Skip to content →

Tag: teaching

Seriously now, where was the Bourbaki wedding?

A few days before Halloween, Norbert Dufourcq (who died december 17th 1990…), sent me a comment, containing lots of useful information, hinting I did get it wrong about the church of the Bourbali wedding in the previous post.

Norbert Dufourcq, an organist and student of Andre Machall, the organist-in-charge at the Saint-Germain-des-Prés church in 1939, the place where I speculated the Bourbaki wedding took place, concluded his comment with :

“P.S. Lieven, you _do_ know about the Schola Cantorum, now, don’t you?!?”.

Euh… actually … no, I did not …

La Schola Cantorum is a private music school in Paris. It was founded in 1894 by Charles Bordes, Alexandre Guilmant and Vincent d’Indy as a counterbalance to the Paris Conservatoire’s emphasis on opera. Its alumni include many significant figures in 20th century music, such as Erik Satie and Cole Porter.

Schola Cantorum is situated 69, rue Saint Jacques, Paris, just around the corner of the Ecole Normal Superieure, home base to the Bourbakis. In fact, closer investigation reveals striking similarities and very close connections between the circle of artists at la Schola and the Bourbaki group.

In december 1934, the exact month the Bourbaki group was formed, a radical reorganisation took place at the Schola, when Nestor Lejeune became the new director. He invited several young musicians, many from the famous Dukas-class, to take up teaching positions at the Schola.

Here’s a picture of part of the Dukas class of 1929, several of its members will play a role in the upcoming events :
from left to right next to the piano : Pierre Maillard-Verger, Elsa Barraine, Yvonne Desportes, Tony Aubin, Pierre Revel, Georges Favre, Paul Dukas, René Duclos, Georges Hugon, Maurice Duruflé. Seated on the right : Claude Arrieu, Olivier Messiaen.

The mid-1930s in Paris saw the emergence of two closely-related groups with a membership which overlapped : La Spirale and La Jeune France. La Spirale was founded in 1935 under the leadership of Georges Migot; its other committee members were Paul Le Flem, his pupil André Jolivet, Edouard Sciortino, Claire Delbos, her husband Olivier Messiaen, Daniel-Lesur and Jules Le Febvre. The common link between almost all of these musicians was their connection with the Schola Cantorum.

On the left : Les Jeunes Musiciens Français : André Jolivet on the Piano. Standing from left to right :
Olivier Messiaen, Yves Baudrier, Daniel-Lesur.

Nigel Simeone wrote this about Messiaen and La Jeune France :
“The extremely original and independent-minded Messiaen had already shown himself to be a rather unexpected enthusiast for joining groups: in December 1932 he wrote to his friend Claude Arrieu about a letter from another musician, Jacques Porte, outlining plans for a new society to be called Les Jeunes Musiciens Français.
Messiaen agreed to become its vice-president, but nothing seems to have come of the project. Six months later, in June 1933, he had a frustrating meeting with Roger Désormière on behalf of the composers he described to Arrieu as ‘les quatre’, all of them Dukas pupils: Elsa Barraine, the recently-deceased Jean Cartan, Arrieu and Messiaen himself; during the early 1930s Messiaen and Arrieu organised concerts featuring all four composers.”

Finally, we’re getting a connection with the Bourbaki group! Norbert Dufourcq mentioned it already in his comment “Messiaen was also a good friend of Jean Cartan (himself a composer, and Henri’s brother)”. Henri Cartan was one of the first Bourbakis and an excellent piano player himself.

The Cartan family picture on the right : standing from left to right, father Elie Cartan (one of the few older French mathematicians respected by the Bourbakis), Henri and his mother Marie-Louise. Seated, the younger children, from left to right : Louis, Helene (who later became a mathematician, herself) and the composer Jean Cartan, who sadly died very young from tuberculoses in 1932…

The december 1934 revolution in French music at the Schola Cantorum, instigated by Messiaen and followers, was the culmination of a process that started a few years before when Jean Cartan was among the circle of revolutionados. Because Messiaen was a fiend of the Cartan family, they surely must have been aware of the events at the Schola (or because it was merely a block away from the ENS), and, the musicians’ revolt may very well have been an example to follow for the first Bourbakis…(?!)

Anyway, we now know the intended meaning of the line “with lemmas sung by the Scholia Cartanorum” on the wedding-invitation. Cartanorum is NOT (as I claimed last time) bad Latin for ‘Cartesiorum’, leading to Descartes and the Saint-Germain-des-Pres church, but is in fact passable Latin (plur. gen.) of CARTAN(us), whence the translation “with lemmas sung by the school of the Cartans”. There’s possibly a double pun intended here : first, a reference to (father) Cartan’s lemma and, of course, to La Schola where the musical Cartan-family felt at home.

Fine, but does this brings us any closer to the intended place of the Bourbaki-Petard wedding? Well, let’s reconsider the hidden ‘clues’ we discovered last time : the phrase “They will receive the trivial isomorphism from P. Adic, of the Order of the Diophantines” might suggest that the church belongs to a a religious order and is perhaps an abbey- or convent-church and the phrase “the organ will be played by Monsieur Modulo” requires us to identify this mysterious Mister Modulo, because Norbert Dufourcq rightfully observed :

“note however that in 1939, it wasn’t as common to have a friend-organist perform at a wedding as it is today: the appointed organists, especially at prestigious Paris positions, were much less likely to accept someone play in their stead.”

The history of La Schola Cantorum reveals something that might have amused Frank Smithies (remember he was one of the wedding-invitation-composers) : the Schola is located in the Convent(!) of the Brittish Benedictines…

In 1640 some Benedictine monks, on the run after the religious schism in Britain, found safety in Paris under the protection of Cardinal Richelieu and Anne of Austria at Val-de-Grace, where the Schola is now housed.

As is the case with most convents, the convent of the Brittish Benedictines did have its own convent church, now called l’église royale Notre-Dame du Val-de-Grâce (remember that one of the possible interpretations for “of the universal variety” was that the name of the church would be “Notre-Dame”…).

This church is presently used as the concert hall of La Schola and is famous for its … musical organ : “In 1853, Aristide Cavaillé-Coll installed a new organ in the Church of Sainte-geneviève which had been restored in its rôle as a place of worship by Prince President Louis-Napoléon. In 1885, upon the decision of President Jules Grévy, this church once again became the Pantheon and, six years later, according to an understanding between the War and Public Works Departments, the organ was transferred to the Val-de-Grâce, under the supervision of the organ builder Merklin. Beforehand, the last time it was heard in the Pantheon must have been for the funeral service of Victor Hugo.
In 1927, a raising was carried out by the builder Paul-Marie Koenig, and the inaugural concert was given by André Marchal and Achille Philippe, the church’s organist. Added to the register of historic monument in 1979, Val-de-Grâce’s “ little great organ ”, as Cavaillé-Coll called it, was restored in 1993 by the organ builders François Delangue and Bernard Hurvy.
The organ of Val-de-Grâce is one the rare parisian surviving witnesses of the art of Aristide Cavaillé-Coll, an instrument that escaped abusive and definitive transformations or modernizations. This explain why, in spite of its relatively modest scale, this organ enjoys quite a reputation, and this, as far as the United States.”

By why would the Val-de-Grace organiste at the time Achille Philip, “organiste titulaire du Val-de-Grâce de 1903 à 1950 et professeur d’orgue et d’harmonie à la Schola Cantorum de 1904 à 1950”, be called ‘Mister Modulo’ in the wedding-invitations line “L’orgue sera tenu par Monsieur Modulo”???

Again, the late Norbert Dufourcq comes to our rescue, proposing a good candidate for ‘Monsieur Modulo’ : “As for “modulo”, note that the organist at Notre-Dame at that time, Léonce de Saint-Martin, was also the composer of a “Suite Cyclique”, though I admit that this is just wordplay: there is nothing “modular” about this work. Maybe a more serious candidate would be Olivier Messiaen (who was organist at the Église de la Trinité): his “modes à transposition limitée” are really about Z/12Z→Z/3Z and Z/12Z→Z/4Z. “

Messiaen’s ‘Modes of limited transposition’ were compiled in his book ‘Technique de mon langage musical’. This book was published in Paris by Leduc, as late as 1944, 5 years after the wedding-invitation.

Still, several earlier works of Messiaen used these schemes, most notably La Nativité du Seigneur, composed in 1935 : “The work is one of the earliest to feature elements that were to become key to Messiaen’s later compositions, such as the extensive use of the composer’s own modes of limited transposition, as well as influence from birdsong, and the meters and rhythms of Ancient Greek and traditional Indian music.”

More details on Messiaen’s modes and their connection to modular arithmetic can be found in the study Implementing Modality in Algorithmic Composition by Vincent Joseph Manzo.

Hence, Messiaen is a suitable candidate for the title ‘Monsieur Modulo’, but would he be able to play the Val-de-Grace organ while not being the resident organist?

Remember, the Val-de-Grace church was the concert hall of La Schola, and its musical organ the instrument of choice for the relevant courses. Now … Olivier Messiaen taught at the Schola Cantorum and the École Normale de Musique from 1936 till 1939. So, at the time of the Bourbaki-Petard wedding he would certainly be allowed to play the Cavaillé-Coll organ.

Perhaps we got it right, the second time around : the Bourbaki-Pétard wedding was held on June 3rd 1939 in the church ‘l’église royale Notre-Dame du Val-de-Grâce’ at 12h?


Grothendieck’s functor of points

A comment-thread well worth following while on vacation was Algebraic Geometry without Prime Ideals at the Secret Blogging Seminar. Peter Woit became lyric about it :

My nomination for the all-time highest quality discussion ever held in a blog comment section goes to the comments on this posting at Secret Blogging Seminar, where several of the best (relatively)-young algebraic geometers in the business discuss the foundations of the subject and how it should be taught.

I follow far too few comment-sections to make such a definite statement, but found the contributions by James Borger and David Ben-Zvi of exceptional high quality. They made a case for using Grothendieck’s ‘functor of points’ approach in teaching algebraic geometry instead of the ‘usual’ approach via prime spectra and their structure sheaves.

The text below was written on december 15th of last year, but never posted. As far as I recall it was meant to be part two of the ‘Brave New Geometries’-series starting with the Mumford’s treasure map post. Anyway, it may perhaps serve someone unfamiliar with Grothendieck’s functorial approach to make the first few timid steps in that directions.

Allyn Jackson’s beautiful account of Grothendieck’s life “Comme Appele du Neant, part II” (the first part of the paper can be found here) contains this gem :

“One striking characteristic of Grothendieck’s
mode of thinking is that it seemed to rely so little
on examples. This can be seen in the legend of the
so-called “Grothendieck prime”.

In a mathematical
conversation, someone suggested to Grothendieck
that they should consider a particular prime number.
“You mean an actual number?” Grothendieck
asked. The other person replied, yes, an actual
prime number. Grothendieck suggested, “All right,
take 57.”

But Grothendieck must have known that 57 is not
prime, right? Absolutely not, said David Mumford
of Brown University. “He doesn’t think concretely.””

We have seen before how Mumford’s doodles allow us to depict all ‘points’ of the affine scheme $\mathbf{spec}(\mathbb{Z}[x]) $, that is, all prime ideals of the integral polynomial ring $\mathbb{Z}[x] $.
Perhaps not too surprising, in view of the above story, Alexander Grothendieck pushed the view that one should consider all ideals, rather than just the primes. He achieved this by associating the ‘functor of points’ to an affine scheme.

Consider an arbitrary affine integral scheme $X $ with coordinate ring $\mathbb{Z}[X] = \mathbb{Z}[t_1,\ldots,t_n]/(f_1,\ldots,f_k) $, then any ringmorphism
$\phi~:~\mathbb{Z}[t_1,\ldots,t_n]/(f_1,\ldots,f_k) \rightarrow R $
is determined by an n-tuple of elements $~(r_1,\ldots,r_n) = (\phi(t_1),\ldots,\phi(t_n)) $ from $R $ which must satisfy the polynomial relations $f_i(r_1,\ldots,r_n)=0 $. Thus, Grothendieck argued, one can consider $~(r_1,\ldots,r_n) $ an an ‘$R $-point’ of $X $ and all such tuples form a set $h_X(R) $ called the set of $R $-points of $X $. But then we have a functor

$h_X~:~\mathbf{commutative rings} \rightarrow \mathbf{sets} \qquad R \mapsto h_X(R)=Rings(\mathbb{Z}[t_1,\ldots,t_n]/(f_1,\ldots,f_k),R) $

So, what is this mysterious functor in the special case of interest to us, that is when $X = \mathbf{spec}(\mathbb{Z}[x]) $?
Well, in that case there are no relations to be satisfied so any ringmorphism $\mathbb{Z}[x] \rightarrow R $ is fully determined by the image of $x $ which can be any element $r \in R $. That is, $Ring(\mathbb{Z}[x],R) = R $ and therefore Grothendieck’s functor of points
$h_{\mathbf{spec}(\mathbb{Z}[x]} $ is nothing but the forgetful functor.

But, surely the forgetful functor cannot give us interesting extra information on Mumford’s drawing?
Well, have a look at the slightly extended drawing below :

What are these ‘smudgy’ lines and ‘spiky’ points? Well, before we come to those let us consider the easier case of identifying the $R $-points in case $R $ is a domain. Then, for any $r \in R $, the inverse image of the zero prime ideal of $R $ under the ringmap $\phi_r~:~\mathbb{Z}[x] \rightarrow R $ must be a prime ideal of $\mathbb{Z}[x] $, that is, something visible in Mumford’s drawing. Let’s consider a few easy cases :

For starters, what are the $\mathbb{Z} $-points of $\mathbf{spec}(\mathbb{Z}[x]) $? Any natural number $n \in \mathbb{Z} $ determines the surjective ringmorphism $\phi_n~:~\mathbb{Z}[x] \rightarrow \mathbb{Z} $ identifying $\mathbb{Z} $ with the quotient $\mathbb{Z}[x]/(x-n) $, identifying the ‘arithmetic line’ $\mathbf{spec}(\mathbb{Z}) = { (2),(3),(5),\ldots,(p),\ldots, (0) } $ with the horizontal line in $\mathbf{spec}(\mathbb{Z}[x]) $ corresponding to the principal ideal $~(x-n) $ (such as the indicated line $~(x) $).

When $\mathbb{Q} $ are the rational numbers, then $\lambda = \frac{m}{n} $ with $m,n $ coprime integers, in which case we have $\phi_{\lambda}^{-1}(0) = (nx-m) $, hence we get again an horizontal line in $\mathbf{spec}(\mathbb{Z}[x]) $. For $ \overline{\mathbb{Q}} $, the algebraic closure of $\mathbb{Q} $ we have for any $\lambda $ that $\phi_{\lambda}^{-1}(0) = (f(x)) $ where $f(x) $ is a minimal integral polynomial for which $\lambda $ is a root.
But what happens when $K = \mathbb{C} $ and $\lambda $ is a trancendental number? Well, in that case the ringmorphism $\phi_{\lambda}~:~\mathbb{Z}[x] \rightarrow \mathbb{C} $ is injective and therefore $\phi_{\lambda}^{-1}(0) = (0) $ so we get the whole arithmetic plane!

In the case of a finite field $\mathbb{F}_{p^n} $ we have seen that there are ‘fat’ points in the arithmetic plane, corresponding to maximal ideals $~(p,f(x)) $ (with $f(x) $ a polynomial of degree $n $ which remains irreducible over $\mathbb{F}_p $), having $\mathbb{F}_{p^n} $ as their residue field. But these are not the only $\mathbb{F}_{p^n} $-points. For, take any element $\lambda \in \mathbb{F}_{p^n} $, then the map $\phi_{\lambda} $ takes $\mathbb{Z}[x] $ to the subfield of $\mathbb{F}_{p^n} $ generated by $\lambda $. That is, the $\mathbb{F}_{p^n} $-points of $\mathbf{spec}(\mathbb{Z}[x]) $ consists of all fat points with residue field $\mathbb{F}_{p^n} $, together with slightly slimmer points having as their residue field $\mathbb{F}_{p^m} $ where $m $ is a divisor of $n $. In all, there are precisely $p^n $ (that is, the number of elements of $\mathbb{F}_{p^n} $) such points, as could be expected.

Things become quickly more interesting when we consider $R $-points for rings containing nilpotent elements.

Leave a Comment

Don’t realize how lucky I am

After a difficult time for us all, PD1 tells me she finally ‘found her drive’ again : she hopes to finish her 2nd(!) master in fine arts this year as well as her teaching-diplome. Besides, she teaches evening arts-classes twice a week, organizes exhibitions, enters competitions, wins prizes … Looking at the time-stamps on her emails, there are simply not enough hours in a day to fulfill her many ambitions.

Yesterday she made a blitz-appearance, on her way to a variety of exciting other encounters.

PD1 : And, what about you? A lot of teaching this year?

me : Yes (sigh), the first semester is really hard. I’ve an obligatory 60 hours course in each of the three bachelor years, and two courses in the masters. Fortunately, the master-students all wanted a different topic, so they only pop in to ask questions when they get stuck with their reading courses. But still, officially I’ll be teaching 300 hours before christmas.

PD1 : Yeah, yeah, officially… But, then there are exercises and so. How much time do you really have to teach in front of a blackboard?

me : Well, let’s see. Wednesday afternoon I have the 2nd year, thursday afternoon the first and friday morning the third year.

PD1 : Is that all?

me : huh? Yes…

PD1 : Wow! You only have to teach three half days a week and can spend all your other time doing mathematics! A pretty good deal isn’t it?

me : Yeah, I guess I don’t realize often enough just how lucky I am …

One Comment

return of the cat ceilidh

I couldn’t believe my eyes. I was watching an episode of numb3rs, ‘undercurrents’ to be precise, and there it was, circled in the middle of the blackboard, CEILIDH, together with some of the key-exchange maps around it…

Only, the plot doesn’t involve any tori-crypto… okay, there is an I-Ching-coded-tattoo which turns out to be a telephone number, but that’s all. Still, this couldn’t just be a coincidence. Googling for ‘ceilidh+numbers‘ gives as top hit the pdf-file of an article Alice in NUMB3Rland written by … Alice Silverberg (of the Rubin-Silverberg paper starting tori-cryptography). Alice turns out to be one of the unpaid consultants to the series. The 2-page article gives some insight into how ‘some math’ gets into the script

Typically, Andy emails a draft of the
script to the consultants. The FBI plot
is already in place, and the writers want
mathematics to go with it. The placeholder “math” in the draft is often nonsense or
jargon; the sort of things people with no
mathematical background might find by
Googling, and think was real math. Since
there’s often no mathematics that makes
sense in those parts of the script, the best
the consultants can do is replace jargon
that makes us cringe a lot with jargon that
makes us cringe a little.

From then on, it’s the Telephone Game.
The consultants email Andy our suggestions (“replace ‘our discrete universes’
with ‘our disjoint universes'”; “replace
the nonsensical ‘we’ve tried everything
-a full frequency analysis, a Vignere
deconstruction- we even checked for
a Lucas sequence’ with the slightly less
nonsensical ‘It’s much too short to try
any cryptanalysis on. If it were longer
we could try frequency analyses, or try
to guess what kind of cryptosystem it is
and use a specialized technique. For example, if it were a long enough Vigenere
cipher we could try a Kasiski test or an
index-of-coincidence analysis’). Andy
chooses about a quarter of my sugges-
tions and forwards his interpretation
of them to the writers and producers.
The script gets changed, and then the
actors ad lib something completely dif-
ferent (‘disjointed universes’: cute, but
loses the mathematical allusion; ‘Kasiski
exam’ : I didn’t mean that kind of ‘test’).

She ends her article with :

I have mixed feelings about NUMB3RS. I still have concerns about the violence, the depiction of women, and the pretense
that the math is accurate. However, if NUMB3RS could interest people in the power of mathematics enough for society
to greater value and support mathematics teaching, learning, and research, and
motivate more students to learnthat would be a positive step.

Further, there is a whole blog dedicated to some of the maths featuring in NUMB3RS, the numb3rs blog. And it was the first time I had to take a screenshot of a DVD, something usually off limits to the, but there is a simple hack to do it…


44 32’28.29″N, 4 05’08.61″E

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.

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 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…


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!

Leave a Comment

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

Leave a Comment

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!

Leave a Comment

Krull & Paris

Category-Cafe ran an interesting post The history of n-categories
claiming that “mathematicians’ histories are largely
‘Royal-road-to-me’ accounts”

To my mind a key
difference is the historians’ emphasis in their histories that things
could have turned out very differently, while the mathematicians tend to
tell a story where we learn how the present has emerged out of the past,
giving the impression that things were always going to turn out not very
dissimilarly to the way they have, even if in retrospect the course was
quite tortuous.

Over the last weeks I’ve been writing up
the notes of a course on ‘Elementary Algebraic Geometry’ that I’ll
be teaching this year in Bach3. These notes are split into three
historical periods more or less corresponding to major conceptual leaps
in the subject : (1890-1920) ideals in polynomial rings (1920-1950)
intrinsic definitions using the coordinate ring (1950-1970) scheme
theory. Whereas it is clear to take Hilbert&Noether as the leading
figures of the first period and Serre&Grothendieck as those of the
last, the situation for the middle period is less clear to me. At
first I went for the widely accepted story, as for example phrased by Miles Reid in the
Final Comments to his Undergraduate Algebraic Geometry course.

rigorous foundations for algebraic geometry were laid in the 1920s and
1930s by van der Waerden, Zariski and Weil (van der Waerden’s
contribution is often suppressed, apparently because a number of
mathematicians of the immediate post-war period, including some of the
leading algebraic geometers, considered him a Nazi collaborator).

But then I read The Rising Sea: Grothendieck
on simplicity and generality I
by Colin McLarty and stumbled upon
the following paragraph

From Emmy Noether’s viewpoint,
then, it was natural to look at prime ideals instead of classical and
generic points—or, as we would more likely say today, to identify
points with prime ideals. Her associate Wolfgang Krull did this. He gave
a lecture in Paris before the Second World War on algebraic geometry
taking all prime ideals as points, and using a Zariski topology (for
which see any current textbook on algebraic geometry). He did this over
any ring, not only polynomial rings like C[x, y]. The generality was
obvious from the Noether viewpoint, since all the properties needed for
the definition are common to all rings. The expert audience laughed at
him and he abandoned the idea.

The story seems to be
due to Jurgen Neukirch’s ‘Erinnerungen an Wolfgang Krull’
published in ‘Wolfgang Krull : Gesammelte Abhandlungen’ (P.
Ribenboim, editor) but as our library does not have this book I would
welcome any additional information such as : when did Krull give this
talk in Paris? what was its precise content? did he introduce the prime
spectrum in it? and related to this : when and where did Zariski
introduce ‘his’ topology? Answers anyone?

Leave a Comment