas from number theory and mathematical analysis. The project will involve an element of numerical analysis via Mathematica.

### Current supervision

## Ambrose Law

Research student

## Grant Jones

Research student

### Past projects

Successful previous PhD students:

- 2017 Dilbak Mohammed (second supervisor)
- 2018 Sally L. Hill (first supervisor)
- 2021 David M. Humphreys (first supervisor)