Wednesday, May 24, 2006

Exceptional Objects

One of my favourite topics in mathematics is the existence of exceptional objects. There have been many articles written on the web about this subject - particularly by John Baez. But I recently came across this article by John Stillwell, in American Mathematical Monthly, that also gives a historical perspective. The connection with classical geometry - Desargue's theorem and Pappus's hexagonal theorem - was also new to me.

Very briefly: mathematics is full of classification theorems. What frequently happens is that objects of a certain type are classified into a bunch of series with a handful of exceptional objects left over. For example the simple finite groups have all been classified and besides the series (such as the cyclic groups of prime order) there are 26 'sporadic' simple groups such as the Monster. The classification of compact Lie groups gives a similar pattern and even the classification of regular polyhedra is similar. Bizarrely, exceptional objects from different branches of mathematics are often related to each other. In fact, the entire classifications from different branches of mathematics are often closely related. And for some reason Dynkin diagrams and octonions seem to play a special role in many of these classifications. The article suggests that the octonions are the "mother of all exceptional objetcs". Anyway, read the article for more information.

I also found this article by the same author on the history of the 120-cell, one of those exceptional regular polyhedra.

1 comment:

Alexandre Borovik said...

You said in your post:

"What frequently happens is that objects of a certain type are classified into a bunch of series with a handful of exceptional objects left over."

What frequently happens is something even more interesting: by extending the language and therefore increasing the number of "objects", it is possible to place old sporadic objects into new infinite families. In the new extended set-up we have a "bunch" of infinite series and a finite number of new sporadic objectts. This extension process continues ad infinitum. See a model-theoretic explanation of this phenomenon in a paper by Gregory Cherlin "Sporadic homogeneous structures".

http://www.rci.rutgers.edu/~cherlin/Publication/sporadic.pdf

His proof is crucially based on the classification of finite simple groups.

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