His work is broad and has theoretical and experimental flavours. He develops the theory of charged signed directed graphs, called digraphs for short. He outlines how these can be used to compute Mahler measures, an interesting number theoretic object which, loosely speaking, measures how big the zeros of a polynomial are. He uses this theory to then find “small” Mahler measures values, for both one and two-variable polynomials, where he finds over 97% of these known values, including a new one (a result which was published jointly with his supervisor, James McKee). 

The thesis ends by initiating a discussion on when polynomials share the same Mahler measure, an area which is still in its infancy. He hopes that the preliminary results he has here can spawn a deeper and richer understanding of the area in the future!

The viva was examined by Jozef Skokan (LSE) and Rainer Dietmann (RHUL), and chaired by Simon Blackburn (RHUL).

The pandemic didn’t stop Josh presenting his work: he spoke at almost a dozen conferences during his 3 years, including at last year’s Young Researchers in Mathematics conference, where he was awarded a prize for a poster presentation. He was an active member of the Maths and Information Security community, and the department is very grateful for his numerous contributions to outreach events, such as presenting at Exploring Maths.