Research
Overview
My interests are in additive combinatorics and adjacent fields, including combinatorics (especially probabilistic or extremal), number theory (especially analytic), probability and statistics, ergodic theory and topological dynamics, aspects of theoretical computer science, and analysis (especially Fourier methods).
A lot of my recent work has focussed on the area known as higher order Fourier analysis, aka the inverse theory of the Gowers norms, and in results surrounding Freiman's theorem and the polynomial Freiman–Ruzsa conjecture.
More generally I am interested in wide variety of problems in all the fields listed above.
Publications and preprints
- Sets of integers with no large sum-free subset. With Ben Green and Sean Eberhard. Annals of Mathematics. 2014; arXiv:1301.4579.
- A solution to the pyjama problem. Inventiones Mathematicae 2015. arXiv:1305.1514.
- Periodic nilsequences and inverse theorems on cyclic groups. Preprint, 2014; arXiv:1404.7742.
- Finding a low-dimensional pice of a set of integers. Int. Math. Res. Not (IMRN) 2017. arXiv:1512.06272.
- Additive triples of bijections, or the toroidal semiqueens problem. With Sean Eberhard and Rudi Mrazović. J. Eur. Math. Soc. (JEMS) 2019. arXiv:1510.05987
- The structure theory of Nilspaces I. With Yonatan Gutman and Péter Varjú. Journal d'Analyse Mathématique 2020. arXiv:1605.08945
- The structure theory of Nilspaces II: Representations as nilmanifolds. With Yonatan Gutman and Péter Varjú. Transactions of the AMS 2019. arXiv:1605.08948
- The structure theory of Nilspaces III: Inverse limit representations and topological dynamics. With Yonatan Gutman and Péter Varjú. Advances in Mathematics 2020; arXiv:1605.08950.
- Good bounds in certain systems of true complexity one. Discrete Analysis 2018. arXiv:1705.06801
- Formulations of the PFR Conjecture over Z. Math. Proc. Cambridge Philos. Soc 2019. arXiv:1709.09738
- A proof of Tomescu's graph coloring conjecture. With Jacob Fox and Xiaoyu He. J. Combin. Theory Ser. B (JCTB) 2019. arXiv:1712.06067.
- Quantitative bounds in the inverse theorem for the Gowers Us+1-norms over cyclic groups. 2018; arXiv:1811.00718.
- An asypmtotic for the Hall–Paige conjecture. With Sean Eberhard and Rudi Mrazović. Advances in Mathematics 2022. arXiv:2003.01798.
- The apparent structure of dense Sidon sets. With Sean Eberhard. 2021; arXiv:2107.05744.
- True complexity and iterated Cauchy–Schwarz. 2021; arXiv:2109.05731.
- Transversals in quasirandom latin squares. With Sean Eberhand and Rudi Mrazović. 2022; arXiv:2209.02180.