How is it determined who gets the Feilds Medal?

Answer 1

The Executive Committee of the IMU (International Mathematical Union) forms the Fields Medal Committee, who award at least two and as many as four medals. Candidates must be younger than 40 years in the year that the award will be made.

It's a very tricky thing to explain to the outside world what makes for excellent mathematics: when expressing awe, mathematicians will say "That's a deep result" or "That's very elegant." There is a combination of hard work and divine inspiration. Somehow the question of who deserves this medal is never terribly controversial: there is widespread agreement in the mathematical community on what constitutes really great work.

Fields Medallists have typically solved long-outstanding problems. The Poincaré Conjecture was a had been already been around for a very long time when I was in graduate school at Berkeley. It was the subject of keen interest by the best mathematicians in the world. And that was in the '70s! Perelman's solving this is something like finding the Holy Grail, and so he was pretty much a slam-dunk to be offered the prize.

The medal is awarded every four years at the ICM (International Congress of Mathematicians). I attended the one in Vancouver, British Columbia in 1974 where the algebraic geometers Enrico Bombieri and David Mumford got the medal.

Answer 2

The Fields Medal is awarded every four years on the occasion of the International Congress of Mathematicians to recognize outstanding mathematical achievement for existing work and for the promise of future achievement.

The Fields Medal Committee is chosen by the Executive Committee of the International Mathematical Union and is normally chaired by the IMU President. It is asked to choose at least two, with a strong preference for four, Fields Medallists, and to have regard in its choice to representing a diversity of mathematical fields. A candidate's 40th birthday must not occur before January 1st of the year of the Congress at which the Fields Medals are awarded.