He studied at local schools in Bristol, Bishop Road Primary School and Fairfield Grammar School, competing in the International Mathematical Olympiad in 1994 and 1995. He entered Trinity College, University of Cambridge in 1995 and completed his B.A. in mathematics in 1998, winning the Senior Wrangler title.
He is the Waynflete Professor of Pure at the University of Oxford. He earned his doctorate under English mathematician Timothy Gowers in 2003, with a thesis entitled Topics in arithmetic combinatorics. He was a research Fellow at Trinity College, Cambridge between 2001 and 2005, before becoming a Professor of at the University of Bristol from January 2005 to September 2006 and then the first Herchel Smith Professor of Pure at the University of Cambridge from September 2006 to August 2013.
He became the Waynflete Professor of Pure at the University of Oxford on 1 Aug 2013. He was also a Research Fellow of the Clay Institute and held various positions at institutes such as Princeton University, University of British Columbia, and Massachusetts Institute of Technology. Green has published several results in both combinatorics and number theory.
These include improving the estimate by Jean Bourgain of the size of arithmetic progressions in sumsets, as well as a proof of the Cameron–Erdős conjecture on sum-free sets of natural numbers. His work in demonstrating that every set of primes of positive relative upper density contains an arithmetic progression of length three then led to his 2004 work with mathematician Terence Tao now known as the Green–Tao theorem. This theorem showed that for all n there exist infinitely many arithmetic progressions of length n in the prime numbers.
American Mathematical Society. Royal Society.