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Tag: Connes

Alain Connes on everything

Published January 6, 2006 by lieven

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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nostalgia

Published May 4, 2005 by lieven

Unlike the
cooler people out there, I haven’t received my
_pre-ordered_ copy (via AppleStore) of Tiger yet. Partly my own fault
because I couldn’t resist the temptation to bundle up with a
personalized iPod Photo!
The good news is that it buys me more time to follow the
housecleaning tips. First, my idea was to make a CarbonCopyClooner–
image of my iBook and put it on the _iMac_ upstairs which I
rarely use these days, do a clean
Tiger install on the iBook and gradually copy over the essential
programs and files I need (and only those!). But reading the
macdev-article, I think it is better to keep my iBook running Panther
and experiment with Tiger on the redundant iMac. (Btw. unless you want
to have a copy of my Mac-installation there will be hardly a point
checking this blog the next couple of weeks as I intend to write down
all details of the Panther/Tiger switch here.)

Last week-end I
started a _Paper-rescue_ operation, that is, to find among the
multiple copies of books/papers/courses, the ones that contain all the
required material to re-TeX them and unfortunately my _archive_
is in a bad state. There is hardly a source-file left of a paper prior
to 1999 when I started putting all my papers on the arXiv.

On the other hand, I do
have saved most of my undergraduate courses. Most of them were still
using postscript-crap like _epsfig_ etc. so I had to convert all
the graphics to PDFs (merely using Preview ) and
modify the epsfig-command to _includegraphics_. So far, I
converted all my undergraduate _differential geometry_ courses
from 1998 to this year and made them available in a uniform
screen-friendly viewing format at TheLibrary/undergraduate.

There are two
ways to read the changes in these courses over the years. (1) as a shift
from _differential_ geometry to more _algebraic_ geometry
and (2) as a shift towards realism wrt.the level of our undegraduate
students. In 1998 I was still thinking
that I could teach them an easy way into Connes non-commutative standard
model but didn’t go further than the Lie group sections (maybe one day
I’ll rewrite this course as a graduate course when I ever get
reinterested in the Connes’ approach). In 1999 I had the illusion that
it might be a good idea to introduce manifolds-by-examples coming from
operads! In 2000 I gave in to the fact
that most of the students which had to follow this course were applied
mathematicians so perhaps it was a good idea to introduce them to
dynamical systems (quod non!). The 2001 course is probably the
most realistic one while still doing standard differential geometry. In
2002 I used the conifold
singularity and conifold transitions (deformations and blow-ups) as
motivation but it was clear that the students did have difficulties with
the blow-up part as they didn’t have enough experience in
_algebraic_ geometry. So the last two years I’m giving an
introduction to algebraic geometry culminating in blow-ups and some
non-commutative geometry.

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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 \pi \epsilon_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

$\zeta(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

$\zeta(a)\zeta(b)=\zeta(a+b)+\zeta(a,b)+\zeta(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
$\zeta(3),\zeta(5),\dots $ are all trancendental!

Drinfeld
introduced the Grothendieck-Teichmuller group-scheme over $\mathbb{Q} $
whose Lie algebra $\mathfrak{grt}_1 $ is conjectured to be the free Lie
algebra on infinitely many generators which correspond in a natural way
to the numbers $\zeta(3),\zeta(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
$\mathbb{Q} $-algebra whose degree $d $-term are the linear combinations
of the numbers $\zeta(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.

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