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Tag: Connes

why mathematicians can’t write

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…

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music of the primes

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.

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noncommutative topology (1)

A couple of days ago Ars Mathematica had a post Cuntz on noncommutative topology pointing to a (new, for me) paper by Joachim Cuntz

A couple of years ago, the Notices of the AMS featured an article on noncommutative geometry a la Connes: Quantum Spaces and Their Noncommutative Topology by Joachim Cuntz. The hallmark of this approach is the heavy reliance on K theory. The first few pages of the article are fairly elementary (and full of intriguing pictures), before the K theory takes over.

A few comments are in order. To begin, the paper is **not** really about noncommutative geometry a la Connes, but rather about noncommutative geometry a la Cuntz&Quillen (based on quasi-free algebras) or, equivalently, a la Kontsevich (formally smooth algebras) or if I may be so bold a la moi (qurves).

About the **intruiging pictures** : it seems to be a recent trend in noncommutative geometry research papers to include meaningless pictures to lure the attention of the reader. But, unlike aberrations such as the recent pastiche by Alain Connes and Mathilde Marcolli A Walk in the Noncommutative Garden, Cuntz is honest about their true meaning

I am indebted to my sons, Nicolas and Michael,
for the illustrations to the examples above. Since
these pictures have no technical meaning, they
are only meant to provide a kind of suggestive
visualization of the corresponding quantum spaces.

As one of these pictures made it to the cover of the **Notices** an explanation was included by the cover-editor

About the Cover :

The image on this month’s cover arose from
Joachim Cuntz’s effort to render into visible art
his own internal vision of a noncommutative
torus, an object otherwise quite abstract. His
original idea was then implemented by his son
Michael in a program written in Pascal. More
explicitly, he says that the construction started
out with a triangle in a square, then translated
the triangle by integers times a unit along a line
with irrational slope; plotted the images thus
obtained in a periodic manner; and stopped
just before the figure started to seem cluttered.
Many mathematicians carry around inside
their heads mental images of the abstractions
they work with, and manipulate these objects
somehow in conformity with their mental imagery. They probably also make aesthetic judgements of the value of their work according to
the visual qualities of the images. These presumably common phenomena remain a rarely
explored domain in either art or psychology.

—Bill Casselman(covers@ams.org)

There can be no technical meaning to the pictures as in the Connes and Cuntz&Quillen approach there is only a noncommutative algebra and _not_ an underlying geometric space, so there is no topology, let alone a noncommutative topology. Of course, I do understand why Cuntz&others name it as such. They view the noncommutative algebra as the ring of functions on some virtual noncommutative space and they compute topological invariants (such as K-groups) of the algebras and interprete them as information about the noncommutative topology of these virtual and unspecified spaces.

Still, it is perfectly possible to associate to a qurve (aka quasi-free algebra or formally smooth algebra) a genuine noncommutative topological space. In this series of posts I’ll explain the little I know of the history of this topic, the thing I posted about it a couple of years ago, why I abandoned the project and the changes I made to it since and the applications I have in mind, both to new problems (such as the birational_classification of qurves) as well as classical problems (such as rationality problems for $PGL_n $ quotient spaces).

Although others have tried to define noncommutative topologies before, I learned about them from Fred Van Oystaeyen. Fred spend the better part of his career constructing structure sheaves associated to noncommutative algebras, mainly to prime Noetherian algebras (the algebras of preference for the majority of non-commutative algebraists). So, suppose you have an ordinary (meaning, the usual commutative definition) topological space X associated to this algebra R, he wants to define an algebra of sections on every open subset $X(\sigma) $ by taking a suitable localization of the algebra $Q_{\sigma}(R) $. This localization is taken with respect to a suitable filter of left ideals $\mathcal{L}(\sigma) $ of R and is defined to be the subalgebra of the classiocal quotient ring $Q(R) $ (which exists because $R$ is prime Noetherian in which case it is a simple Artinian algebra)

$Q_{\sigma}(R) = { q \in Q(R)~|~\exists L \in \mathcal{L}(\sigma)~:~L q \subset R } $

(so these localizations are generalizations of the usual Ore-type rings of fractions). But now we come to an essential point : if we want to glue this rings of sections together on an intersection $X(\sigma) \cap X(\tau) $ we want to do this by ‘localizing further’. However, there are two ways to do this, either considering $~Q_{\sigma}(Q_{\tau}(R)) $ or considering $Q_{\tau}(Q_{\sigma}(R)) $ and these two algebras are only the same if we impose fairly heavy restrictions on the filters (or on the algebra) such as being compatible.

As this gluing property is essential to get a sheaf of noncommutative algebras we seem to get stuck in the general (non compatible) case. Fred’s way out was to make a distinction between the intersection $X_{\sigma} \cap X_{\tau} $ (on which he put the former ring as its ring of sections) and the intersection $X_{\tau} \cap X_{\sigma} $ (on which he puts the latter one). So, the crucial new ingredient in a noncommutative topology is that the order of intersections of opens matter !!!

Of course, this is just the germ of an idea. He then went on to properly define what a noncommutative topology (and even more generally a noncommutative Grothendieck topology) should be by using this localization-example as guidance. I will not state the precise definition here (as I will have to change it slightly later on) but early version of it can be found in the Antwerp Ph.D. thesis by Luc Willaert (1995) and in Fred’s book Algebraic geometry for associative algebras.

Although _qurves_ are decidedly non-Noetherian (apart from trivial cases), one can use Fred’s idea to associate a noncommutative topological space to a qurve as I will explain next time. The quick and impatient may already sneak at my old note a non-commutative topology on rep A but please bear in mind that I changed my mind since on several issues…

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