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Tag: symmetry

football representation theory

Unless
you never touched a football in your life (that’s a _soccer-ball_
for those of you with an edu account) you will know that the world
championship in Germany starts tonight. In the wake of it, the field of
‘football-science’ is booming. The BBC runs its The
Science of Football-site
and did you know the following?

Research indicates that watching such a phenomenon is not
only exciting, it can be good for our health too. The Scottish
researchers found that there were 14% fewer psychiatric admissions in
the weeks after one World Cup than before it started.

But, would you believe that some of the best people in the field
(Kostant and Sternberg to name a few) have written papers on the
representation theory of a football? Perhaps this becomes more plausible
when you realize that a football has the same shape as the buckyball aka Carbon60.
Because the football (or buckyball) is a truncated icosahedron, its
symmetry group is $A_5$, the smallest of all simple groups and its
representations explain some physical properties of the buckyball. Some
of these papers are freely available and are an excellent read. In fact,
I’m thinking of using them in my course on representations of finite
groups, nxt year. Mathematics and the Buckyball by Fan
Chung and Schlomo Sternberg is a marvelous introduction to
representation theory. Among other things they explain how Schur’s
lemma, Frobenius reciprocity and Maschke’s theorem are used to count the
number of lines in the infra red buckyball spectrum! The Graph of the
Truncated Icosahedron and the Last Letter of Galois
by Bertram
Kostant explains the observation, first made by Galois in his last
letter to Chevalier, that $A_{5} = PSL_2(\mathbb{F}_5)$ embeds into
$PSL_{2}(\mathbb{F}_{11})$ and applies this to the buckyball.

In effect, the model we are proposing for C60is such that
each carbon atom can be labeled by an element of order 11 in PSl(2,11)
in such a fashion that the carbon bonds can be expressed in terms of the
group structure of PSl(2,11). It will be seen that the twelve pentagons
are exactly the intersections of M with the twelve Borel sub- groups of
PSl(2,11). (A Borel subgroup is any subgroup which is conjugate to the
group PSl(2,11) defined in (2).) In particular the pentagons are the
maximal sets of commuting elements in M. The most subtle point is the
natural existence of the hexagonal bonds. This will arise from a group
theoretic linkage of any element of order 11 in one Borel subgroup with
a uniquely defined element of order 11 in another Borel subgroup.

These authors consequently joined forces to write Groups and the
Buckyball
in which they give further applications of the Galois
embeddings to the electronic spectrum of the buckyball. Another
account can be found in the Master Thesis by Joris Mooij called The
vibrational spectrum of Buckminsterfullerene – An application of
symmetry reduction and computer algebra
. Plenty to read should
tonight’s match Germany-Costa Rica turn out to be boring…

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symmetry and the monster

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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Alain Connes on everything

A few
days ago, Ars Mathematica wrote :

Alain Connes and Mathilde Marcolli have posted a
new survey paper on Arxiv A walk in the
noncommutative garden
. There are many contenders for the title of
noncommutative geometry, but Connes’ flavor is the most
successful.

Be that as it may, do
not print this 106 page long paper! Browse through it
if you have to, be dazzled by it if you are so inclined, but I doubt it
is the eye-opener you were looking for if you gave up on reading
Connes’ book Noncommutative
Geometry
…. Besides, there is much better
_Tehran-material_ on Connes to be found on the web : An interview
with Alain Connes
, still 45 pages long but by all means : print it
out, read it in full and enjoy! Perhaps it may contain a lesson or two
for you. To wet your appetite a few quotes

It is
important that different approaches be developed and that one
doesn’t try to merge them too fast. For instance in noncommutative
geometry my approach is not the only one, there are other approaches
and it’s quite important that for these approaches there is no
social pressure to be the same so that they can develop
independently. It’s too early to judge the situation for instance
in quantum gravity. The only thing I resent in string theory is that
they put in the mind of people that it is the only theory that can
give the answer or they are very close to the answer. That I resent.
For people who have enough background it is fine since they know all
the problems that block the road like the cosmological constant, the
supersymmetry breaking, etc etc…but if you take people who are
beginners in physics programs and brainwash them from the very start
it is really not fair. Young physicists should be completely free,
but it is very hard with the actual system.

And here for some (moderate) Michael Douglas bashing :

Physicists tend to shift often and work on the
last fad. I cannot complain because at some point around 98 that fad was
NCG after my paper with Douglas and Schwarz. But after a while when
I saw Michael Douglas and asked him if he had thought more about
these problems the answer was no because it was no longer the last
fad and he wanted to work on something else. In mathematics one
sometimes works for several years on a problem but these young
physicists have a very different type of working habit. The unit of
time in mathematics is about 10 years. A paper in mathematics which is
10 years old is still a recent paper. In physics it is 3 months. So
I find it very difficult to cope with constant
zapping.

To the suggestion that he is the
prophet (remember, it is a Tehran-interview) of noncommutative geometry
he replies

It is flattering but I don’t think
it is a good thing. In fact we are all human beings and it is a
wrong idea to put a blind trust in a single person and believe in
that person whatever happens. To give you an example I can tell you
a story that happened to me. I went to Chicago in 1996, and gave a
talk in the physics department. A well known physicist was there and
he left the room before the talk was over. I didn’t meet this
physicist for two years and then, two years later, I gave the same
talk in the Dirac Forum in Rutherford laboratory near Oxford. This
time the same physicist was attending, looking very open and convinced
and when he gave his talk later he mentioned my talk quite
positively. This was quite amazing because it was the same talk and
I had not forgotten his previous reaction. So on the way back to
Oxford, I was sitting next to him in the bus, and asked him openly
how can it be that you attended the same talk in Chicago and you
left before the end and now you really liked it. The guy was not a
beginner and was in his forties, his answer was “Witten was seen
reading your book in the library in Princeton”! So I don’t want
to play that role of a prophet preventing people from thinking on
their own and ruling the sub ject, ranking people and all that. I
care a lot for ideas and about NCG because I love it as a branch of
mathematics but I don’t want my name to be associated with it as a
prophet.

and as if that was not convincing
enough, he continues

Well, the point is that what
matters are the ideas and they belong to nobody. To declare that
some persons are on top of the ladder and can judge and rank the
others is just nonsense mostly produced by the sociology (in fact by the
system of recommendation letters). I don’t want that to be true in
NCG. I want freedom, I welcome heretics.

But please, read it all for yourself and draw your own conclusions.

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