classical Fourier analysis. Green and Tao used higher order Fourier analysis to present a new method for counting the number of solutions to simultaneous equations in certain sets of integers, including in the primes. This generalises the classical approach using Hardy–Littlewood circle method. Many aspects of this theory, including the quantitative aspects of the inverse theorem for the Gowers norms, are still the subject of ongoing research. Green has also collaborated with Emmanuel Breuillard on topics in group theory. In particular, jointly with Terence Tao, they proved a structure theorem for approximate groups, generalising the Freiman-Ruzsa theorem on sets of integers with small doubling. Green also has work, joint with Kevin Ford and Sean Eberhard, on the theory of the symmetric group, in particular on what proportion of its elements fix a set of size . Green and Tao also have a paper on algebraic combinatorial geometry, resolving the Dirac-Motzkin conjecture (see Sylvester–Gallai theorem). In particular they prove that, given any collection of points in the plane that are not all collinear, if is large enough then there must exist at least lines in the plane containing exactly two of the points. Kevin Ford, Ben Green, Sergei Konyagin, James Maynard and Terence Tao, initially in two separate research groups and then in combination, improved the lower bound for the size of the longest gap between two consecutive primes of size at most . The form of the previously best-known bound, essentially due to Rankin, had not been improved for 76 years. More recently Green has considered questions in arithmetic Ramsey theory. Together with Tom Sanders he proved that, if a sufficiently large finite field of prime order is coloured with a fixed number of colours, then the field has elements such that all have the same colour. Green has also been involved with the new developments of Croot-Lev-Pach-Ellenberg-Gijswijt on applying the polynomial method to bound the size of subsets of a finite vector space without solutions to linear equations. He adapted these methods to prove, in function fields, a strong version of Sárközy's theorem. Awards and honours Green has been a Fellow of the Royal Society since 2010, and a Fellow of the American Mathematical Society since 2012. Green was chosen by the German Mathematical Society to deliver a Gauss Lectureship in 2013. He has received several awards: 2004: Clay Research Award 2005: Salem Prize 2005: Whitehead Prize 2007: SASTRA