My buddy Ryan sent me an incredibly exciting article this morning entitled "Major Math Problem Is Believed Solved By Reclusive Russian" … sounds mysterious, doesn’t it? The major math problem the article is referring to is the 6th Millennium Problem called The Poincare’ Conjecture. Named after Henri Ponicare, a french mathematician born in the 19th century, this millennium problem deals with concepts of topology.
Topology is an area of mathematics that studies general properties of surfaces and other objects. Topology has a really interesting way of looking at things. Imagine that you have a football and a soccer ball made out of clay. In topology, a football is the same as a soccer ball, because one can be morphed into the other – a ball is a ball. Coffee and donuts – the classic example … As Paul Erdos, Hungarian mathematician, said "A mathematician is a machine for turning coffee into theorems". In topology, a donut is also the same as a coffee cup – given a clay donut, you can move most of the clay to make the cup and leave a small outter ring to make the handle.
Another example of topological thinking would be to take the Subway map of Boston. Note the intersections between the different lines, and the actual subway stops. Printing this map on a stretchable piece of rubber paper, you can stretch and morph this map into any shape you want. No matter how you deform the map, all the intersections and stops remain the same, and so we can say that the map’s configuration has not changed. This same process can be applied to a circuit board in a computer.
So, the funny thing about the "reclusive russian", Grigori Perelman, is that he did not post a paper of his proof in a mathematical journal to claim the $1 million dollar prize. The proof was posted on the internet, and many pieces were left out.
The sketchiness may reflect how a genius interacts with mortals. Dr. Perelman may believe some things are so obvious he needn’t bother to explain them step by step, say mathematicians. If readers are too dumb to fill in the blanks, he doesn’t care. Or, he has better things to do than justify every tortuous step, as proofs must.
Others have taken it upon themselves to explicate his work — and find no major flaws. Like Torah commentaries, they dwarf the original. Dr. Perelman’s 2003 paper is 22 pdf pages; the 2002 paper is 39. But "Notes on Perelman’s Papers," in which Prof. Kleiner and John Lott of the University of Michigan explain them almost line-by-line, is 192 pages. A book on the papers is expected to top 300 pages. A "complete proof" of Poincaré, based on Dr. Perelman’s breakthrough and published last month in the Asian Journal of Mathematics (which Prof. Milnor describes as throwing "a monkey wrench" into the question of who gets credit), is 328 pages long.