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Tag: monster

way too ambitious

Published January 30, 2007 by lievenlb

Student-evaluation sneak preview : I am friendly and
extremely helpful but have a somewhat chaotic teaching style and am way
too ambitious as regards content… I was about to deny vehemently
all assertions (except for the chaotic bit) but may have to change my
mind after reading this report on
Mark Rowan’s book ‘Symmetry and the monster’ (see also
my post
)

Oxford University Press considers this book
“a must-read for all fans of popular science”. In his blog,
Lieven le Bruyn, professor of algebra and geometry at the University of
Antwerp, suggests that “Mark Ronan has written a beautiful book
intended for the general public”. However, he goes on to say:
“this year I’ve tried to explain to an exceptionally
good second year of undergraduates, but failed miserably Perhaps
I’ll give it another (downkeyed) try using Symmetry and the
Monster as reading material”.

As an erstwhile
mathematician, I found the book more suited to exceptional maths
undergraduates than to the general public and would strongly encourage
authors and/or publishers to pass such works before a few fans of
popular science before going to press.

Peggie Rimmer,
Satigny.

Well, this ‘exceptionally good
year’ has moved on and I had to teach a course ‘Elementary
Algebraic Geometry’ to them last semester. I had the crazy idea to
approach this in a historical perspective : first I did the
Hilbert-Noether period (translating geometry to ideal theory of
polynomial rings), then the Krull-Weil-Zariski period (defining
everything in terms of coordinate rings) to finish off with the
Serre-Grothendieck period (introducing scheme theory)… Not
surprisingly, I lost everyone after 1920. Once again there were
complaints that I was expecting way too much from them etc. etc. and I
was about to apologize and promise I’ll stick to a doable course
next year (something along the lines of Miles Reid’s
‘Undergraduate Algebraic Geometry’) when one of the students
(admittedly, probably the best of this ‘exceptional year’)
decided to do all exercises of the first two chapters of Fulton’s
‘Algebraic Curves’ to become more accustomed to the subject.
Afterwards he told me “You know, I wouldn’t change the
course too much, now that I did all these exercises I realize that your
course notes are not that bad after all…”. Yeah, thanks!

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symmetry and the monster

Published June 2, 2006 by lievenlb

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.

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simple groups

Published October 24, 2004 by lievenlb

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.

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