In the
[previous post](http://www.neverendingbooks.org/index.php?p=309) we have
seen that it is important to have lots of mobile pieces around in the
endgame and that it is hard for a computer-program to evaluate a
position correctly. In fact, we illustrated this with a position which
‘clearly’ looks much better for Black (the computer) whereas it is
already lost! In fact, the computer lost this particular game already 7
plies earlier. Consider the position

$\xymatrix@=.3cm @!C @R=.7cm{.& & & & & & & & & & & & & \\ & & & \SBlack \connS & & \bull{d}{5} \conn & & \bull{e}{5} \conn & & \bull{f}{5} \conn & & \bull{g}{5} \conn & & \bull{h}{5} \conn & & \SWhite \connS & & \SWhite \connS & & \SWhite \conneS & & & \\ & & \SBlack \connS & & \SBlack \connS & & \Black{6} \connS & & \bull{e}{4} \conn& & \bull{f}{4} \conn & & \bull{g}{4} \conn & & \bull{h}{4} \conn & & \SWhite \connS & & \SWhite \connS & & \SWhite \conneS & & \\ & \SBlack \connbeginS & & \SBlack \connS & & \BDvonn{2} \connS & & \bull{d}{3} \conn & & \SBlack \connS & & \BDvonn{3} \connS & & \White{4} \connS & & \SWhite \connS & & \Dvonn \connS & & \SWhite \connS & & \SWhite \connendS & . \\ & & \Black{5} \connbeginS & & \SBlack \connS & & \SBlack \connS & & \bull{d}{2} \conn & & \SBlack \connS & & \bull{f}{2} \conn & & \bull{g}{2} \conn & & \SWhite \connS & & \SWhite \connS & & \SWhite \connendS & & \\ & & & \bull{a}{1} \con & & \bull{b}{1} \con & & \Black{5} \conS & & \bull{d}{1} \con & & \bull{e}{1} \con & & \bull{f}{1} \con & & \bull{g}{1} \con & & \bull{h}{1} \con & & \White{2} & & & \\ .& & & & & & & & & & & & & }$

Probably, Black lost the
game with its last move d1-f3 thereby disconnecting its pieces into two
clusters. White (the human player) must already have realized at this
moment he had a good chance of winning (as indicated in the previous
post) by letting Black run out of moves by building large stacks on the
third row, White building a stack of the appropriate size which then
jumps on the largest Black stack on the final move. Btw. this technique
is called *sharpshooting* in Dvonn-parlance

The concept
of manipulating the height of a stack so that it can land precisely on a
critical space. It’s a matter of counting and one-digit addition. Notice
that this doesn’t necessarily mean putting your own stacks atop one
another – the best sharpshooting moves are moves which also neutralize.
To counter a sharpshooting move is called “spoiling”.

But
for this strategy to have a chance, White must keep the Black stacks
containing the Dvonn pieces on the third row. At the moment the stack on
c3 can move to c1 or to c5 and with his next move White counters this

To spoil a move or
prevent a lifting move by moving atop the enemy stack. Even if the
opponent has enough control to retake the stack, he cannot move it
because it has become taller.

So, White sacrifies his
height 4 stack on g3 with the move g3-c3. Black must take back
immediately (if not, White moves c3-i3 and all Black’s material in the
farmost right cluster is lost) but now the previously mobile Black
height 2 stack at c3 has become an immobile (or *old stack*) height 7
stack which has no option but to stay on c3 (clearly Black will never
move it to j3…). Next, White performs a similar startegy to
neutralize the *young* height 3 Black stack on f3 by overloading it by 2
and hence after the forced recapture it becomes a height 6 Black stack
which must remain on f3 forever. Here are the actual moves 1) g3-c3
b2-c3 2) h2-h3 b4-c5 3) h3-f3 e2-f3 and we end up with the
situation we analyzed last time, that is

$\xymatrix@=.3cm @!C @R=.7cm{.& & & & & & & & & & & & & \\ & & & \Black{2} \connS & & \bull{d}{5} \conn & & \bull{e}{5} \conn & & \bull{f}{5} \conn & & \bull{g}{5} \conn & & \bull{h}{5} \conn & & \SWhite \connS & & \SWhite \connS & & \SWhite \conneS & & & \\ & & \bull{b}{4} \conn & & \SBlack \connS & & \Black{6} \connS & & \bull{e}{4} \conn& & \bull{f}{4} \conn & & \bull{g}{4} \conn & & \bull{h}{4} \conn & & \SWhite \connS & & \SWhite \connS & & \SWhite \conneS & & \\ & \SBlack \connbeginS & & \SBlack \connS & & \BDvonn{7} \connS & & \bull{d}{3} \conn & & \SBlack \connS & & \BDvonn{6} \connS & & \bull{g}{3} \conn & & \bull{h}{3} \conn & & \Dvonn \connS & & \SWhite \connS & & \SWhite \connendS & . \\ & & \Black{5} \connbeginS & & \bull{b}{2} \conn & & \SBlack \connS & & \bull{d}{2} \conn & & \bull{e}{2} \conn & & \bull{f}{2} \conn & & \bull{g}{2} \conn & & \bull{h}{2} \conn & & \SWhite \connS & & \SWhite \connendS & & \\ & & & \bull{a}{1} \con & & \bull{b}{1} \con & & \Black{5} \conS & & \bull{d}{1} \con & & \bull{e}{1} \con & & \bull{f}{1} \con & & \bull{g}{1} \con & & \bull{h}{1} \con & & \White{2} & & & \\ . & & & & & & & & & & & & & }$

## Latexrender and dvonn boards

In order
to blog a bit about Dvonn-strategy, I made myself a simple Dvonn
LaTeX-template which works very well on paper but which gets mutilated
by Latexrender, for example the first situation of the looks
like

$~\xymatrix@=.3cm @!C @R=.7cm{ & & \Black{2} \connS & & \bull{d}{5} \conn & & \bull{e}{5} \conn & & \bull{f}{5} \conn & & \bull{g}{5} \conn & & \bull{h}{5} \conn & & \SWhite \connS & & \SWhite \connS & & \SWhite \conneS & & \\ & \bull{b}{4} \conn & & \SBlack \connS & & \Black{6} \connS & & \bull{e}{4} \conn& & \bull{f}{4} \conn & & \bull{g}{4} \conn & & \bull{h}{4} \conn & & \SWhite \connS & & \SWhite \connS & & \SWhite \conneS & \\ \SBlack \connbeginS & & \SBlack \connS & & \BDvonn{7} \connS & & \bull{d}{3} \conn & & \SBlack \connS & & \BDvonn{6} \connS & & \bull{g}{3} \conn & & \bull{h}{3} \conn & & \Dvonn \connS & & \SWhite \connS & & \SWhite \connendS \\ & \Black{5} \connbeginS & & \bull{b}{2} \conn & & \SBlack \connS & & \bull{d}{2} \conn & & \bull{e}{2} \conn & & \bull{f}{2} \conn & & \bull{g}{2} \conn & & \bull{h}{2} \conn & & \SWhite \connS & & \SWhite \connendS & \\ & & \bull{a}{1} \con & & \bull{b}{1} \con & & \Black{5} \conS & & \bull{d}{1} \con & & \bull{e}{1} \con & & \bull{f}{1} \con & & \bull{g}{1} \con & & \bull{h}{1} \con & & \White{2} & &}$

The
reason behind this unwanted clipping is that Latexrender uses
**convert** to take the relevant part of a ps-page containing only the
TeXed formula on an empty page by performing clipping and then converts
it into a GIF-file (or any other format you desire). The obvious way
round this is to enlarge my template by adding two additional rows and
columns and putting visible nonsense there (such as dots) to enlarge the
relevant part so that no clipping is done of essential info. But then
(1) the picture generated becomes even larger than that above and (2) I
don’t want you to see the extra nonsensical dots… The essential line
in the **class.latexrender.php** file is

$command =$this->_convert_path." -density ".$this->_formula_density. " -trim -transparent \"#FFFFFF\" ".$this->_tmp_filename.".ps ". $this->_tmp_filename.".".$this->_image_format; 

So
I needed to delve into the [manual pages for the convert command](http://amath.colorado.edu/computing/software/man/convert.html)
of the ImageMagick-package. To my surprise, the *-trim* option (which I
got around my second problem using the *crop* option and around the
first by using the very useful *geometry* option. The latter is also
useful if you find that the size of the output of Latexrender is not
compatible with the size of your regular text. Of course you can amend
this somewhat by using the *extarticle* documentclass (as suggested) but
if you want to further adjust it, use for example

 -geometry 86%

to size the output to exactly 86% (or whatever you need).
So, whenever I want to do some Dvonn-blogging from now on I’ll change my
class.latexrender.php file as follows

 $command =$this->_convert_path." -crop 0x0-10% -crop 0x0+10% -density ".$this->_formula_density. " -geometry 80% -transparent \"#FFFFFF\" ".$this->_tmp_filename.".ps ". $this->_tmp_filename.".".$this->_image_format; 

which
produces the output

$\xymatrix@=.3cm @R=.7cm{.& & & & & & & & & & & & & \\ & & & \Black{2} \connS & & \bull{d}{5} \conn & & \bull{e}{5} \conn & & \bull{f}{5} \conn & & \bull{g}{5} \conn & & \bull{h}{5} \conn & & \SWhite \connS & & \SWhite \connS & & \SWhite \conneS & & & \\ & & \bull{b}{4} \conn & & \SBlack \connS & & \Black{6} \connS & & \bull{e}{4} \conn& & \bull{f}{4} \conn & & \bull{g}{4} \conn & & \bull{h}{4} \conn & & \SWhite \connS & & \SWhite \connS & & \SWhite \conneS & & \\ & \SBlack \connbeginS & & \SBlack \connS & & \BDvonn{7} \connS & & \bull{d}{3} \conn & & \SBlack \connS & & \BDvonn{6} \connS & & \bull{g}{3} \conn & & \bull{h}{3} \conn & & \Dvonn \connS & & \SWhite \connS & & \SWhite \connendS & . \\ & & \Black{5} \connbeginS & & \bull{b}{2} \conn & & \SBlack \connS & & \bull{d}{2} \conn & & \bull{e}{2} \conn & & \bull{f}{2} \conn & & \bull{g}{2} \conn & & \bull{h}{2} \conn & & \SWhite \connS & & \SWhite \connendS & & \\ & & & \bull{a}{1} \con & & \bull{b}{1} \con & & \Black{5} \conS & & \bull{d}{1} \con & & \bull{e}{1} \con & & \bull{f}{1} \con & & \bull{g}{1} \con & & \bull{h}{1} \con & & \White{2} & & & \\ . & & & & & & & & & & & & & }$

which (I hope) you will
find slightly better…

## dvonn (1) mobility

[Dvonn](http://www.gipf.com/dvonn $is the fourth game in the [Gipf Project](http://www.gipf.com/project_gipf/index.html) and the most mathematical of all six. It is a very fast (but subtle) game with a simple [set of rules](http://www.gipf.com/dvonn/rules/rules.html). Here is a short version DVONN is a stacking game. It is played on an elongated hexagonal board, with 23 white, 23 black and 3 red DVONN-pieces. In the beginning the board is empty. The players first place the DVONN-pieces on the board and next their own pieces. Then they start stacking pieces on top of each other. A single piece may be moved 1 space in any direction, a stack of two pieces may be moved two spaces, etc. A stack must always be moved as a whole and a move must always end on top of another piece or stack. If pieces or stacks lose contact with the DVONN pieces, they must be removed from the board. The game ends when no more moves can be made. The players put the stacks they control on top of each other and the one with the highest stack is the winner. That’s all! All this will become clearer once we fix a specific end-game, for example$\xymatrix@=.3cm @!C @R=.7cm{ & &
\Black{2} \connS & & \bull{d}{5} \conn & & \bull{e}{5} \conn & &
\bull{f}{5} \conn & & \bull{g}{5} \conn & & \bull{h}{5} \conn & &
\SWhite \connS & & \SWhite \connS & & \SWhite \conneS & & \\ &
\bull{b}{4} \conn & & \SBlack \connS & & \Black{6} \connS & &
\bull{e}{4} \conn& & \bull{f}{4} \conn & & \bull{g}{4} \conn & &
\bull{h}{4} \conn & & \SWhite \connS & & \SWhite \connS & & \SWhite
\conneS & \\ \SBlack \connbeginS & & \SBlack \connS & & \BDvonn{7}
\connS & & \bull{d}{3} \conn & & \SBlack \connS & & \BDvonn{6} \connS &
& \bull{g}{3} \conn & & \bull{h}{3} \conn & & \Dvonn \connS & & \SWhite
\connS & & \SWhite \connendS \\ & \Black{5} \connbeginS & &
\bull{b}{2} \conn & & \SBlack \connS & & \bull{d}{2} \conn & &
\bull{e}{2} \conn & & \bull{f}{2} \conn & & \bull{g}{2} \conn & &
\bull{h}{2} \conn & & \SWhite \connS & & \SWhite \connendS & \\ & &
\bull{a}{1} \con & & \bull{b}{1} \con & & \Black{5} \conS & &
\bull{d}{1} \con & & \bull{e}{1} \con & & \bull{f}{1} \con & &
\bull{g}{1} \con & & \bull{h}{1} \con & & \White{2} & &} $with White to move. Some comments about notation : the left-slanted columns are denoted by letters from a (left) to k (right) and the rows are labeled 1 to 5 from bottom to top (surprisingly this ‘standard’ webgame-notation differs from the numbering on my Dvonn-board where the rows are labeled from top to bottom…). So, for example, the three spots on the upper right are k3,k4 and k5 (there are no k1 or k2 spots). The three Dvonn pieces are colored red and in the course of the game a stack may land on a Dvonn piece and so stacks containing a Dvonn piece are denoted with a red halo. For example, the symbol on spot f3 stands for for a stack of 6 pieces, one of which is a red Dvonn piece, under the control of Black (that is, the top-piece is Black). Further note that a piece or stack can only move if it is not surrounded by 6 other pieces or stacks (so the White pieces on j3 and j4 cannot (yet) move). A piece can only move by one step in either line-direction provided there is another piece or stack on that position. The same applies for stacks : an height 3 stack for example can move in each lin-direction by exactly 3 steps provided there is a piece or stack to jump onto. For example, the height 6 stack on d4 can only move to j4 whereas the height 6 stack on f3 cannot move at all! Similarly, the two black height 5 stacks are immobile. At the moment black has all its stacks defended, that is, if White should be able to jump onto one of them (which White cannot at the moment), Black can use one of its neighbouring pieces to take the stack back under its control. So, any computer program would ‘evaluate’ the position as favourable for Black : Black has stacks of total height 34 safely under control (there are no immediate threats to be seen : the [horizon effect](http://www.comp.lancs.ac.uk/computing/research/aai-aied/people/paulb/old243prolog/subsection3_7_5.html) in such programs) whereas White can only claim potential stacks of total height 13… Still, Black has already lost the game. White has more pieces which are quite mobile as opposed to the immobile black stacks, so Black will soon run out of moves to make and his end position will have some large stacks on the third row. All white has to do is to let Black run out of moves and then continue (Dvonn forces each player to make a move if they still can and to pass the move otherwise, so the most mobile player can still continue long after the other player was forced to stop) to build a White stack of the appropriate height on the third row to jump on the highest Black stack with its last move! Here is how the play continued : 1) j2-k3 ; a3-b3 2) i1-k3 ; c5-c3 3) i2-i3 ; c2-c3 4) i3-k3 ; d4-j4 5) j3-j4 ; e3-f3 6) i4-j4 ; c4-b3 to arrive at the position where Black is no longer able to make any moves at all$\xymatrix@=.3cm
@!C @R=.7cm{ & & \bull{c}{5} \conn & & \bull{d}{5} \conn & & \bull{e}{5}
\conn & & \bull{f}{5} \conn & & \bull{g}{5} \conn & & \bull{h}{5} \conn
& & \SWhite \connS & & \SWhite \connS & & \SWhite \conneS & & \\ &
\bull{b}{4} \conn & & \bull{c}{4} \conn & & \bull{d}{4} \conn & &
\bull{e}{4} \conn& & \bull{f}{4} \conn & & \bull{g}{4} \conn & &
\bull{h}{4} \conn & & \bull{i}{4} \connS & & \White{9} \connS & &
\SWhite \conneS & \\ \bull{a}{3} \connbegin & & \Black{3} \connS & &
\BDvonn{10} \connS & & \bull{d}{3} \conn & & \bull{e}{3} \conn & &
\BDvonn{7} \connS & & \bull{g}{3} \conn & & \bull{h}{3} \conn & &
\bull{i}{3} \conn & & \bull{j}{3} \conn & & \WDvonn{6} \connendS \\ &
\Black{5} \connbeginS & & \bull{b}{2} \conn & & \bull{c}{2} \conn & &
\bull{d}{2} \conn & & \bull{e}{2} \conn & & \bull{f}{2} \conn & &
\bull{g}{2} \conn & & \bull{h}{2} \conn & & \bull{i}{2} \conn & &
\bull{j}{2} \connend & \\ & & \bull{a}{1} \con & & \bull{b}{1} \con & &
\bull{c}{1} \con & & \bull{d}{1} \con & & \bull{e}{1} \con & &
\bull{f}{1} \con & & \bull{g}{1} \con & & \bull{h}{1} \con & &
\bull{i}{1} & &} $Note that all pieces and stacks no longer connected to a Dvonn piece must be removed. So, for example, after the third move by Black, the Black height 5 stacks on c1 was removed. All white now has to do is to built an height 8 stack on k3 and jump onto the height 10 Black stack on c3 to win the game. The (only) way to do this is by 7. j5-k5 and 8. k5-k3 to finish with 9. k3-c3 with final position (note again that the White right-hand pieces and stacks are no longer connected to a Dvonn piece and are hence removed)$\xymatrix@=.3cm @!C @R=.7cm{ & & \bull{c}{5} \conn & & \bull{d}{5}
\conn & & \bull{e}{5} \conn & & \bull{f}{5} \conn & & \bull{g}{5} \conn
& & \bull{h}{5} \conn & & \bull{i}{5} \conn & & \bull{j}{5} \conn & &
\bull{k}{5} \conne & & \\\ & \bull{b}{4} \conn & & \bull{c}{4} \conn &
& \bull{d}{4} \conn & & \bull{e}{4} \conn& & \bull{f}{4} \conn & &
\bull{g}{4} \conn & & \bull{h}{4} \conn & & \bull{i}{4} \conn & &
\bull{j}{4} \conn & & \bull{k}{4} \conne & \\\ \bull{a}{3} \connbegin
& & \Black{3} \connS & & \WDvonn{18} \connS & & \bull{d}{3} \conn & &
\bull{e}{3} \conn & & \BDvonn{7} \connS & & \bull{g}{3} \conn & &
\bull{h}{3} \conn & & \bull{i}{3} \conn & & \bull{j}{3} \conn & &
\bull{k}{3} \connend \\\ & \Black{5} \connbeginS & & \bull{b}{2} \conn
& & \bull{c}{2} \conn & & \bull{d}{2} \conn & & \bull{e}{2} \conn & &
\bull{f}{2} \conn & & \bull{g}{2} \conn & & \bull{h}{2} \conn & &
\bull{i}{2} \conn & & \bull{j}{2} \connend & \\\ & & \bull{a}{1} \con &
& \bull{b}{1} \con & & \bull{c}{1} \con & & \bull{d}{1} \con & &
\bull{e}{1} \con & & \bull{f}{1} \con & & \bull{g}{1} \con & &
\bull{h}{1} \con & & \bull{i}{1} & & } \$

So White wins with 18 to
Black’s 15. This shows that it is important to maintain mobility and
also that it is possible to win a Dvonn-game from computers. In fact,
the above end-game was played against a computer-program (Black). The
entire game can be found
[here](http://www.littlegolem.net/jsp/game/game.jsp?gid=426457&nmove=91)
.

## Alain Connes on everything

A few
days ago, Ars Mathematica wrote :

Alain Connes and Mathilde Marcolli have posted a
new survey paper on Arxiv A walk in the
noncommutative garden
. There are many contenders for the title of
noncommutative geometry, but Connes‚Äô flavor is the most
successful.

Be that as it may, do
not print this 106 page long paper! Browse through it
if you have to, be dazzled by it if you are so inclined, but I doubt it
is the eye-opener you were looking for if you gave up on reading
Connes’ book Noncommutative
Geometry
…. Besides, there is much better
_Tehran-material_ on Connes to be found on the web : An interview
with Alain Connes
, still 45 pages long but by all means : print it
out, read it in full and enjoy! Perhaps it may contain a lesson or two
for you. To wet your appetite a few quotes

It is
important that different approaches be developed and that one
doesn‚Äôt try to merge them too fast. For instance in noncommutative
geometry my approach is not the only one, there are other approaches
and it‚Äôs quite important that for these approaches there is no
social pressure to be the same so that they can develop
independently. It‚Äôs too early to judge the situation for instance
in quantum gravity. The only thing I resent in string theory is that
they put in the mind of people that it is the only theory that can
give the answer or they are very close to the answer. That I resent.
For people who have enough background it is fine since they know all
the problems that block the road like the cosmological constant, the
supersymmetry breaking, etc etc…but if you take people who are
beginners in physics programs and brainwash them from the very start
it is really not fair. Young physicists should be completely free,
but it is very hard with the actual system.

And here for some (moderate) Michael Douglas bashing :

Physicists tend to shift often and work on the
last fad. I cannot complain because at some point around 98 that fad was
NCG after my paper with Douglas and Schwarz. But after a while when
these problems the answer was no because it was no longer the last
fad and he wanted to work on something else. In mathematics one
sometimes works for several years on a problem but these young
physicists have a very different type of working habit. The unit of
time in mathematics is about 10 years. A paper in mathematics which is
10 years old is still a recent paper. In physics it is 3 months. So
I find it very difficult to cope with constant
zapping.

To the suggestion that he is the
prophet (remember, it is a Tehran-interview) of noncommutative geometry
he replies

It is flattering but I don‚Äôt think
it is a good thing. In fact we are all human beings and it is a
wrong idea to put a blind trust in a single person and believe in
that person whatever happens. To give you an example I can tell you
a story that happened to me. I went to Chicago in 1996, and gave a
talk in the physics department. A well known physicist was there and
he left the room before the talk was over. I didn‚Äôt meet this
physicist for two years and then, two years later, I gave the same
talk in the Dirac Forum in Rutherford laboratory near Oxford. This
time the same physicist was attending, looking very open and convinced
and when he gave his talk later he mentioned my talk quite
positively. This was quite amazing because it was the same talk and
I had not forgotten his previous reaction. So on the way back to
Oxford, I was sitting next to him in the bus, and asked him openly
how can it be that you attended the same talk in Chicago and you
left before the end and now you really liked it. The guy was not a
beginner and was in his forties, his answer was ‚ÄúWitten was seen
reading your book in the library in Princeton‚Äù! So I don‚Äôt want
to play that role of a prophet preventing people from thinking on
their own and ruling the sub ject, ranking people and all that. I
care a lot for ideas and about NCG because I love it as a branch of
mathematics but I don‚Äôt want my name to be associated with it as a
prophet.

and as if that was not convincing
enough, he continues

Well, the point is that what
matters are the ideas and they belong to nobody. To declare that
some persons are on top of the ladder and can judge and rank the
others is just nonsense mostly produced by the sociology (in fact by the
system of recommendation letters). I don‚Äôt want that to be true in
NCG. I want freedom, I welcome heretics.

Published
Categorized as stories

## nostalgia

Unlike the
cooler people out there, I haven’t received my
_pre-ordered_ copy (via AppleStore) of Tiger yet. Partly my own fault
because I couldn’t resist the temptation to bundle up with a
personalized iPod Photo!
The good news is that it buys me more time to follow the
housecleaning tips
. First, my idea was to make a CarbonCopyClooner
image of my iBook and put it on the _iMac_ upstairs which I
rarely use these days, do a clean
Tiger install
on the iBook and gradually copy over the essential
programs and files I need (and only those!). But reading the
macdev-article, I think it is better to keep my iBook running Panther
and experiment with Tiger on the redundant iMac. (Btw. unless you want
to have a copy of my Mac-installation there will be hardly a point
checking this blog the next couple of weeks as I intend to write down
all details of the Panther/Tiger switch here.)

Last week-end I
started a _Paper-rescue_ operation, that is, to find among the
multiple copies of books/papers/courses, the ones that contain all the
required material to re-TeX them and unfortunately my _archive_
is in a bad state. There is hardly a source-file left of a paper prior
to 1999 when I started putting all my papers on the arXiv.

On the other hand, I do
have saved most of my undergraduate courses. Most of them were still
using postscript-crap like _epsfig_ etc. so I had to convert all
the graphics to PDFs (merely using Preview ) and
modify the epsfig-command to _includegraphics_. So far, I
converted all my undergraduate _differential geometry_ courses
from 1998 to this year and made them available in a uniform

There are two
ways to read the changes in these courses over the years. (1) as a shift
from _differential_ geometry to more _algebraic_ geometry
and (2) as a shift towards realism wrt.the level of our undegraduate
students. In 1998 I was still thinking
that I could teach them an easy way into Connes non-commutative standard
model but didn’t go further than the Lie group sections (maybe one day
I’ll rewrite this course as a graduate course when I ever get
reinterested in the Connes’ approach). In 1999 I had the illusion that
it might be a good idea to introduce manifolds-by-examples coming from
operads! In 2000 I gave in to the fact
that most of the students which had to follow this course were applied
mathematicians so perhaps it was a good idea to introduce them to
dynamical systems (quod non!). The 2001 course is probably the
most realistic one while still doing standard differential geometry. In
2002 I used the conifold
singularity and conifold transitions (deformations and blow-ups) as
motivation but it was clear that the students did have difficulties with
the blow-up part as they didn’t have enough experience in
_algebraic_ geometry. So the last two years I’m giving an
introduction to algebraic geometry culminating in blow-ups and some
non-commutative geometry.

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