Thursday, March 29, 2007

Higher Math - Sophus Lie and the 248th dimensions!!

A calculation the size of Manhattan

This is some pretty intense math. The "problem" has been around for around 100 years. The solution took a dedicated team (from around the globe) 4 years and a supercomputer to solve. The solution is in numbers/code that if written out in the tiniest print would blanket Manhattan. For comparison the human genome is about 1GB in a similar coding. This is 60GB!!

Now WHAT exactly is the solution? Well I started looking at what the problem was and started to get tangled in a web of math. I think that generally, it is about a system to classify symmetry. The Lie groups, of which E8 is the Everest, start out with very simple 1 dimensional symmetry (one group) and 2 dimensional (4 groups) on up to this E8 solution which is 248 dimensional being the symmetries of a particular 57 dimensional object. Start to explore some of the terms on Wikipedia and you quickly begin to see that the most basic understanding of this E8 fellow is not a depth easily fathomed!!! I think one would spend years just to understand the question. The solution is mathematical but is making physicists and geometry folks are getting giddy about it. (pretty picture of the E8 root system)

One theme that keeps coming up and is manifest here too is that many of the really great achievements are born and fostered in small groups of great friends.

Here there were a larger than usual group of 20 people who gathered annually altogether and more often in smaller groups.

Get this - Sophus Lie, the Norwegian mathematician who formulated Lie groups including E8, developed the theories with Felix Klein (of Klein Bottle fame).

and over to literature:
The Inklings (including C.S. Lewis, J.R.R. Tolkien) read aloud and critiqued/heckled eachother's work - with what grand results! English literature finally had mythology.

Mary Shelly (who wrote Frankenstein), Lord Byron (wrote Don Juan), Percy Shelley and others seemed to be a clique of sorts.

and over to politics:
OOOPS!! Most examples of this leave a sour taste!
... but not all. Consider the framers of the US constitution. Consider the advances in Athens.

It seems to be of use to have someone to "bounce things off of". It may have the effect of magnifying your stature since the public only first sees your ideas after they have been subjected to a round of constructive criticism.

The message to me is to never take lightly the criticisms of friends and colleagues - they will usually have your best interest at heart.

As for that exception above - politics. It seems that the best work gets done before there is too much at stake. Once the taxes are high enough and the country big enough that the participants are abnormally interested in that power they do not care about the best interest of anyone or anything but themselves. They end up out-shouting the good ones and even punishing them for "embarrassing [i.e. actual] good ideas".