# Modern Mathematicians: 21st Century Breakthroughs

Recently I was asked what I though was the most important breakthrough in 21st century mathematics, and to be completely honest with you I couldn’t immediately point one out. Although I am familiar with mathematicians who lived in the 20th, 19th, 18th, etc. centuries, as I have read extensively about them in a range of books, I realised that it is extremely hard to engage with 21st century mathematics, primarily because it’s so advanced and daunting to read for a beginner. Therefore, I’ve decided to make a series where I talk about some breakthroughs in mathematics throughout the 21st century.

Terence Tao

Although this is not technically a specific ‘breakthrough’, in my eyes Terence Tao how been one of the most important mathematicians in the 21st century. Terence Tao is an Australian mathematician who was a recipient of a Fields Medal in 2006 and the 2014 Breakthrough Prize in Mathematics. His mathematical contributions span many fields: harmonic analysis, partial differential equations, combinatorics, ergodic Ramsey theory, random matrix theory, analytic number theory, compressed sensing, etc.

However, perhaps his best-known accomplishment is the Green-Tao theorem. This was proved by Ben Green and Tao in 2004, and stated that a sequence of prime numbers contains arbitrarily long arithmetic progressions. In other words, that for any given n, there is some m and k such that the sequence,

m, m+k, m+2k, …, m+k(n-1)

are all prime numbers. While sequences like this of any length exist, no one has found one of more than 25 primes, since the primes by then are more than 18 digits long.

Another one of his great achievements was when on May 2013, after the mathematician Yitang Zhang proved that there are infinitely many gaps of prime numbers that do not exceed 70 million, Terence Tao joined with the Polymath project, an online collaboration of volunteer mathematicians to reduce this bound. By early 2014, the figure had been reduced from 70,000,000 to a mere 246.

Terence Tao also maintains a blog where he posts all his latest research. You can visit it here.