Millennium Prize: the Poincaré Conjecture

The problem’s been solved … but the sweet treats were declined. Back to the Cutting Board

MILLENNIUM PRIZE SERIES: The Millennium Prize Problems are seven mathematics problems laid out by the Clay Mathematics Institute in 2000. They’re not easya correct solution to any one results in a US$1,000,000 prize being awarded by the institute.

Russian mathematician Grigori “Grisha” Perelman was awarded the Prize on March 18 last year for solving one of the problems, the Poincaré conjecture – as yet the only one that’s been solved. Famously, he turned down the $1,000,000 Millennium Prize.

Over the coming weeks, each of these problems will be illuminated by experts from the Australian Mathematical Sciences Institute (AMSI) member institutions.

Here, Hyam Rubinstein discusses the now-resolved Poincaré Conjecture. Enjoy.

In 1904, French mathematician Henri Poincaré asked a key question about three-dimensional spaces (“manifolds”).

Imagine a piece of rope, so that firstly a knot is tied in the rope and then the ends are glued together. This is what mathematicians call a knot. A link is a collection of knots that are tangled together.

It has been observed that DNA, which is coiled up within cells, occurs in closed knotted form.

Complex molecules such as polymers are tangled in knotted forms. There are deep connections between knot theory and ideas in mathematical physics. The outsides of a knot or link in space give important examples of three-dimensional spaces.

Torus. Fropuff