### Grigori Perelman

July 6, 2010

In a previous article (Math Burnout, August 23, 2006), I wrote about the reclusive Russian mathematician, Grigori Perelman. Perelman was an employee of the Staklov Institute of Mathematics, Saint Petersburg, but he resigned his position several years ago and now lives with his mother in that same city. In 2006, Perelman refused the award of a Fields Medal,[1-3] considered the Mathematics equivalent of the Nobel Prize, and now he's refused the million dollar Clay Mathematics Institute prize for his proof of the
Poincaré Conjecture.[4] Perelman eschewed conventional journal publication by posting his proof in three internet posting in 2002 and 2003.

What is the Poincaré Conjecture, and why was its proof worth a million dollars? In 1904, Henri Poincaré, a preeminent mathematician whose life bridged the nineteenth and twentieth centuries, had a conjecture on topology. The conjecture essentially states that the only three dimensional surfaces that can be mapped to a sphere are those with no holes and no boundary (e.g., a coffee cup with a handle cannot be mapped to a sphere). The "no boundary" part means that you can walk all over the surface, reaching every point, and not be stopped by anything. As time went by and a proof (or disproof) was not forthcoming, the conjecture became one of the glamour problems in mathematics, having a cachet similar to a proof of Fermat's Last Theorem.

*Henri Poincaré*

The Millennium Prize Problems were established in 2000 as an incentive to solve the seven most vexing problems in mathematics:

- P versus NP
- The Hodge conjecture
- The Poincaré conjecture
- The Riemann hypothesis
- Yang-Mills existence and mass gap
- Navier-Stokes existence and smoothness
- The Birch and Swinnerton-Dyer conjecture