Perfect Rigor

OK, are you ready for something a little more challenging than my usual Monday morning posts?  Try to get your mind around this!

This book totally fascinated me.  Math captivates me, though I can not grasp the depths of it and this book covers an aspect that I can not get my mind around- by far.

Math was always just a subject to get through when I was school age; it wasn’t until I began teaching it that its spell took hold of me.

The time I was explaining the concept of infinite and I was totally dumbfounded;  I knew what infinite meant but to say it out loud and explain it to my children caught me up short.  I was in awe that God was in our math.  Or reading an article about the possibility that there is no such thing as a prime number gave me just the slightest peek into realms of which I had never dreamt .  These ideas are nothing compared to the concepts in Perfect Rigor.

This may seem tedious, but read this and be totally amazed at the vastness and complexity of, of what?  Of God and His creation of all things, things so deep and complex that the creation of the earth seems like finger play.

In 1904 Henri Poincaré published a paper on three-dimensional manifolds.  What is a manifold?  It is an object, or a space, existing in the mathematician’s imagination- whether or not something like it can actually be observed in reality-that can be divided into many separate neighborhoods.  Each neighborhood, taken separately, has a basic Euclidean geometry or can be explained through it, but all the neighborhoods together add up to  something much more complicated.  The best example of a manifold is the Earth as portrayed through a series of maps, each showing only a small part of its surface.

Imagine a map of Manhattan, for example: its Euclidean nature is obvious.  When maps are put together in an atlas, their parallel lines continue not to cross and their triangles maintain their 180-degree nature.  But if we used the maps to try to replicate the actual surface of the Earth, we would start with something that looked like a many-many faceted disco ball, and then we would smooth out the edges and ultimately get a globe that reflected the Earth’s curved complexity- and if we extended Manhattan’s First Avenue and Second Avenue, they would cross.

What makes one manifold different from another is its having a hole.  To a topologist a ball, a box, a blob are all the same but a bagel is different.  If a very tight rubber band is placed around a ball, it will find a way to contract and slip off the ball.  A bagel is different. If you could thread the rubber band through the hole and then reconnect it, it will stay around the bagel never slipping off.

So…  a rubber band can be slipped off of the box, ball or blob, which makes them essentially similar to one another in a way that the bagel is not.  In the language of topology, diffeomorphic to one another.  This means you can reshape them into any other and then back again.

This more or less, brings us to the point of being able to understand the Poincaré Conjecture, an innocent sounding question: if a three-dimensional manifold is smooth and simply connected, then is it diffeomorphic to a three-dimensional sphere?

At the dawn of the 1960′s, several mathematician’s proved the Poincaré Conjecture for dimensions five and higher.  In 1982, Michael Freedman published a proof of the conjecture for dimension four.

Consider the enormity of solving the fourth dimension.  Perhaps one of the problems with four-dimensional spaces is that, unlike higher dimensional ones, they are not quite abstractions; it seems that we humans may very well inhabit a three-dimensional space embedded in four-dimensions, even if we can not wrap our minds around it.  And it is here that there is just no denying God; three-dimensional humans embedded in four-dimensions.

But… experts say there is one living man,the American geometer, William Thurston, who can imagine four dimensions, “When you see him or talk to him, he is often staring out into space and you can see that he sees these pictures,” said John Morgan, a professor at Columbia University.  Morgan watched Thurston attempt to solve the Poincaré and when he didn’t get it, Morgan was sure no one would.

Enter Grisha Perelman.

In almost a hundred years and with many, many mathematicians working on the conjecture no one had ever solved it.  Perelman simply posted his proof for the Poincaré Conjecture in an open forum on the internet in 2002.

Prior to the post about a dozen mathematicians received an email from Perelman asking them to take a look at his post, every recipient had been working on some aspect of the problem for many years.  Perelman had proven in half of his paper what one mathematician had unsuccessfully  worked on for two decades.  The Clay Institute had offered a prize of one million dollars for the solution and Perelman, without fanfare simply posted the solution to the million dollar question on a public internet site.

Perelman had formulated the proof to the conjecture so clearly in his mind that in 2003 he was able, to reduce the original copy by eight pages down to twenty-two pages, later his third revision would be a mere seven pages.  His original had taken only three weeks to write- less time than it had taken the other mathematicians to read. Perelam had the ability to absorb a problem in its entirety and then boil it down to an essence that proved simpler than anyone assumed.

Especially amazing to me is that we inhabit a three-dimensional world and that was the hardest part to solve of the conjecture; the fourth-dimension and higher were easier to solve, but right where we live was the dimension that had mathematicians stymied for almost a century.

If you are looking for a great book for someone you know who loves math, Perfect Rigor is the one.  Read it and fall prostrate in total adoration of God.

*Please note: if any of this writing seems to be out of my league ,consider those portions totally plagiarized from the author, Masha Gessen.