neverendingbooks – Page 187

why mathematicians can’t write

Published March 29, 2006 by lievenlb

The Music of the
Primes will attract many young people to noncommutative geometry a
la Connes. It would be great if someone would spend a year trying to
write a similar pamphlet in favour of noncommutative _algebraic_
geometry, but as I mentioned before chances are not very high as most
mathematicians are unwilling to sacrifice precision and technical detail
for popular success. Still, perhaps we should reconsider this position.
A fine illustration why most mathematicians cannot write books for a
bigger audience is to be found in the preface to the book “The
problems of mathematics” (out of print or at least out of
amazon.com) by the Warwick mathematician Ian Stewart.
Below I quote a fraction from his ‘An interview with a
mathematician…’

(I)nterviewer : … So,
Mathematician : what delights do you have in store for us?
(M)athematician : I thought I’d say a bit about how you can get a TOP
but non-DIFF 4-manifold by surgery on the Kummer surface. You see,
there’s this fascinating cohomology intersection form related to the
exceptional Lie algebra $E_8$, and…
(I) : That’s
fascinating.
(M) : Thank you.
(I) : Is all that
gobbledegook really significant?
(M) : Of course! It’s one of the
most important discoveries of the last decade!
(I) : Can you
explain it in words ordinary mortals can understand?
(M) : Look,
buster, if ordinary mortals could understand it, you wouldn’t need
mathematicians to do the job for you, right?
(I) : I don’t want
the technical details. Just a general feeling for what’s going on.
(M) : You can’t get a feeling for what’s going on without
understanding the technical details.
(I) : Why not?
(M) :
Well, you just can’t.
(I) : Physicists seem to manage.
(M)
: But they work with things from everyday experience…
(I) :
Sure. ‘How gluon antiscreening affects the colour charge of a
quark.’ ‘Conduction bands in Gallium Arsenide.’ Trip over
‘em all the time on the way to work, don’t you?
(M) : Yes,
but…
(I) : I’m sure that the physicists find all the
technical details just as fascinating as you do. But they don’t let them
intrude so much.
(M) : But how can I explain things properly if I
don’t give the details?
(I) : How can anyone else
understand them if you do?
(M) : But if I skip the fine
points, some of the things I say won’t be completely true! How can I
talk about manifolds without mentioning that the theorems only work if
the manifolds are finite-dimensional paracompact Hausdorff with empty
boundary?
(I) : Lie a bit.
(M) : Oh, but I couldn’t do
that!
(I) : Why not? Everybody else does.
(M) : But, I
must tell the truth!
(I) : Sure. But you might be prepared to
bend it a little, if it helps people understand what you’re doing.
(M) : Well…

music of the primes

Published March 28, 2006 by lievenlb

Let me
admit it : i was probably wrong in this post to
advise against downloading A walk in the noncommutative
garden by Alain Connes and Matilde Marcolli. After all, it seems
that Alain&Matilde are on the verge of proving the biggest open
problem in mathematics, the Riemann
hypothesis using noncommutative geometry. At least, this is the
impression one gets from reading through The music of the
primes, why an unsolved problem in mathematics matters by Oxford
mathematician Prof.
Marcus du Sautoy… At the moment I’ve only read the first
chapter (_Who wants to be a millionaire?_) and the final two
chapters (_From orderly zeros to quantum chaos_ and _The
missing piece of the jigsaw_) as I assume I’ll be familiar with most
of the material in between (and also, I’m saving these chapters for some
vacation reading). From what I’ve read, I agree most with the final
review at amazon.co.uk

Fascinating
and infuriating, October 5, 2004
Reviewer: pja_jennings
from Southampton, Hants. United Kingdom
This is a book I found
fascinating and infuriating in turns. It is an excellent layman’s
history of number theory with particular reference to prime numbers and
the Riemann zeta function. As such it is well worth the reading.
However I found that there are certain elements, more of style than
anything else, that annoyed me. Most of the results are handed to us
without any proof whatsoever. All right, some of these proofs would be
obviously well beyond the layman, but one is described as being
understandable by the ancient Greeks (who started the whole thing) so
why not include it as a footnote or appendix?
Having established
fairly early on that the points where a mathematical function
“reaches sea level” are known as zeros, why keep reverting
to the sea level analogy? And although the underlying theme throughout
the book is the apparent inextricable link between the zeta function’s
zeros and counting primes, the Riemann hypothesis, I could find no
clear, concise statement of exactly what Riemann said.
Spanning
over 2000 years, from the ancient Greeks to the 21st century, this is a
book I would thoroughly recommend.

Books on Fermat’s last
theorem (and there are some nice ones, such as Alf Van der Poorten’s
Notes on
Fermat’s last theorem) can take Wiles’ solution as their focal
point. Failing a solution, du Sautoy constructs his book around an
April’s Fool email-message by Bombieri in which he claimed that a young
physicist did prove the Riemann hypothesis after hearing a talk by Alain
Connes. Here’s du Sautoy’s account (on page 3)

According
to his email, Bombieri has been beaten to his prize. ‘There are
fantastic developments to Alain Connes’s lecture at IAS last wednesday.’
Bombieri began. Several years previously, the mathematical world had
been set alight by the news that Alain Connes had turned his attention
to trying to crack the Riemann Hypothesis. Connes is one of the
revolutionaries of the subject, a benign Robespierre of mathematics to
Bombieri’s Louis XVI. He is an extraordinary charismatic figure whose
fiery style is far from the image of the staid, awkward mathematician.
He has the drive of a fanatic convinced of his world-view, and his
lectures are mesmerising. Amongst his followers he has almost cult
status. They will happily join him on the mathematical barricades to
defend their hero against any counter-offensive mounted from the ancien
regime’s entrenched positions.

Contrary to physics,
mathematics doesn’t produce many books aimed at a larger public. To a
large extend this is caused by most mathematicians’ unwillingness to
sacrifice precision and technical detail. Hence, most of us would never
be able to come up with something like du Sautoy’s description of Weil’s
work on the zeta function of curves over finite fields (page 295)

It was while exploring some of these related landscapes that
Weil discovered a method that would explain why points at sea level in
them like to be in a straight line. The landscapes where Weil was
successful did not have to do with prime numbers, but held the key to
counting how many solutions an equation such as $y^2=x^3-x$ will have if
you are working on one of Gauss’s clock calculators.

But,
it is far too easy to criticize people who do want to make the effort.
Books such as this one will bring more young people to mathematics than
any high-publicity-technical-paper. To me, the chapter on quantum chaos
was an eye-opener as I hadn’t heard too much about all of this before.
Besides, du Sautoy accompanies this book with an interesting website musicofprimes and several of
his articles for newspapers available from his homepage are
a good read (in case you wonder why the book-cover is full of joggers
with a prime number on their T-shirt, you might have a look at Beckham in his
prime number). The music of the
primes will definitely bring many students to noncommutative
geometry and its possible use to proving the Riemann Hypothesis.

7 Comments

why nag? (1)

Published March 26, 2006 by lievenlb

Let us
take a hopeless problem, motivate why something like non-commutative
algebraic geometry might help to solve it, and verify whether this
promise is kept.

Suppose we want to know all solutions in invertible
matrices to the braid relation (or Yang-Baxter equation)

$X Y X = Y X Y$

All such solutions (for varying size of matrices)
form an additive Abelian category $\mathbf{rep}~B_3$ , so a big step forward would be to know all its
simple solutions (that is, those whose matrices cannot be brought in
upper triangular block form). A literature check shows that even this
task is far too ambitious. The best result to date is the classification
due to Imre Tuba and
Hans Wenzl of simple solutions of which the matrix size is at most
5.

For fixed matrix size n, finding solutions in $\mathbf{rep}~B_3$ is the same as solving a system of $n^2$ cubic
polynomial relations in $2n^2$
unknowns, which quickly becomes a daunting task. Algebraic geometry
tells us that all solutions, say $\mathbf{rep}_n~B_3$ form an affine closed subvariety of $n^2$ -dimensional affine space. If we assume that $\mathbf{rep}_n~B_3$ is a smooth variety (that is, a manifold) and
if we know one solution explicitly, then we can use the tangent space in
this point to linearize the problem and to get at all solutions in a
neighborhood.

So, here is an idea : assume that $\mathbf{rep}~B_3$ itself would be a non-commutative manifold, then
we might linearize our problem by considering tangent spaces and obtain
new solutions out of already known ones. But, what is a non-commutative
manifold? Well, by the above we at least require that for all integers n
the commutative variety $\mathbf{rep}_n~B_3$ is a commutative manifold.

But, there
is still some redundancy in our problem : if $(X,Y)$ is a
solution, then so is any conjugated pair $(g^{-1}Xg,g^{-1}Yg)$ where $g \in GL_n$ is a basechange matrix. In categorical terms, we are only
interested in isomorphism classes of solutions. Again, if we fix the
size n of matrix-solutions, we consider the affine variety $\mathbf{rep}_n~B_3$ as a variety with a $GL_n$ -action
and we like to classify the orbits of simple solutions. If $\mathbf{rep}_n~B_3$ is a manifold then the theory of Luna slices
provides a method, both to linearize the problem as well as to reduce
its complexity. Instead of the tangent space we consider the normal
space N to the $GL_n$ -orbit
(in a suitable solution). On this affine space, the stabilizer subgroup
$GL(\alpha)$ acts and there is a natural one-to-one
correspondence between $GL_n$ -orbits
in $\mathbf{rep}_n~B_3$ and $GL(\alpha)$ -orbits in the normal space N (at least in a
neighborhood of the solution).

So, here is a refinement of the
idea : we would like to view $\mathbf{rep}~B_3$ as a non-commutative manifold with a group action
given by the notion of isomorphism. Then, in order to get new isoclasses
of solutions from a constructed one we want to reduce the size of our
problem by considering a linearization (the normal space to the orbit)
and on it an easier isomorphism problem.

However, we immediately
encounter a problem : calculating ranks of Jacobians we discover that
already $\mathbf{rep}_2~B_3$ is not a smooth variety so there is not a
chance in the world that $\mathbf{rep}~B_3$ might be a useful non-commutative manifold.
Still, if $(X,Y)$ is a
solution to the braid relation, then the matrix $(XYX)^2$
commutes with both X and Y.

If $(X,Y)$ is a
simple solution, this means that after performing a basechange, $C=(XYX)^2$ becomes a scalar matrix, say $\lambda^6 1_n$ . But then, $(X_1,Y_1) = (\lambda^{-1}X,\lambda^{-1}Y)$ is a solution to

$XYX = YXY , (XYX)^2 = 1$

and all such solutions form a
non-commutative closed subvariety, say $\mathbf{rep}~\Gamma$ of $\mathbf{rep}~B_3$ and if we know all (isomorphism classes of)
simple solutions in $\mathbf{rep}~\Gamma$ we have solved our problem as we just have to
bring in the additional scalar $\lambda \in \mathbb{C}^*$ .

Here we strike gold : $\mathbf{rep}~\Gamma$ is indeed a non-commutative manifold. This can
be seen by identifying $\Gamma$
with one of the most famous discrete infinite groups in mathematics :
the modular group $PSL_2(\mathbb{Z})$ . The modular group acts by Mobius
transformations on the upper half plane and this action can be used to
write $PSL_2(\mathbb{Z})$ as the free group product $\mathbb{Z}_2 \ast \mathbb{Z}_3$ . Finally, using
classical representation theory of finite groups it follows that indeed
all $\mathbf{rep}_n~\Gamma$ are commutative manifolds (possibly having
many connected components)! So, let us try to linearize this problem by
looking at its non-commutative tangent space, if we can figure out what
this might be.

Here is another idea (or rather a dogma) : in the
world of non-commutative manifolds, the role of affine spaces is played
by $\mathbf{rep}~Q$ the representations of finite quivers Q. A quiver
is just on oriented graph and a representation of it assigns to each
vertex a finite dimensional vector space and to each arrow a linear map
between the vertex-vector spaces. The notion of isomorphism in $\mathbf{rep}~Q$ is of course induced by base change actions in all
of these vertex-vector spaces. (to be continued)

neverendingbooks Posts

why mathematicians can’t write

music of the primes

why nag? (1)