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Tag: teaching

Krull & Paris

The
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?

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teaching mathematics

Tracking an email address from a subscribers’ list to the local news bulletin of a tiny village somewhere in the French mountains, I ended up at the Maths department of Wellington College.

There I found the following partial explanation as to why I find it increasingly difficult to convey mathematics to students (needless to say I got my math-education in the abstract seventies…)

“Teaching Maths in 1950:

A logger sells a truckload of lumber for £ 100. His cost of production is 4/5 of the price. What is his profit?

Teaching Maths in 1960:

A logger sells a truckload of lumber for £ 100. His cost of production is 4/5 of the price, or £80. What is his profit?

Teaching Maths in 1970:

A logger exchanges a set A of lumber for a set M of money. The cardinality of set M is 100. Each element is worth one dollar. The set C the cost of production, contains 20 fewer elements than set M. What is the cardinality of the set P of profits?

Teaching Maths in 1980:

A logger sells a truckload of lumber for £ 100. His cost of production is £80 and his profit is £20. Your assignment: Underline the number 20.

Teaching Maths in 1990:

By cutting down beautiful forest trees, the logger makes £20. What do you think of this way of making a living? How did the forest birds and squirrels feel as the logger cut down the
trees? (There are no wrong answers.)

Teaching Maths in 2000:

Employer X is at loggerheads with his work force. He gives in to union pressure and awards a pay increase of 5% above inflation for the next five years.

Employer Y is at loggerheads with his work force. He refuses to negotiate and insists that salaries be governed by productivity and market forces.

Is there a third way to tackle this problem? (Yes or No).”

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micro-sudoku

One
cannot fight fashion… Following ones own research interest is a
pretty frustrating activity. Not only does it take forever to get a
paper refereed but then you have to motivate why you do these things
and what their relevance is to other subjects. On the other hand,
following fashion seems to be motivation enough for most…
Sadly, the same begins to apply to teaching. In my Geometry 101 course I
have to give an introduction to graphs&groups&geometry. So,
rather than giving a standard intro to graph-theory I thought it would
be more fun to solve all sorts of classical graph-problems (Konigsberger
bridges
, Instant
Insanity
, Gas-
water-electricity
, and so on…) Sure, these first year
students are (still) very polite, but I get the distinct feeling that
they think “Why on earth should we be interested in these old
problems when there are much more exciting subjects such as fractals,
cryptography or string theory?” Besides, already on the first day
they made it pretty clear that the only puzzle they are interested in is
Sudoku.
Next week I’ll have to introduce groups and I was planning to do
this via the Rubik
cube
but I’ve learned my lesson. Instead, I’ll introduce
symmetry by considering micro-
sudoku
that is the baby 4×4 version of the regular 9×9
Sudoku. The first thing I’ll do is work out the number of
different solutions to micro-Sudoku. Remember that in regular Sudoku
this number is 6,670,903,752,021,072,936,960 (by a computer search
performed by Bertram
Felgenhauer
). For micro-Sudoku there is an interesting
(but ratther confused) thread on the
Sudoku forum
and after a lot of guess-work the consensus seems to be
that there are precisely 288 distinct solutions to micro-Sudoku. In
fact, this is easy to see and uses symmetry. The symmetric group $S_4$
acts on the set of all solutions by permuting the four numbers, so one
may assume that a solution is in the form where the upper-left 2×2
block is 12 and 34 and the lower right 2×2 block consists of the
rows ab and cd. One quickly sees that either this leeds to a
unique solution or so does the situation with the roles of b and c
changed. So in all there are $4! \\times \\frac{1}{2} 4!=24 \\times 12 =
288$ distinct solutions. Next, one can ask for the number of
_essentially_ different solutions. That is, consider the action
of the _Sudoku-symmetry group_ (including things such as
permuting rows and columns, reflections and rotations of the grid). In
normal 9×9 Sudoku this number was computed by Ed Russell
and Frazer Jarvis
to be 5,472,730,538 (again,heavily using the
computer). For micro-Sudoku the answer is that there are just 2
essentially different solutions and there is a short nice argument,
given by ‘Nick70′ at the end of the above mentioned thread. Looking a bit closer one verifies easily that the
two Sudoku-group orbits have different sizes. One contains 96 solutions,
the other 192 solutions. It will be interesting to find out how these
calculations will be received in class next week…

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