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Category: stories

THE rationality problem

This morning, Esther Beneish
arxived the paper The center of the generic algebra of degree p that may contain the most
significant advance in my favourite problem for over 15 years! In it she
claims to prove that the center of the generic division algebra of
degree p is stably rational for all prime values p. Let me begin by
briefly explaining what the problem is all about. Consider one n by n
matrix A which is sufficiently general, then it will have all its
eigenvalues distinct, but then it is via the Jordan normal form theorem uniquely
determined upto conjugation (that is, base change) by its
characteristic polynomial. In
other words, the conjugacy class of a sufficiently general n by n matrix
depends freely on the coefficients of the characteristic polynomial
(which are the n elementary symmetric functions in the eigenvalues of
the matrix). Now what about couples of n by n matrices (A,B) under
simultaneous conjugation (that is all couples of the form $~(g A
g^{-1}, g B g^{-1}) $ for some invertible n by n matrix g) ??? So,
does there exist a sort of Jordan normal form for couples of n by n
matrices which are sufficiently general? That is, are there a set of
invariants for such couples which determine it is freely upto
simultaneous conjugation?

For couples of 2 by 2 matrices, Claudio Procesi rediscovered an old
result due to James Sylvester saying
that this is indeed the case and that the set of invariants consists of
the five invariants Tr(A),Tr(B),Det(A),Det(B) and Tr(AB). Now, Claudio
did a lot more in his paper. He showed that if you could prove this for
couples of matrices, you can also do it for triples, quadruples even any
k-tuples of n by n matrices under simultaneous conjugation. He also
related this problem to the center of the generic division algebra of
degree n (which was introduced earlier by Shimshon Amitsur in a rather
cryptic manner and for a while he simply refused to believe Claudio’s
description of this division algebra as the one generated by two
_generic_ n by n matrices, that is matrices filled with independent
variables). Claudio also gave the description of the center of this
algebra as a field of lattice-invariants (over the symmetric group S(n)
) which was crucial in subsequent investigations. If you are interested
in the history of this problem, its connections with Brauer group
problems and invariant theory and a short description of the tricks used
in proving the results I’ll mention below, you might have a look at the
talk Centers of Generic Division Algebras, the rationality problem 1965-1990
I gave in Chicago in 1990.

The case of couples of 3 by 3 matrices was finally
settled in 1979 by Ed Formanek and a
year later he was able to solve also the case of couples of 4 by 4
matrices in a fabulous paper. In it, he used solvability of S(4) in an
essential way thereby hinting at the possibility that the problem might
no longer have an affirmative answer for larger values of n. When I read
his 4×4 paper I believed that someone able to prove such a result must
have an awesome insight in the inner workings of matrices and decided to
dedicate myself to this problem the moment I would get a permanent
job… . But even then it is a reckless thing to do. Spending all of
your time to such a difficult problem can be frustrating as there is no
guarantee you’ll ever write a paper. Sure, you can find translations of
the problem and as all good problems it will have connections with other
subjects such as moduli spaces of vectorbundles and of quiver
representations, but to do the ‘next number’ is another matter.

Fortunately, early 1990, together with
Christine Bessenrodt we were
able to do the next two ‘prime cases’ : couples of 5 by 5 and couples of
7 by 7 matrices (Katsylo and Aidan Schofield had already proved that if
you could do it for couples of k by k and l by l matrices and if k and l
were coprime then you could also do it for couples of kl by kl matrices,
so the n=6 case was already done). Or did we? Well not quite, our
methods only allowed us to prove that the center is stably rational
that is, it becomes rational by freely adjoining extra variables. There
are examples known of stably rational fields which are NOT rational, but
I guess most experts believe that in the case of matrix-invariants
stable rationality will imply rationality. After this paper both
Christine and myself decided to do other things as we believed we had
reached the limits of what the lattice-method could do and we thought a
new idea was required to go further. If today’s paper by Esther turns
out to be correct, we were wrong. The next couple of days/weeks I’ll
have a go at her paper but as my lattice-tricks are pretty rusty this
may take longer than expected. Still, I see that in a couple of weeks
there will be a meeting in
Atlanta were Esther
and all experts in the field will be present (among them David Saltman
and Jean-Louis Colliot-Thelene) so we will know one way or the other
pretty soon. I sincerely hope Esther’s proof will stand the test as she
was the only one courageous enough to devote herself entirely to the
problem, regardless of slow progress.

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group think

The
moment I read about it, I ordered the book, but received at least three
emails from Amazon.co.uk apologizing for not being able to find me a
copy of Lee Smolin’s The trouble with physics.
A very
considerate review of the book can be found at Background Independence, Christine Dantas’
old blog. Btw. I’m happy Christine has set up a new blog called
Theorema Egregium. Here’s the section
in her review that convinced me to have a look at the book myself.

I do not wish to make public some of my old, deep own
feelings about what I think science is and how it should be conducted.
There are of course certain points that I often do make public, but
there are some others that tormented me for quite a long time now, and
are so personal and even of emotive nature that I would rather keep them
to myself. But this is the fact per se that should be mentioned here,
since this is the contribution that I feel I can give on examining his
book: I found out that he was addressing some of my personal views and
doubts, of course from his own perspective and wisdom, but it was like
talking to an old friend who followed my own career in science and
understood what troubled me most for all those years. So this book is
for you if you want to be challenged over your own vision of science and
how you fit in it.

Finally, after all these months, just
before going on vacation I discovered a copy in one of my favourite
bookshops in Antwerp and took it along. I dont know Christine’s
favourite chapters of the book but I feel somehow I’ll be not too far
off mark in believing that chapter 16 “How Do You Fight Sociology?” will
be among them. This chapter (just 27 pages) should be read and reread by
all scientists. In it, Lee Smolin describes community behaviour of
certain scientific groups (he has the stringtheory-community in mind but
I’m sure anyone will recognise some of its behavior in groups closer to
ones own research-interests. I certainly did…). Here we go (citing
from page 284)

1. _Tremendous self-confidence_ ,
leading to a sense of entitlement and of belonging to an elite of
experts.
2. _An unusually monolithic community_ , with a
strong sense of consensus, whether driven by evidence or not, and an
unusual uniformity of views on open questions. These views seem related
to the existence of a hierarchical structure in which ideas of a few
leaders dictate the viewpoint, strategy, and direction of the field,
3. In some cases, a _sense of identification with the group_ ,
akin to identification with a religious faith or political platform.
4. A strong sense of the _boundary between the group and other
experts_ .
5. A _disregard for and disinterest in_ the
ideas, opinions, and work of experts who are not part of the group, and
a preference for talking only with other members of the commnity.
6. A tendency to _interpret evidence optimistically_ , to believe
exaggerated or incorrect statements of results, and to disregard the
possibility that the theory might be wrong. This is coupled with a
tendency to _believe results are true because they are widely
believed’_ even if one has not checked (or even seen) the proof
oneself.
7. A lack of appreciation for the extent to which a
research program ought to involve risk.

Although spotting
such behaviour can be depressing and/or frustrating, Smolin’s analysis
is that such groups are doomed to perish sooner or later for it is
exactly the kind of behaviour sociologists and psychologists recognize
as groupthink, following the Yale psychologist Irving Janis, “a mode
of thinking that people engage in when they are deeply involved in a
cohesive in-group, when the members’ strivings for unanimity override
their motivation to realistically appraise alternative courses of
action”. Groupthink is responsable for failures of decision making by
groups of experts such as the “failure of NASA to prevent the Challenger
disaster, the failure of the West to anticipate the collapse of the
Soviet Union, the failure of the American automobile companies to
feresee the demand for smaller cars, and most recently – and perhaps
most calamitously – the Bush administration’s rush to war on the basis
of a false belief that Iraq had weapons of mass destruction.” (Smolin,
page 286). An aspect of these groupthinking science-groups that worries
me most of all is their making of exagerated claims to potential
applications, not supported (yet) by solid proof. Short-time effect may
be to attract more people to the subject and to keep doubting followers
on board, but in the long term (at least if the claimed results remain
out of reach) this will destroy the subject itself (and, sadly enough,
also closeby subjects making no outrageous claims…). My advice to
people making such claims is : do a Perelman! Rather than doing a
PR-job, devote yourself for as long as it takes to prove your hopes,
somewhere in splendid isolation and come back victoriously. I have a
spare set of keys if you are in search for the perfect location…

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44 32’28.29″N, 4 05’08.61″E

Twenty
five years ago I was amazed that writing merely “Le
Travers,Sablieres,France” on an envelop did the job. Today I’m even more
surprised that typing just “Le Travers,Sablieres” into Google Maps or Google earth brings you there in seconds with an
offset of about 100 meters!

Actually, the Google mark may be more accurate as it depicts the spot on
an old mule-path entering ‘le hameau de travers’ which consists of two
main buildings : ‘le by’ just below us and what we call ‘the travers’
but locals prefer to call ‘le jarlier’ or ‘garlelier’ or whathever (no
consistent spelling for the house-name yet). If you are French and know
the correct spelling, please leave a comment (it may have to do
something with making baskets and/or pottery).

I’ve always
thought the building dated from the late 18th century, but now they tell
me part of it may actually be a lot older. How they decide this is
pretty funny : around the buildings is a regular grid of old chestnut
trees and as most of them are around 400 years old, so must be the
core-building, which was extended over time to accomodate the growing
number of people and animals, until some 100 yrs ago when the place was
deserted and became ruins…

The first
few days biking conditions were excellent. If you ever come to visit or
will be in the neighborhood and are in for an easy (resp. demanding,
resp. tough) one and a half hour ride here, are some suggestions.

Start/end
point is always the end of the loose green path in the middle (le
travers). An easy but quite nice route to get a feel for the
surroundings is the yellowish loop (gooing back over blue/green) from
Sablieres to Orcieres and gooing back along camping La Drobie. Slighly
more demanding is the blue climb to over 900 meters to Peyre (and back).
By far the nicest (but also hardest) small tour is the green one
(Dompnac-Pourcharesse-St.Melany). If you want to study
these routes in more detail using GoogleEarth here is the kmz-file. Btw.
this file was obtained from my GPS gpx-file using
GPS-visualizer. Two and a half years
ago I managed to connect the
place via a slow dial-up line and conjectured that broadband-internet
would never come this far. I may have to reconsider that now as the
village got an offer from Numeo.fr to set-up a
wireless (??!!) broadband-network with a pretty low subscription… But,
as no cell-phone provider has yet managed to cover this area, I’m a bit
doubtful about Numeo’s bizness-plan. Still, it would be great. Now, all
I have to do is to convince the university-administration that my online
teaching is a lot better than my in-class-act and Ill be taking up
residence here pretty soon…

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