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icosahedral group

Published October 21, 2004 by lieven

In my geometry 101 course I'm doing the rotation-symmetry groups
of the Platonic solids right now. This goes slightly slower than
expected as it turned out that some secondary schools no longer give a
formal definition of what a group is. So, a lot of time is taken up
explaining permutations and their properties as I want to view the
Platonic groups as subgroups of the permutation groups on the vertices.
To prove that the _tetrahedral group_ is isomorphic to $A_{4}$ was pretty
straigthforward and I'm half way through proving that the
_octahedral group_ is just $S_{4}$ (using the duality of the octahedron
with the cube and using the $4$ body diagonals of the cube).
Next
week I have to show that the _icosahedral group_ is isomorphic to $A_{5}$
which is a lot harder. The usual proof (that is, using the duality
between the icosahedron and the dodecahedron and using the $5$ cubes
contained in the dodecahedron, one for each of the diagonals of a face)
involves too much calculations to do in one hour. An alternative road is
to view the icosahedral group as a subgroup of $S_{6}$ (using the main
diagonals on the $12$ vertices of the icosahedron) and identifying this
subgroup as $A_{5}$ . A neat exposition of this approach is given by John Baez in his
post Some thoughts on
the number $6$ . (He also has another post on the icosahedral group
in his Week 79's
finds in mathematical physics).

But
probably I'll go for an “In Gap we
thrust”-argument. Using the numbers on the left, the rotation by
$72^{o}$ counter-clockwise in the top face we get the permutation in
$S_{20}$
$(1, 2, 3, 4, 5) (6, 8, 10, 12, 14) (7, 9, 11, 13, 15) (16, 17, 18, 19, 20)$
and the
rotation by $72^{o}$ counterclockwise along the face $(1, 2, 8, 7, 8)$ gives
the permutation
$(1, 6, 7, 8, 2) (3, 5, 15, 16, 9) (4, 14, 20, 17, 10) (12, 13, 19, 18, 11)$
GAP
calculates that the subgroup $d o d e$ of $S_{20}$ generated by these two
elements is $60$ (the correct number) and with $I s S i m p l e g r o u p (d o d e);$ we
find that this group must be simple. Finally using
$I s o m o r p h i s m T y p e I n f o F i n i t e S i m p l e g r o u p (d o d e);$
we get the required
result that the group is indeed isomorphic to $A_{5}$ . The time saved I
can then use to tell something about the classification project of
finite simple groups which might be more inspiring than tedious
calculations…

a cosmic Galois group

Published October 6, 2004 by lieven

Are
there hidden relations between mathematical and physical constants such
as

$\frac{e^{2}}{4 π ϵ_{0} h c} \sim \frac{1}{137}$

or are these numerical relations mere accidents? A couple of years
ago, Pierre Cartier proposed in his paper A mad day’s work : from Grothendieck to Connes and
Kontsevich : the evolution of concepts of space and symmetry that
there are many reasons to believe in a cosmic Galois group acting on the
fundamental constants of physical theories and responsible for relations
such as the one above.

The Euler-Zagier numbers are infinite
sums over $n_{1} > n_{2} >! > n_{r} \geq 1$ of the form

$ζ (k_{1}, \dots, k_{r}) = \sum n_{1}^{- k_{1}} \dots n_{r}^{- k_{r}}$

and there are polynomial relations with rational coefficients between
these such as the product relation

$ζ (a) ζ (b) = ζ (a + b) + ζ (a, b) + ζ (b, a)$

It is
conjectured that all polynomial relations among Euler-Zagier numbers are
consequences of these product relations and similar explicitly known
formulas. A consequence of this conjecture would be that
$ζ (3), ζ (5), \dots$ are all trancendental!

Drinfeld
introduced the Grothendieck-Teichmuller group-scheme over $Q$
whose Lie algebra ${grt}_{1}$ is conjectured to be the free Lie
algebra on infinitely many generators which correspond in a natural way
to the numbers $ζ (3), ζ (5), \dots$ . The Grothendieck-Teichmuller
group itself plays the role of the Galois group for the Euler-Zagier
numbers as it is conjectured to act by automorphisms on the graded
$Q$ -algebra whose degree $d$ -term are the linear combinations
of the numbers $ζ (k_{1}, \dots, k_{r})$ with rational coefficients and
such that $k_{1} + \dots + k_{r} = d$ .

The Grothendieck-Teichmuller
group also appears mysteriously in non-commutative geometry. For
example, the set of all Kontsevich deformation quantizations has a
symmetry group which Kontsevich conjectures to be isomorphic to the
Grothendieck-Teichmuller group. See section 4 of his paper Operads and motives in
deformation quantzation for more details.

It also appears
in the renormalization results of Alain Connes and Dirk Kreimer. A very
readable introduction to this is given by Alain Connes himself in Symmetries Galoisiennes
et renormalisation. Perhaps the latest news on Cartier’s dream of a
cosmic Galois group is the paper by Alain Connes and Matilde Marcolli posted
last month on the arXiv : Renormalization and
motivic Galois theory. A good web-page on all of this, including
references, can be found here.

anyone interested

Published September 15, 2004 by lieven

I've been here before! I mean, I did try to set up
non-commutative algebra&geometry sites before and sooner or later
they always face the same basic problems :

a :
dyspnoea : one person does not have enough fresh ideas
to keep a mathematical site updated daily so that it continues to be of
interest (at least, I'm not one of those who can).

b :
claustrophobia : the topic of non-commutative algebra
& non-commutative geometry is too wide to be covered (cornered) by
one person. More (and differing) views are needed for balance and
continued interest.

c : paranoia : if one is
not entirely naive one has to exercise some restraint trying to protect
ones research plans (or those of students) so the most interesting ideas
never even get posted!

By definition, I cannot solve problems
a) and b) on my own. All I can hope is that, now that the basic
technological problems (such as including LaTeX-code in posts) are
solved, other people are willing to contribute. For this reason I
'depersonalized' this blog : I changed the title, removed all
personal links in the sidebar and so on. I want to open up this site
(but as I said, I've tried this before without much success) to
anyone working in non-commutative algebra and/or non-commutative
geometry who is willing to contribute posts on at least a monthly basis
(or fortnightly, weekly, daily…) for the foreseeable future. At
the moment the following 'categories' of posts are available
(others can be added on request) :

courses : if you want
to tell about your topic of interest in small daily or weekly pieces.
columns : if you want to ventilate an opinion on something
related (even vaguely) to na&g.
nc-algebra : for anything
on non-commutative algebra not in the previous categories.
nc-geometry : for anything on non-commutative geometry not in the
previous categories.
this blog : for suggestions or
explanations on the technology of this site.

Mind you,
I am not looking for people who seek a forum to post
their questions (such people can still add questions as comments to
related posts) but rather for people active in na&g with a personal
opinion on relevance and future of the topic.
If you are
interested in contributing, please email me and we will work
something out. I'll also post information for authors (such as, how
to include tex, how to set restrictions etc.) in a _sticky_ post
soon.

Now, problem c) : in running sites for our master class
on noncommutative geometry I've noticed that some people are more
willing to post lectures notes etc. if they know that there is some
control on who can download their material. For this reason there will
be viewing restrictions on certain posts. Such posts will get a
padlock-sign next to them in the 'recent posts' sidebar (they
will not show up in your main page, if you are not authorized to see
them). I will add another sticky on all of this soon. For now, if you
would only be willing to contribute if there was this safeguard, rest
assured, it will be there soon. All others can of course already sign-up
or wait whether any of these plans (resp. day-dreams) ever work
out….

update (febr 2007) : still waiting
but the padlock idea is abandoned.

Tag: geometry

icosahedral group

a cosmic Galois group

anyone interested