Mark

Ronan has written a beautiful book intended for the general public

on Symmetry and the Monster. The

book’s main theme is the classification of the finite simple groups. It

starts off with the introduction of groups by Galois, gives the

classifivcation of the finite Lie groups, the Feit-Thompson theorem and

the construction of several of the sporadic groups (including the

Mathieu groups, the Fischer and Conway groups and clearly the

(Baby)Monster), explains the Leech lattice and the Monstrous Moonshine

conjectures and ends with Richard Borcherds proof of them using vertex

operator algebras. As in the case of Music of the

Primes it is (too) easy to be critical about notation. For example,

whereas groups are just called symmetry groups, I don’t see the point of

calling simple groups ‘atoms of symmetry’. But, unlike du Sautoy,

Mark Ronan stays close to mathematical notation, lattices are just

lattices, characer-tables are just that, j-function is what it is etc.

And even when he simplifies established teminology, for example

‘cyclic arithmetic’ for modular arithmetic, ‘cross-section’

for involution centralizer, ‘mini j-functions’ for Hauptmoduln

etc. there are footnotes (as well as a glossary) mentioning the genuine

terms. Group theory is a topic with several colourful people

including the three Johns John Leech, John

McKay and John Conway

and several of the historical accounts in the book are a good read. For

example, I’ve never known that the three Conway groups were essentially

discovered in just one afternoon and a few telephone exchanges between

Thompson and Conway. This year I’ve tried to explain some of

monstrous moonshine to an exceptionally good second year of

undergraduates but failed miserably. Whereas I somehow managed to give

the construction and proof of simplicity of Mathieu 24, elliptic and

modular functions were way too difficult for them. Perhaps I’ll give it

another (downkeyed) try using ‘Symmetry and the Monster’ as

reading material. Let’s hope Oxford University Press will soon release a

paperback (and cheaper) version.

# Tag: monster

I

found an old copy (Vol 2 Number 4 1980) of the The Mathematical Intelligencer with on its front

cover the list of the 26 _known_ sporadic groups together with a

starred added in proof saying

- added in

proof … the classification of finite simple groups is complete.

there are no other sporadic groups.

(click on the left picture to see a larger scanned image). In it is a

beautiful paper by John Conway “Monsters and moonshine” on the

classification project. Along the way he describes the simplest

non-trivial simple group $A_5 $ as the icosahedral group. as well as

other interpretations as Lie groups over finite fields. He also gives a

nice introduction to representation theory and the properties of the

character table allowing to reconstruct $A_5 $ only knowing that there

must be a simple group of order 60.

A more technical account

of the classification project (sketching the main steps in precise

formulations) can be found online in the paper by Ron Solomon On finite simple

groups and their classification. In addition to the posts by John Baez mentioned

in this

post he has a few more columns on Platonic solids and their relation to Lie

algebras, continued here.

Yesterday morning I thought that I could use some discussions I had a

week before with Markus Reineke to begin to make sense of one

sentence in Kontsevich’ Arbeitstagung talk Non-commutative smooth

spaces :

It seems plausible that Borcherds’ infinite rank

algebras with Monstrous symmetry can be realized inside Hall-Ringel

algebras for some small smooth noncommutative

spaces

However, as I’m running on a 68K RAM-memory, I

didn’t recall the fine details of all connections between the monster,

moonshine, vertex algebras and the like. Fortunately, there is the vast

amount of knowledge buried in the arXiv and a quick search on Borcherds gave me a

list of 17 papers. Among

these there are some delightful short (3 to 8 pages) expository papers

that gave me a quick recap on things I once must have read but forgot.

Moreover, Richard Borcherds has the gift of writing at the same time

readable and informative papers. If you want to get to the essence of

things in 15 minutes I can recommend What

is a vertex algebra? (“The answer to the question in the title is

that a vertex algebra is really a sort of commutative ring.”), What

is moonshine? (“At the time he discovered these relations, several

people thought it so unlikely that there could be a relation between the

monster and the elliptic modular function that they politely told McKay

that he was talking nonsense.”) and What

is the monster? (“3. It is the automorphism group of the monster

vertex algebra. (This is probably the best answer.)”). Borcherds

maintains also his homepage on which I found a few more (longer)

expository papers : Problems in moonshine and Automorphic forms and Lie algebras. After these

preliminaries it was time for the real goodies such as The

fake monster formal group, Quantum vertex algebras and the like.

After a day of enjoyable reading I think I’m again ‘a point’

wrt. vertex algebras. Unfortunately, I completely forgot what all this

could have to do with Kontsevich’ remark…