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Category: stories

The best rejected proposal ever

The Oscar in
the category The Best Rejected Research Proposal in Mathematics
goes to … Alexander Grothendieck
for his proposal Esquisse d’un Programme, Grothendieck\’s research program from 1983, written as
part of his application for a position at the CNRS, the French
equivalent of the NSF. An English translation is

Here is one of the problems discussed :
Give TWO non-trivial elements of
$Gal(\overline{\mathbb{Q}}/\mathbb{Q}) $
the _absolute_
Galois group of the algebraic closure of the rational numbers
$\overline{\mathbb{Q}} $, that is the group of all
$\mathbb{Q} $-automorphisms of $\overline{\mathbb{Q}} $. One element
most of us can give (complex-conjugation) but to find any other
element turns out to be an extremely difficult task.

To get a handle on
this problem, Grothendieck introduced his _’Dessins d’enfants’_
(Children’s drawings). Recall from last session the pictures of the
left and right handed Monsieur Mathieu

The left hand side drawing was associated to a map
$\mathbb{P}^1_{\mathbb{C}} \rightarrow \mathbb{P}^1_{\mathbb{C}} $ which was
defined over the field $\mathbb{Q} \sqrt{-11} $ whereas the right side
drawing was associated to the map given when one applies to all
coefficients the unique non-trivial automorphism in the Galois group
$Gal(\mathbb{Q}\sqrt{-11}/\mathbb{Q}) $ (which is
complex-conjugation). Hence, the Galois group
$Gal(\mathbb{Q}\sqrt{-11}/\mathbb{Q}) $ acts _faithfully_ on the
drawings associated to maps $\mathbb{P}^1_{\mathbb{Q}\sqrt{-11}} \rightarrow
\mathbb{P}^1_{\mathbb{Q}\sqrt{-11}} $ which are ramified only over
the points ${ 0,1,\infty } $.

Grothendieck’s idea was to
extend this to more general maps. Assume that a projective smooth curve
(a Riemann surface) X is defined over the algebraic numbers
$\overline{\mathbb{Q}} $ and assume that there is a map $X
\rightarrow \mathbb{P}^1_{\mathbb{C}} $ ramified only over the points
${ 0,1,\infty } $, then we can repeat the procedure of last time and
draw a picture on X consisting of d edges (where d is the degree
of the map, that is the number of points lying over another point of
$\mathbb{P}^1_{\mathbb{C}} $) between white resp. black points (the
points of X lying over 1 (resp. over 0)).

Call such a drawing a
‘dessin d\’enfant’ and look at the collection of ALL dessins
d’enfants associated to ALL such maps where X runs over ALL curves
defined over $\overline{\mathbb{Q}} $. On this set, there is an action
of the absolute Galois group
$Gal(\overline{\mathbb{Q}}/\mathbb{Q}) $ and if this action would be
faithful, then this would give us insight into this
group. However, at that time even the existence of a map $X \rightarrow
\mathbb{P}^1 $ ramified in the three points ${ 0,1,\infty } $
seemed troublesome to prove, as Grothendieck recalls in his proposal

In more erudite terms, could it be true that
every projective non-singular algebraic curve defined over a number
field occurs as a possible ‚ modular curve‚ parametrising
elliptic curves equipped with a suitable rigidification? Such a
supposition seemed so crazy that I was almost embarrassed to submit
it to the competent people in the domain. Deligne when I consulted
him found it crazy indeed, but didn’t have any counterexample up
his sleeve. Less than a year later, at the International Congress in
Helsinki, the Soviet mathematician Bielyi announced exactly that result,
with a proof of disconcerting simplicity which fit into two little
pages of a letter of Deligne ‚ never, without a doubt, was such a
deep and disconcerting result proved in so few lines!

the form in which Bielyi states it, his result essentially says that
every algebraic curve defined over a number field can be obtained as
a covering of the projective line ramified only over the points 0,
1 and infinity. This result seems to have remained more or less
unobserved. Yet, it appears to me to have considerable importance. To
me, its essential message is that there is a profound identity
between the combinatorics of finite maps on the one hand, and the
geometry of algebraic curves defined over number fields on the
other. This deep result, together with the algebraic- geometric
interpretation of maps, opens the door onto a new, unexplored world within reach of all, who pass by without seeing it.

Belyi’s proof is indeed relatively easy
(full details can be found in the paper Dessins d’enfants on the
Riemann sphere
by Leila
Schneps). Roughly it goes as follows : as both X and the map are
defined over $\overline{\mathbb{Q}} $ the map is only ramified over
(finitely many) $\overline{\mathbb{Q}} $-points. Let S be the finite
set of all Galois-conjugates of these points and consider the polynomial

$f_0(z_0) = \prod_{s \in S} (z_0 -s) \in
\mathbb{Q}[z_0] $

Now, do a
resultant trick. Consider the
polynomial $f_1(z_1) = Res_{z_0}(\frac{d f_0}{d
z_0},f_0(z_0)-z_1) $ then the roots of $f_1(z_1) $ are exactly the
finite critical values of $f_0 $, $f_1 $ is again defined over
$\mathbb{Q} $ and has lower degree (in $z_1 $) than $f_0 $ in $z_1 $.
Continue this trick a finite number of times untill you have constructed
a polynomial $f_n(z_n) \in \mathbb{Q}[z_n] $ of degree zero.

the original map with the maps $f_j $ in succession yields that all
ramified points of this composition are
$\mathbb{Q} $-points! Now, we only have to limit the number of
these ramified $\mathbb{Q} $-points (let us call this set T) to three.

Take any three elements of T, then there always exist integers $m,n
\in \mathbb{Z} $ such that the three points go under a linear
fractional transformation (a Moebius-function associated to a matrix in
$PGL_2(\mathbb{Q}) $) to ${ 0,\frac{m}{m+n},1 } $. Under the
transformation $z \rightarrow \frac{(m+n)^{m+n}}{m^m
n^n}z^m(1-z)^n $ the points 0 and 1 go to 0 and
$\frac{m}{m+n} $ goes to 1 whence the ramified points of the
composition are one less in number than T. Continuing in this way we
can get the set of ramified $\mathbb{Q} $-points of a composition at
most having three elements and then a final Moebius transformation gets
them to ${ 0,1,\infty } $, done!

As a tribute for this clever
argument, maps $X \rightarrow \mathbb{P}^1 $ ramified only in 0,1 and
$\infty $ are now called Belyi morphisms. Here is an example of
a Belyi-morphism (and the corresponding dessin d’enfants) associated to
one of the most famous higher genus curves around : the Klein
(if you haven’t done
so yet, take your time to go through this marvelous pre-blog post by
John Baez).

One can define the Klein quartic as the plane projective
curve K with defining equation in
$\mathbb{P}^2_{\\mathbb{C}} $ given by $X^3Y+Y^3Z+Z^3X = 0 $ K has
a large group of automorphism, namely the simple group of order
168 $G = PSL_2(\mathbb{F}_7) =
SL_3(\mathbb{F}_2) $ It is a classical fact (see for example
the excellent paper by Noam Elkies The Klein quartic in number theory) that the quotient map $K \rightarrow K/G =
\mathbb{P}^1_{\mathbb{C}} $ is ramified only in the points
0,1728 and $\infty $ and the number of points of K lying over them
are resp. 56, 84 and 24. Now, compose this map with the Moebius
transormation taking ${ 0,1728,\infty } \rightarrow { 0,1,\infty } $
then the resulting map is a Belyi-map for the Klein quartic. A
topological construction of the Klein quartic is fitting 24 heptagons
together so that three meet in each vertex, see below for the gluing
data-picture in the hyperbolic plane : the different heptagons are given
a number but they appear several times telling how they must fit

The resulting figure has exactly $\frac{7 \times 24}{2} =
84 $ edges and the 84 points of K lying over 1 (the white points in
the dessin) correspond to the midpoints of the edges. There are exactly
$\frac{7 \times 24}{3}=56 $ vertices corresponding to the 56 points
lying over 0 (the black points in the dessin). Hence, the dessin
d\’enfant associated to the Klein quartic is the figure traced out by
the edges on K. Giving each of the 168 half-edges a
different number one assigns to the white points a permutation of order
two and to the three-valent black-points a permutation of order three,
whence to the Belyi map of the Klein quartic corresponds a
168-dimensional permutation representation of $SL_2(\mathbb{Z}) $,
which is not so surprising as the group of automorphisms is
$PSL_2(\mathbb{F}_7) $ and the permutation representation is just the
regular representation of this group.

Next time we will see how
one can always associate to a curve defined over
$\overline{\mathbb{Q}} $ a permutation representation (via the Belyi
map and its dessin) of one of the congruence subgroups $\Gamma(2) $ or
$\Gamma_0(2) $ or of $SL_2(\mathbb{Z}) $ itself.


the father of all beamer talks

Who was the first mathematician to give a slide show talk? I don’t have the
definite answer to this question, but would like to offer a strong
candidate : Hermann Minkowski gave the talk “Zur Geometrie der Zahlen” (On the
geometry of numbers) before the third ICM in 1904 in Heidelberg and even
the title page of his paper in the proceedings indicates that he did
present his talk using slides (Mit Projektionsbildern auf einer

of these eight slides would be hard to improve using LaTeX

What concerns
us today is the worst of all slides, the seventh, where Minkowski tries
to depict his famous questionmark function $?(x) $, sometimes also called
the _devil’s staircase_

The devil’s
staircase is a fractal curve and can be viewed as a mirror (taking a
point on the horizontal axis to the point on the vertical axis through
the function value) having magical simplifying properties : – it takes
rational numbers to _dyadic numbers_, that is those of the form
$n.2^{-m}$ with $n,m \in \mathbb{Z} $. – it takes quadratic
_irrational_ numbers to rational numbers. So, iterating this
mirror-procedure, the devil’s staircase is a device solving the main
problem of Greek Mathematics : which lengths can be constructed using
ruler and compass? These _constructible numbers_ are precisely those
real numbers which become after a finite number of devil-mirrors a
dyadic number. The proofs of these facts are not very difficult but
they involve a piece of long-forgotten mathematical technology :
_continued fractions_. By repeted approximations using the
floor-function (the largest natural number less than or equal to the real
number), every positive real number can be written as

$a = a_0 +
\frac{1}{a_1 + \frac{1}{a_2 + \frac{1}{a_3 + \frac{1}{\dots}}}} $

with all $a_i $ natural numbers. So, let us just denote from now on this
continued fraction of a by the expression

$a = \langle
a_0;a_1,a_2,a_3,\dots \rangle $

Clearly, a is a rational number if
(and also if but this requires a small argument using the Euclidian
algorithm) the above description has a tail of zeroes at the end and
(slightly more difficult) $a$ is a real quadratic irrational number
(that is, an element of a quadratic extension field
$\mathbb{Q}\sqrt{n} $) if and only if the continued fraction-expression
has a periodic tail. There is a lot more to say about
continued-fraction expressions and I’ll do that in another
‘virtual-course-post’ (those prepended with a (c): sign). For the
impatient let me just say that two real numbers will lie in the same
$GL_2(\mathbb{Z}) $-orbit (under the action via Moebius-transformations)
if and only if their continued fraction expressions have the same tails
eventually (which has applications in noncommutative geometry as in the
work of Manin and Marcolli but maybe I’ll come to this in the (c):

Right, now we can define the mysterious devil-stair function
$?(x) $. If a is in the real interval $[0,1] $ and if $a \in
\mathbb{Q} $ then $a = \langle 0;a_1,a_2,\dots,a_n,0,0,\dots
\rangle $ and we define $?(a) = 2 \sum_{k=1}^{n} (-1)^k
2^{-(a_1+a_2+\dots+a_k)} $ and if a is irrational with continued
fraction expression $a = \langle 0;a_1,a_2,a_3,\dots \rangle $, then

$?(a) = 2 \sum_{k=1}^{\infty} (-1)^{k+1} 2^{-(a_1+a_2+\dots+a_k)} $

perhaps easier description is that with the above continued-fraction
expression, the _binary_ expansion of $?(a) $ has the following form

$?(a) = 0,0 \dots 01 \dots 1 0 \dots 0 1 \dots 1 0 \dots 0 1 \dots
1 0 \dots $

where the first batch of zeroes after the comma has length
$a_1-1 $, the first batch of ones has length $a_2 $ the next batch of
zeroes length $a_3 $ and so on.

It is a pleasant exercise to verify that
this function does indeed have the properties we claimed before. A
recent incarnation of the question mark function is in Conway’s game of
_contorted fractions_. A typical position consists of a finite number of
boxed real numbers, for example the position might be

$\boxed{\pi} + \boxed{\sqrt{2}} + \boxed{1728} +
\boxed{-\frac{1}{3}} $

The Rules of the game are : (1) Both
players L and R take turns modifying just one of the numbers such that
the denominator becomes strictly smaller (irrational numbers are
supposed to have $\infty$ as their ‘denominator’). And if the boxed
number is already an integer, then its absolute value must decrease.
(2) Left must always _decrease_ the value of the boxed number, Right
must always increase it. (3) The first player unable to move looses
the game. To decide who wins a particular game, one needs to compute
the value of a position $\boxed{x} $ according to the rules of
combinatorial game theory (see for example the marvelous series of four
books Winning Ways for your Mathematical Plays. It turns out that this CG-value is no other than $?(x)$
… And, Conway has a much improved depiction of the devil-staircase in
his book On Numbers And Games

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noncommutative geometry : a medieval science?

According to a science article in the New York Times, archeologists have discovered “signs of advanced math” in medieval mosaics. An example of a quasi-crystalline Penrose pattern was found at the Darb-i Imam shrine in Isfahan, Iran.

“A new study shows that the Islamic pattern-making process, far more intricate than the laying of one‚Äôs bathroom floor, appears to have involved an advanced math of quasi crystals, which was not understood by modern scientists until three decades ago. Two years ago, Peter J. Lu, a doctoral student in physics at Harvard University, was transfixed by the geometric pattern on a wall in Uzbekistan. It reminded him of what mathematicians call quasi-crystalline designs. These were demonstrated in the early 1970s by Roger Penrose, a mathematician and cosmologist at the University of Oxford. Mr. Lu set about examining pictures of other tile mosaics from Afghanistan, Iran, Iraq and Turkey, working with Paul J. Steinhardt, a Princeton cosmologist who is an authority on quasi crystals and had been Mr. Lu’s undergraduate adviser. ”

Penrose tilings are one of the motivating examples of Alain Connes’ book as there is a $C^* $-algebra associated to it. In fact, the algebra is AF ( a limit of semi-simple finite dimensional algebras) so is even a formally smooth algebra in Kontsevichian noncommutative geometry (it is remarkable how quickly one gets used to silly terminology…). However, the Penrose algebra is simple, so rather useless from the point of view of finite dimensional representations… Still, Connesian noncommutative geometry may be a recent incarnation of the medieval Tehran program (pun intended). Thanks to easwaran for the link (via Technorati).

Added, March 1 : I haven’t looked at the Connes-Marcolli paper A walk in the noncommutative garden for a while but now that I do, I see that they mentioned the above already at the end of their section on Tilings (page 32). They also include clearer pictures.

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time for selfcriticism

problem with criticizing others is that you have to apply the same
standards to your own work. So, as of this afternoon, I do agree with
all those who said so before : my book is completely unreadable and
should either be dumped or entirely rewritten!

Here’s what happened :
Last week I did receive the contract to publish _noncommutative
geometry@n_ in a reputable series. One tiny point though, the editors
felt that the title was somewhat awkward and would stand out with
respect to the other books in the series, so they proposed as an
alternative title _Noncommutative Geometry_. A tall order, I thought,
but then, if others are publishing books with such a title why
shouldn’t I do the same?

The later chapters are quite general, anyway,
and if I would just spice them up a little adding recent material it
might even improve the book. So, rewriting two chapters and perhaps
adding another “motivational chapter” aimed at physicists… should
be doable in a month, or two at the latest which would fit in nicely
with the date the final manuscript is due.

This week, I got myself once
again in writing mode : painfully drafting new sections at a pace of 5
to 6 pages a day. Everything was going well. Today I wanted to finish
the section on the “one quiver to rule them all”-trick and was
already mentally planning the next section in which I would give details
for groups like $PSL_2(\mathbb{Z}) $ and $GL_2(\mathbb{Z}) $, all I
needed was to type in a version of the proof of the last proposition.

The proof uses a standard argument, which clearly should be in the book
so I had to give the correct reference and started browsing through the
print-out of the latest version (about 600 pages long..) but… _I
could not find it!???_ And, it was not just some minor technical lemma,
but a result which is crucial to the book’s message (for the few who
want to know, the result is the construction and properties of the local
quiver at a semi-simple representation of a Quillen-smooth algebra). Of
course, there is a much more general result contained in the book, but
you have to be me (or have to be drilled by me) to see the connection…
Not good at all! I’d better sleep on this before taking further

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wednesday bookshelf

The Newtonian
is a free online book by Nick Evans. If you want a (somewhat
over-positively) review about it, head here. For me, it just didn’t work but then I’m
an avid consumer of thrillers and like to be surprised. I read though to
page 80 (of 138) which is pretty good as I usually lay down an
unreadable book much sooner (most are lend from the public library, so
I’d rather start a new one as soon as it becomes tiresome). But then I
wanted to ‘learn about the frontier of particle physics within a fast
paced crime adventure’. One exchange is pretty telling though (p.

“Rule is don”t get involved in popularising
science.” “That”s a bit harsh. We have to tell people what we”re doing
if we want them to fund us,” said Carl. “Yeah, but it”s all lies. The
theories are all mathematics, yet we say all words. That”s why we get
these nut cases writing to us sometimes with “new theoriesiÃÇ. They”ve
just read a popular science book and mistaken it for the core of
science. They change some words and think they have a new theory. They
don”t understand that you have to get the numbers right for every
experiment you can think of. It”s just misleading.” “Well, OK,”conceded
Carl, “but you can make the point that you”re just reporting a
simplification and that if people want the real thing, they”d need
maths.”Everyone sought inspiration in their pints again.

So, that’s why it is impossible to write about mathematics for a
general public! Anyway, a bit frustrated I went to my favorite bookshops
and read a blurb which sounded all too familiar.

Oxford, 2006: a young woman is found brutally
murdered, her throat cut. Her heart has been removed and in its place
lies an apparently ancient gold coin. Twenty-four hours later, another
woman is found. The MO is identical, except that this time her brain has
been removed, and a silver coin lies glittering in the bowl of her
skull. The police are baffled but when police photographer, Philip
Bainbridge and his estranged lover, Laura Niven become involved, they
discover that these horrific, ritualistic murders are not confined to
the here and now. And a shocking story begins to emerge which
intertwines Sir Isaac Newton, one of seventeenth-century England’s most
powerful figures, with a deadly conspiracy which echoes down the years
to the present day, as lethal now as it was then. Before long those
closest to Laura are in danger, and she finds herself the one person who
can rewrite history; the only person who can stop the killer from
striking again…

The first half of the
is rather promising (and at least its well written).
Mathematics even makes a short appearance as daugther Jo is studying
maths. Not that she contributes much of her talent to the story (apart
from contributing to solving one riddle) and in the end it seems to be
much more important to date the right boyfriend than to do math! The
book really becomes laughable when the couple is trapped in a maze 100
feat below the Bodleian library. Pure Indiana Jones-remake : expecting
pitfalls? here they come! what else do we remember of the movie?
arrow-traps! sure enough they appear. Oh, this must be that movie, so
whats next? well, sure enough (p. 343 “A massive block of stone crashed
down from the lintel of the archway, landing squarely with a thump on
the dusty floor. They were sealed in.” Presumably the authors
inspiration dried out!

Mind you, I do not want to be negative on
principle. Sometimes (too rare) I do read an enjoyable, interesting,
intelligent thriller. The last one Ive read was

Brother Grimm by Craig Russell is a must read if you are into
serial-killer stories. Heres the synopsis

A girl’s
body lies, posed, on the pale sand of a Hamburg beach, a message
concealed in her hand. ‘I have been underground, and now it is
time for me to return home…’ Jan Fabel, of the Hamburg murder
squad, struggles to interpret the twisted imagery of a dark and brutal
mind. Four days later, a man and a woman are found deep in woodland,
their throats slashed deep and wide, the names ‘Hansel’ and
‘Gretel’, in the same, tiny, obsessively neat writing, rolled
tight and pressed into their hands. As it becomes clear that each new
crime is a grisly reference to folk stories collected almost two hundred
years ago by the Brothers Grimm, the hunt is on for a serial killer who
is exploring our darkest, most fundamental fears – a predator who kills
and then disappears into the shadows. He is a monster we all learned to
fear in childhood.

An original point of view, an
unorthodox setting (Hamburg!), interesting and real life personages what
more do you want? Oh, you want to learn something from reading a
bestseller? To me, this book was an eye-opener. Ever wondered why all
these serial-killer-books become bestsellers? Here’s the answer (p.


p>”Weiss toke a novel from the bookshell before him!! ‘Today
we continuously reinvent these tales. The same stories, new characters.
This is a bestseller – a story about the hunt for a serial killer who
ritually dismembers his victims. These are our fairy tales
today. These are our fables, our Maerchen. Instead of elves and kobolts
and hungry wolves lurking in the dark corners of the woods, we have
cannibals and dissectors and abductors lurking in the dark corners of
our cities.
It is in our nature to guise our evil as something
extraordinary or something different: books and films about aliens,
sharks, vampires, ghosts, witches. The fact of the matter is that there
is one beast that is more dangerous, more predatory, tan any other in
the history of nature. Us.”

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way too ambitious

Student-evaluation sneak preview : I am friendly and
extremely helpful but have a somewhat chaotic teaching style and am way
too ambitious as regards content… I was about to deny vehemently
all assertions (except for the chaotic bit) but may have to change my
mind after reading this report on
Mark Rowan’s book ‘Symmetry and the monster’ (see also
my post

Oxford University Press considers this book
“a must-read for all fans of popular science”. In his blog,
Lieven le Bruyn, professor of algebra and geometry at the University of
Antwerp, suggests that “Mark Ronan has written a beautiful book
intended for the general public”. However, he goes on to say:
“this year I’ve tried to explain to an exceptionally
good second year of undergraduates, but failed miserably Perhaps
I’ll give it another (downkeyed) try using Symmetry and the
Monster as reading material”.

As an erstwhile
mathematician, I found the book more suited to exceptional maths
undergraduates than to the general public and would strongly encourage
authors and/or publishers to pass such works before a few fans of
popular science before going to press.

Peggie Rimmer,

Well, this ‘exceptionally good
year’ has moved on and I had to teach a course ‘Elementary
Algebraic Geometry’ to them last semester. I had the crazy idea to
approach this in a historical perspective : first I did the
Hilbert-Noether period (translating geometry to ideal theory of
polynomial rings), then the Krull-Weil-Zariski period (defining
everything in terms of coordinate rings) to finish off with the
Serre-Grothendieck period (introducing scheme theory)… Not
surprisingly, I lost everyone after 1920. Once again there were
complaints that I was expecting way too much from them etc. etc. and I
was about to apologize and promise I’ll stick to a doable course
next year (something along the lines of Miles Reid’s
‘Undergraduate Algebraic Geometry’) when one of the students
(admittedly, probably the best of this ‘exceptional year’)
decided to do all exercises of the first two chapters of Fulton’s
‘Algebraic Curves’ to become more accustomed to the subject.
Afterwards he told me “You know, I wouldn’t change the
course too much, now that I did all these exercises I realize that your
course notes are not that bad after all…”. Yeah, thanks!

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the secret life of numbers

Just read/glanced through another math-for-the-masses book : [The secret life of numbers]( by [George G.
Szpiro]( The subtitle made me buy the book : **50 easy pieces on how
mathematicians work and think** Could be fun, I thought, certainly after
reading the Amazon-blurb :

Most of us picture
mathematicians laboring before a chalkboard, scribbling numbers and
obscure symbols as they mutter unintelligibly. This lighthearted (but
realistic) sneak-peak into the everyday world of mathematicians turns
that stereotype on its head. Most people have little idea what
mathematicians do or how they think. It’s often difficult to see how
their seemingly arcane and esoteric work applies to our own everyday
lives. But mathematics also holds a special allure for many people. We
are drawn to its inherent beauty and fascinated by its complexity – but
often intimidated by its presumed difficulty. \”The Secret Life of
Numbers\” opens our eyes to the joys of mathematics, introducing us to
the charming, often whimsical side, of the

Please correct me when I’m wrong,
but I found just one out of 50 pieces which remotely fulfills this
promise : ‘Cozy Zurich’ ((on the awesome technical support a lecturer
in Zurich is rumoured to receive)). Still, there are some other pieces
worth reading, 1. ‘A puzzle by any other name’ ((On the
Collatz problem)) 2. ‘Twins, cousins and sexy primes’ ((How
reasearch into the twin primes problem led to the discovery of a
Pentium-bug)) 3. ‘Proving the proof’ ((On Kepler’s problem)) 4.
‘Has Poincare’s conjecture finally been solved’ ((Of course it has
been)) 5. ‘Late tribute to a tragic hero’ ((On Abel’s life and
prize)) 6. ‘God’s gift to science?’ ((Stephen Wolfram
bashing)) to single out a few, embedded in a soup made out of the
usual suspects (knots, chaos, RSA etc.). But, all in all, I fear the
book doesn’t fulfill its promises and once again it demonstrates how
little ‘math-substance’ one is able to put in a book for a general
audience. But let us end with a quote from the preface that I really
like :

Whenever a socialite shows off his flair
at a coctail party by reciting a stanza from an obscure poem, he is
considered well-read and full of wit. Not much ado can be made with the
recitation of a mathematical formula, however. At most, one may expect a
few pitying glances and the title ‘party’s most nerdy guest’. To the
concurring nods of the cocktail crowd, most bystanders will admit that
they are no good at math, never have been, and never will be.
Actually, this is quite astonishing. Imagine your lawyer
telling you that he is no good at spelling, your dentist proudly
proclaiming that she speaks n foreign language, and your financial
advisor admitting with glee that he always mixes up Voltaire with
Moliere. With ample reason one would consider such people as ignorant.
Not so with mathematics. Shortcomings in this intellectual discipline
are met with understanding by everyone.

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Shamelessly (if that is a proper word in english/american e.? it should
be…) copied from ‘view source’ from Uncertain Principlesdelurk_terr.jpgJanet reminds me that this has
been declared National De-Lurking Week. If you’re in
the habit of reading this blog, but don’t usually comment,
here’s a made-up holiday you can celebrate by leaving a comment
here. You’ll need to put in a name (it needn’t be yours) and
an email address (I promise it won’t be spammed as a result), but
then you can type anything you like (within reason) into the comment
box, and post it here.

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attention-span : one chat line

spend so much time on teaching than this semester and never felt so
depressed afterwards. The final test for the first year course on
grouptheory (60 hrs. going from nothing to Jordan-Holder and the Sylow
theorems) included the following question :

Question :
For a subgroup $H \subset G $ define the normalizer to be the
subgroup $N_G(H) = \{ g \in G~:~gHg^{-1} = H \} $. Complete the
statement of the result for which the proof is given

theorem : Let P be a Sylow subgroup of
a finite group G and suppose that H is a subgroup of G which
contains the normalizer $N_G(P) $. Then …

proof :
Let $u \in N_G(H) $. Now, $P \subset N_G(P) \subset H $
whence $uPu^{-1} \subset uHu^{-1} = H $. Thus, $uPu^{-1} $, being of the
same order as P is also a Sylow subgroup op H. Applying the Sylow
theorems to H we infer that there exists an element $h \in H $ such
$h(uPu^{-1})h^{-1} = P $. This means that $hu \in N_G(P) $.
Since, by hypotheses, $N_G(P) \subset H $, it follows that $hu \in H $.
As $h \in H $ it follows that $u \in H $, finishing the proof.

majority of the students was unable to do this… Sure, the result was
not contained in their course-notes (if it were I\’m certain all of them
would be able to give the correct statement as well as the full proof
by heart. It makes me wonder how much they understood
of the proof of the Sylow-theorems.) They (and others) blame it on the
fact that not every triviality is spelled out in my notes or on my
\’chaotic\’ teaching-style. I fear the real reason is contained in the

But, I\’m still lucky to be working with students
who are interested in mathematics. I assume it can get a lot worse (but
also a lot funnier)

and what about this one :

If you are (like me) in urgent need for a smile, try out
this newsvine article for more

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mathematics & unhappiness

Sociologists are a constant source of enlightenment as CNN keeps reminding

Kids who are turned
off by math often say they don’t enjoy it, they aren’t good
at it and they see little point in it. Who knew that could be a formula
for success?
The nations with the best scores have the
least happy, least confident math students, says a study by the
Brookings Institution’s Brown Center on Education Policy.
Countries reporting higher levels of enjoyment and confidence
among math students don’t do as well in the subject, the study
The eighth-grade results reflected a common
pattern: The 10 nations whose students enjoyed math the most all scored
below average. The bottom 10 nations on the enjoyment scale all

As this study is based on the 2003 Trends in International
Mathematics and Science Studies
and as “we” scored best
of all western countries
probably explains all the unhappy faces in my first-year class on group
theory. However, they seemed quite happy the first few weeks.
Fortunately, this is proof, at least according to the mountain of wisdom, that I’m on the right track

If too many students are too happy in the math
classes, be sure that it is simply because not much is expected from
them. It can’t be otherwise. If teaching of mathematics is
efficient, it is almost guaranteed that a large group or a majority must
dislike the math classes. Mathematics is hard and if it is not hard, it
is not mathematics.

Right on! But then, why is
it that people willing to study maths enter university in a happy mood?
Oh, I get it, yes, it must be because in secondary school not much was
expected of them! Ouf! my entire world is consistent once again.
But then, hey wait, the next big thing that’s inevitably going to
happen is that in 2007 “we” will be tumbling down this world
ranking! And, believe it or not, that is precisely what
all my colleagues are eagerly awaiting to happen. Most of us are willing
to bet our annual income on it. Belgium was among the first countries to
embrace in the sixties-early seventies what was then called
“modern mathematics’ (you know: Venn-diagrams, sets,
topology, categories (mind you, just categories not the n-stuff ) etc.) Whole
generations of promising Belgian math students were able in the late
70ties, 80ties and early 90ties to do what they did mainly because of
this (in spite of graduating from ‘just’ a Belgian
university, only some of which make it barely in the times top 100 ). But
then, in the ’90ties politicians decided that mathematics had to
be sexed-up, only the kind of mathematics that one might recognize in
everyday life was allowed to be taught. For once, I have to
agree with motl.

Also, the attempts to connect mathematics with
the daily life are nothing else than a form of lowering of the
standards. They are a method to make mathematics more attractive for
those who like to talk even if they don’t know what they’re
talking about. They are a method to include mathematics between the
social and subjective sciences. They give a wiggle room to transform
happiness, confidence, common sense, and a charming personality into
good grades.

Indeed, the major problem we are
facing today in first year classes is that most students have no formal
training at all! An example : last week I did a test after three weeks
of working with groups. One of the more silly questions was to ask them
for precise definitions of very basic concepts (groups, subgroups,
cyclic groups, cosets, order of an element) : just 5 out of 44 were able
to do this! Most of them haven’t heard of sets at all. It seems
that some time ago it was decided that sets no longer had a place in
secondary school, so just some of them had at least a few lessons on
sets in primary school (you know the kind (probably you won’t but
anyway) : put all the green large triangles in the correct place in the
Venn diagram and that sort of things). Now, it seems that politicians
have decided that there is no longer a place for sets in primary schools
either! (And if we complain about this drastic lowering of
math-standards in schools, we are thrown back at us this excellent 2003
international result, so the only hope left for us is that we will fall
down dramatically in the 2007 test.) Mind you, they still give
you an excellent math-education in Belgian primary and secondary schools
provided you want to end up as an applied mathematician or (even worse)
a statistician. But I think that we, pure mathematicians, should
seriously consider recruiting students straight from Kindergarten!

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