# You, too, can collaborate with Fields Medalists!

Timothy Gowers

First some background. The Fields Medal is the most famous prize in pure mathematics.  It is given to between two and four mathematicians during the International Congress of Mathematians (ICM).  The ICM is held only once every four years (or less often if there is a world war going on), so on average less than one Fields Medal per year is given out.  Perhaps most famously, in order to be eligible for the Fields Medal you cannot be over 40 years of age in the year when the prize is awarded (so some mathematicians are now out of luck).

Terence Tao

In 1998 Timothy Gowers received the Fields Medal for his research in functional analysis and combinatorics.  In 2006 Terence Tao received the medal for “for his contributions to partial differential equations, combinatorics, harmonic analysis and additive number theory“.

Both Gowers and Tao are most famous for their results in combinatorics having to do with finding arbitrary long arithmetic progressions (Remember, an arithmetic progression is a sequence of numbers which looks like a, a+b, a+2b, a+3b, a+4b, …, where a and b are two fixed numbers.  So 3,5,7,9,11, … is an arithmetic progression where a=3 and b=2).  In particular, there is the very famous Green-Tao Theorem which says that you can find an arithmetic progression as long as you want where all the numbers in the list are prime numbers.  Of course, just because you know a long arithmetic progression exists doesn’t mean it’s easy to find.  Even though by the Green-Tao theorem we know that arbitrary long arithmetic progressions of primes exist, so far the longest list people have actually found has only 25 numbers.

Why are we bringing this up just now?  Well, because both Timothy Gowers and Terence Tao have blogs, for one.  More than that, Gowers has suggested that people join him on his blog in an experiment to do mathematical research by having a public collaboration where anyone can join in.  The problem he would like to solve is give a combinatorial proof of the Hales-Jewett Theorem.  Actually, Gowers’s goal is less ambitious:

It is not the case that the aim of the project is to find a combinatorial proof of the density Hales-Jewett theorem when $k=3$. I would love it if that was the result, but the actual aim is more modest: it is either to prove that a certain approach to that theorem (which I shall soon explain) works, or to give a very convincing argument that that approach cannot work.

This open collaboration is going on right now on the blogs of Gowers and Tao (they’re friends and have common math interests, so Tao has joined in on Gowers’s experiment).  And it is open to everybody! So if you’d like to have a go at doing real research with  some of the best mathematicians in the world, then they’re ready to listen to what you have to say.  Unfortunately, since you’re a few days behind, you’ll have to read what people have already discussed and get caught up.  Fortunately, Gowers is providing summaries of the discussion so you don’t have to read the 100′s of posts that have already happened.  In any case, it’s cool to see real math research happening live in front of you.  Maybe this is the wave of the future in mathematics?

P.S. We’ve added links to Gowers’s and Tao’s blogs to the Blogroll.  Keep an eye out over there for other interesting blogs we find, or let us know if you know of one we should add to the list.

If you win one of these, then don't forget to mention the OU Math Blog at your award ceremony.