millenium problems

Good at maths?  Really good?  Then maybe one day you’ll be able to claim a $1million prize for solving some of the fiendishly difficult and important maths problems out there.  In 2000, the Clay Institute offered the reward for any mathematician who was able to crack 7 mathematical problems.  In 13 years only one of them have been solved – the Poincare conjecture.  For the rest, the money is still available. 

Summary of the problems (from here)

1 Birch and Swinnerton-Dyer conjecture Euclid geometry for the 21st century, involving things called abelian points and zeta functions and both finite and infinite answers to algebraic equations

2 Poincaré conjecture The surface of an apple is simply connected. But the surface of a doughnut is not. How do you start from the idea of simple connectivity and then characterise space in three dimensions?

See a short news clip explaining why the solver of the Poincare conjecture turned down a million dollar prize.

3 Navier-Stokes equation The answers to wave and breeze turbulence lie somewhere in the solutions to these equations

4 P vs NP problem Some problems are just too big: you can quickly check if an answer is right, but it might take the lifetime of a universe to solve it from scratch. Can you prove which questions are truly hard, which not?

5 Riemann hypothesis Involving zeta functions, and an assertion that all “interesting” solutions to an equation lie on a straight line. It seems to be true for the first 1,500 million solutions, but does that mean it is true for them all?

6 Hodge conjecture At the frontier of algebra and geometry, involving the technical problems of building shapes by “gluing” geometric blocks together

7 Yang-Mills and Mass gap A problem that involves quantum mechanics and elementary particles. Physicists know it, computers have simulated it but nobody has found a theory to explain it.

There is some more discussion about some of the great unsolved maths problems here.  All of these problems would have significant implications to the world if solved – and even problems which initially appear abstract and not related to “real life” have an amazing tendency to be later found to describe some physical feature of the universe.  Is this proof that the universe has an underlying mathematical structure?