Publications, preprints and formal writings

Publications

$ \zeta(\{ \, \{2\}^m, 1, \{2\}^m, 3 \}^n, \{2\}^m) / \pi^{4n + 2m(2n+1)} $ is rational. In: Journal of Number Theory 148 (2015), pp. 463-477. arXiv: 1306.6775 [math.NT]. doi: 10.1016/j.jnt.2014.09.028
With H. Bachmann: Generalized Jacobi-Trudi determinants and evaluations of Schur multiple zeta values. In: European Journal of Combinatorics 87 (2020), pp. 103-133. arXiv: 1908.05061 [math.NT]. doi: 10.1016/j.ejc.2020.103133
An analogue of cyclic insertion for mutiple zeta star values. In: Kyushu Journal of Mathematics 74 (2020), pp. 337-352. arXiv: 1806.10053 [math.NT] doi:10.2206/kyushujm.74.337
The alternating block decomposition of iterated integrals, and cyclic insertion on multiple zeta values. In: The Quarterly Journal of Mathematics, Volume 72, Issue 3, September 2021, Pages 975–1028 arXiv: 1703.03784 [math.NT]. doi:10.1093/qmath/haaa056
With H. Gangl and D. Radchenko: On functional equations for Nielsen polylogarithms. In: Communications in Number Theory and Physics, Vol. 15, No. 2 (2021), pp. 363-454. arXiv: 1908.04770 [math.NT]. doi:10.4310/CNTP.2021.v15.n2.a4
With C. Duhr and H. Gangl: Clean single-valued polylogarithms. In: SIGMA 17 (2021), 107, 34 pages, Special Issue on Algebraic Structures in Perturbative Quantum Field Theory in honor of Dirk Kreimer for his 60th birthday. arXiv: 2104.04344 [math.NT]. doi:10.3842/SIGMA.2021.107
With H. Gangl, L. Lai, C. Xu, J. Zhao: On two conjectures of Sun concerning Apéry-like series. In: Forum Mathematicum, Volume 35 Issue 6 (2023), pp. 1533-1547. arXiv:2210.14704 [math.NT] doi:10.1515/forum-2022-0325
With H. Gangl and D. Radchenko: Functional equations of polygonal type for multiple polylogarithms in weights 5, 6 and 7. In: Pure and Applied Mathematics Quarterly, Vol. 19, No. 1 (2023), pp. 85-93. arXiv: 2012.09840 [math.NT]. doi: 10.4310/PAMQ.2023.v19.n1.a5
On the evaluation of the alternating multiple $ t $ value $ t(\{\overline{1}\}^a, 1, \{\overline{1}\}^b) $. In: The Ramanujan Journal (online first). arXiv: 2112.15349 [math.NT]. doi:10.1007/s11139-023-00788-0
(with H. Gangl, D. Radchenko, and D. Rudenko) On the Goncharov depth conjecture and polylogarithms of depth two. In: Selecta Mathematica New Series, Vol. 30, #27 (2024) arXiv:2210.11938 [math.NT]. doi:10.1007/s00029-024-00918-6
(With A. Keilthy) Evaluation of the multiple zeta values $\zeta(2,\ldots,2,4,2,\ldots,2) $ and period polynomial relations. In: Forum of Mathematics, Sigma, Vol 12. #e46 (2024) arXiv:2210.03616 [math.NT] doi:10.1017/fms.2024.16

Preprints

A review of Dan's reduction procedure for multiple polylogarithms, 42 pages, arXiv: 1703.03961 [math.NT]. Submitted.
- Also: Mathematics worksheets which verify these results.
With H. Gangl and D. Radchenko: Explicit formlas for Grassmannian polylogarithms, 28 pages. arXiv: 1909.13869 [math.NT]. Submitted
Third order coefficient of the area expansion via $ \zeta(3) $. Appendix A in L. Heller, S. Heller, M. Traizet: Complete families of embedded high genus CMC surfaces in the 3-sphere 2108.10214 [math.DG].
On motivic multiple $ t $ values, Saha's basis conjecture, and generators of alternating MZV's, 53 pages. arXiv: 2112.14613 [math.NT]
(With M. E. Hoffman) Symmetry results for multiple $ t $-values, 36 pages. arXiv: 2204.14183 [math.NT]
(with K. C. Au, M. E. Hoffman) Differential operators and a depth reduction for the alternating multiple zeta values $ \zeta(1,\ldots,1,\overline{2m}) $. 12 pages. arXiv:2312.17148 [math.NT]
On the evaluations of multiple $S$ and $T$ values of the form $S(\overset{{}_{(-)}}{2}, 1, \ldots, 1, \overset{{}_{(-)}}{1})$ and $T(\overset{{}_{(-)}}{2}, 1, \ldots, 1, \overset{{}_{(-)}}{1})$, (answers to questions of Xu, Yan, and Zhao). 22 pages. arXiv:2403.04727 [math.NT]
(with K. C. Au) Creative telescoping and generating functions of (variants of) multiple zeta values. 20 pages. arXiv:2404.16199 [math.NT]
Symmetries of weight 6 multiple polylogarithms and Goncharov's Depth Conjecture. 70 pages. arXiv:2405.13853 [math.NT]
(with L. Heller, S. Heller, M. Traizet) Minimal surfaces and alternating multiple zetas. 85 pages. arXiv:2407.07130 [math.DG, math.NT]
(with A. Matveiakin, D. Radchenko, D. Rudenko) The Hopf algebra of multiple polylogarithms . 23 pages. arXiv:2411.15071 [math.AG]
(with L. Mauth, A. Medvedovsky) On the parity of coefficients of eta powers. 40 pages. arXiv:22411.17638 [math.NT]
(with A. Medvedovsky, P. Moree) Euler-Kronecker constants of modular forms: beyond Dirichlet $L$-series. 42 pages. arXiv:2412.01803 [math.NT]
Explicit linear dependence congruence relations for the partition function modulo 4. 11 pages. arXiv:2412.17459 [math.NT]

Thesis

The final version of my thesis is available in the Durham e-Theses repository. Alternatively a version of my thesis is hosted here (this version has the correct page numbering, and PDF bookmarks).

On my thesis page, you can also view mathematica worksheets containing various supporting calculations.

Fourth Year Project

My fourth year project was on Primes of the form $ x^2 + ny^2 $, supervised by Dr Jens Funke. The project involved understanding how Class Field Theory can be used to derive conditions describing which primes can be represented by a given quadratic forms. See more details about my project, including the written work.

I was invited to present a poster for the project at the Durham Rising Stars 2012 Symposia.

The project was entered into the 2012 SET awards, where it made it through to the final three shortlist for mathematics, but ultimately did not win.

Talks and other writings

You may want to view a complete list of talks I have given. The talks page includes abstracts, notes, slides and additional content where available.

Or you may want to view a list of my other writings (reports, notes, musings, rough ideas,…) during my PhD, and MMath.

Did you know that $ 12^2 + 33^2 = 1233 $?