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 ‚ÄòL‚Äô of lumber for a set
‚ÄòM‚Äô of money. The cardinality of set ‚ÄòM‚Äô is 100. Each
element is worth one dollar. The set ‚ÄòC‚Äô, the cost of
production, contains 20 fewer elements than set ‚ÄòM‚Äô. What is
the cardinality of the set ‚ÄòP‚Äô 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).”

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…

irrelevant list

As
there is no way to recover from the previous post, allow me a slow
restart by listing some of the a-typical things done this week :

• Ate more chocolate than during the last five years

• Drove the car more than during the rest of the year (minus
vacations)

• Didn't do any bicycle exercise

• Only checked email in the morning (at best)

• Didn't do any math (apart from helping
PseudonymousDaughter2)

• Didn't go in to university at
all

• Drank even more coffee than usual

• Regardless, felt exhausted every evening

• Did far
less web-surfing (but managed to find
this

• Cooked fast and way too
cholestorol-rich meals

• Ate even more chocolates

Fortunately, the semester (and teaching)
starts tomorrow!

driven by ambition and sloth

Here’s a part of yesterday’s post by bitch ph.d. :

But first of all I have to figure out what the hell I’m going to teach my graduate students this semester, and really more to the point, what I am not going to bother to try to cram into this class just because it’s my first graduate class and I’m feeling like teaching everything I know in one semester is a realistic and desireable possibility. Yes! Here it all is! Everything I have ever learned! Thank you, and goodnight!

Ah, the perpetual motion machine of last-minute course planning, driven by ambition and sloth!.

I’ve had similar experiences, even with undergraduate courses (in Belgium there is no fixed curriculum so the person teaching the course is responsible for its contents). If you compare the stuff I hoped to teach when I started out with the courses I’ll be giving in a few weeks, you would be more than disappointed.
The first time I taught _differential geometry 1_ (a third year course) I did include in the syllabus everything needed to culminate in an outline of Donaldson’s result on exotic structures on $\mathbb{R}^4$ and Connes’ non-commutative GUT-model (If you want to have a good laugh, here is the set of notes). As far as I remember I got as far as classifying compact surfaces!
A similar story for the _Lie theory_ course. Until last year this was sort of an introduction to geometric invariant theory : quotient variety of conjugacy classes of matrices, moduli space of linear dynamical systems, Hilbert schemes and the classification of $GL_n$-representations (again, smile! here is the set of notes).
Compared to these (over)ambitious courses, next year’s courses are lazy sunday-afternoon walks! What made me change my mind? I learned the hard way something already known to the ancient Greeks : mathematics does not allow short-cuts, you cannot expect students to run before they can walk. Giving an over-ambitious course doesn’t offer the students a quicker road to research, but it may result in a burn-out before they get even started!

NOG master class update

Yesterday I made a preliminary program for the first two months
of the masterclass non-commutative geometry. It is likely that
the program will still undergo changes as at the moment I included only
the mini-courses given by Bernhard
Keller
and Markus Reineke but several other people have
already agreed to come and give a talk. For example, Jacques Alev (Reims),
Tom Lenagan (Edinburgh),
Shahn Majid (London),
Giovanna Carnovale (Padua) among others. And in
may, Fred assures me, Maxim Kontsevich will give a couple of talks. />
As for the contents of the two courses I will be
teaching I changed my mind slightly. The course non-commutative
geometry
I teach jointly with Markus Reineke and making the program
I realized that I have to teach the full 22 hours before he will start
his mini-course in the week of March 15-19 to explain the few
things
he needs, like :

To derive all the
counting of points formulas, I only need from your course:

the definition of formally smooth algebras basic properties, like
being
hereditary
– the definition of the component
semigroup
– the fact that dim Hom-dim Ext is constant along
components. This I need
even over finite fields $F_q$, but I
went through your proof in “One quiver”,
and it works. The
key fact is that even over $F_q$, the infinitesimal lifting /> property implies smoothness in the sense Dimension of variety =
dimension of
(schematic) tangent space in any $F_q$-valued
point. But I think it’s fine for
the students if you do all
this over C, and I’ll only sketch the (few)
modifications for
algebras over $F_q$.

So my plan is to do all of
this first and leave the (to me) interesting problem of trying to
classify formally smooth algebras birationally to the second
course projects in non-commutative geometry which fits the title
as a lot of things still need to be done. The previous idea to give in
that course applications of non-commutative orders to the resolution of
singularities (in particular of quotient singularities) as very roughly
explained in my three talks on non-commutative geometry@n I now
propose to relegate to the friday afternoon seminar. I’ll be
happy to give more explanations on all this (in particular more
background on central simple algebras and the theory of (maximal)
orders) if other people work through the main part of the paper in the
seminar. In fact, all (other) suggestions for seminar-talks are welcome