RESEARCH

I represented Australia at the International Mathematical Olympiad in 2005, earning a bronze medal. I did my undergraduate degree at the University of Melbourne, in Commerce and Science, majoring in mathematics, statistics, economics, and finance. Some Australian universities would offer an Honours year, which is like a taught Master's, and I did one of these in 2011. Aside from six subjects, one would write a thesis, usually expository in nature. I took this opportunity to teach myself number theory. An introduction to algebraic number theory, and the class number formula, supervised by Craig Westerland, taught me a bit about algebraic and analytic number theory, and a bit about the geometry of numbers. 

The academic year begins in February in Australia, so any Aussie planning to do a PhD in the Northern Hemisphere has to decide what to do with the eight-month gap. I stayed in Melbourne to do a research Master's with Alex Ghitza in 2012, writing a thesis on Analytic and computational approaches to comparing the Fourier coefficients of Hecke eigenforms (Papers 1, 4). The basic question is as follows: how many initial Fourier coefficients are needed to distinguish two modular forms within a family? The usual answer is the Sturm bound, but in some cases one can do better. We considered newforms, and modular forms modulo a prime, using analytic and computational methods. With regards to the former (Paper 4), we proposed a stability conjecture, and related to this we also proposed a higher-level generalisation of Maeda's conjecture. See Humphries's paper for the Maaß form case.

I was intrigued by the Goldbach conjecture, and also interested in analytic number theory, so I went all the way to Bristol to do my PhD with Trevor Wooley on the circle method (awarded September 2016). During this time I worked on a variety of problems in diophantine equations (Paper 3), diophantine inequalities (Papers 2, 5, 6, 7, 8), and arithmetic combinatorics (Paper 11). My dissertation, Shifts, averages and restriction of forms in several variables, was awarded the Faculty of Science Doctoral Prize for the Physical Sciences. The main outcomes were as follows:

• A higher-degree analogue of Roth's 3AP theorem, also valid for prime powers (Paper 11)

• A diophantine inequalities analogue of a celebrated theorem of Birch on systems of diophantine equations (Paper 8)

• The zeroes of a cubic form are evenly-distributed in an archimedean sense (Paper 7).

My study of diophantine inequalities led me quite naturally to diophantine approximation, for which York is a stronghold: I went there to do a postdoc with Sanju Velani and Victor Beresnevich. Paper 10, in which I prove a conjecture made by Jing-Jing Huang, counts rational points near planar curves, which is relevant to the metric theory of diophantine approximation. I've particularly liked exploring connections between additive/extremal combinatorics and diophantine approximation/uniform distribution (Papers 12, 13, 14). Paper 12 concerns the metric theory of multiplicative diophantine approximation, motivated by the infamous Littlewood conjecture; I was able to develop and apply the correspondence between Bohr sets and generalised arithmetic progressions in this context (the initial observation was made by Tao in a blog post).

I thoroughly enjoyed the Analytic Number Theory semester (January to May 2017) at the Mathematical Sciences Research Institute, in sunny Berkeley, where I was a postdoc. During this time I improved my Erdős number to 2 (Paper 11 with Carl Pomerance), and also worked on other collaborations there (Papers 13, 17, 18). Paper 18 characterises a Ramsey-theoretic property of diagonal equations in sufficiently many variables. One consequence is that for any finite colouring of the integers there are monochromatic Pythagorean quintuples. The paper introduces homogeneous sets within Ben Green's Fourier-analytic transference framework; Chapman has since written about this technique.

After MSRI I went back to York to continue my work on diophantine approximation. Many problems in the area can be recast in terms of the geometry of numbers (Papers 15, 16). Paper 15 generalises Paper 12 to higher dimensions. Instead of continued fractions and the three-distance theorem, we use reduced successive minima and diophantine transference inequalities to establish the correspondence between Bohr sets and generalised arithmetic progressions in this context. We essentially solved the Lebesgue theory of multiplicative diophantine approximation on planar lines in Paper 26. The divergence theory entailed demonstrating effective asymptotic equidistribution for unipotent one-parameter orbits in the space of unimodular lattices in three dimensions; for the convergence theory we used Bohr sets to estimate the relevant sums of reciprocals of fractional parts.

Then I went to Oxford on an EPSRC Fellowship in 2018. Entitled New techniques for old problems in number theory, this supports my research on the topics of diophantine approximation, arithmetic combinatorics, and enumerative Galois theory. A 1936 conjecture of van der Waerden is that irreducible polynomials with non-full Galois group are less prevalent than reducible polynomials; in Paper 17 we use diophantine equations to solve the cubic and quartic cases. Along the way, we determine the order of magnitude of the number of monic, irreducible D4 quartics with integer coefficients between -H and H. While at Oxford I was also a Junior Research Fellow at Wolfson College.

I was hired as an Assistant Professor at the University of Warwick in September 2019. Gallagher's theorem is a strong form of Littlewood's conjecture that holds for almost all pairs of reals. In Paper 25, we establish a fully-inhomogeneous version of Gallagher's theorem; my collaborator was given an award for this work. We also discover a surprising threshold for Liouville fibre refinements, and solve cases of an inhomogeneous variant of the Duffin–Schaeffer conjecture.

There was a number theory programme running virtually, hosted by the Mittag-Leffler Institute from January to April 2021, for which I was a Junior Fellow. There were three parts: Number Theory and Probability, Analytic Number Theory, and Rational Points. In Paper 22, we settle van der Waerden's 1936 conjecture on Galois groups of random polynomials, save for the alternating group and degrees 7, 8, 10.