# Tag: Connes

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

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

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](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) . 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 I saw Michael Douglas and asked him if he had thought more about 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. But please, read it all for yourself and draw your own conclusions. 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 screen-friendly viewing format at TheLibrary/undergraduate. 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. Are there hidden relations between mathematical and physical constants such as$\frac{e^2}{4 \pi \epsilon_0 h c} \sim \frac{1}{137} $or are these numerical relations mere accidents? A couple of years ago, Pierre Cartier proposed in his paper A mad day’s work : from Grothendieck to Connes and Kontsevich : the evolution of concepts of space and symmetry that there are many reasons to believe in a cosmic Galois group acting on the fundamental constants of physical theories and responsible for relations such as the one above. The Euler-Zagier numbers are infinite sums over$n_1 > n_2 > ! > n_r \geq 1 $of the form$\zeta(k_1,\dots,k_r) = \sum n_1^{-k_1} \dots n_r^{-k_r} $and there are polynomial relations with rational coefficients between these such as the product relation$\zeta(a)\zeta(b)=\zeta(a+b)+\zeta(a,b)+\zeta(b,a) $It is conjectured that all polynomial relations among Euler-Zagier numbers are consequences of these product relations and similar explicitly known formulas. A consequence of this conjecture would be that$\zeta(3),\zeta(5),\dots $are all trancendental! Drinfeld introduced the Grothendieck-Teichmuller group-scheme over$\mathbb{Q} $whose Lie algebra$\mathfrak{grt}_1 $is conjectured to be the free Lie algebra on infinitely many generators which correspond in a natural way to the numbers$\zeta(3),\zeta(5),\dots $. The Grothendieck-Teichmuller group itself plays the role of the Galois group for the Euler-Zagier numbers as it is conjectured to act by automorphisms on the graded$\mathbb{Q} $-algebra whose degree$d $-term are the linear combinations of the numbers$\zeta(k_1,\dots,k_r) $with rational coefficients and such that$k_1+\dots+k_r=d $. The Grothendieck-Teichmuller group also appears mysteriously in non-commutative geometry. For example, the set of all Kontsevich deformation quantizations has a symmetry group which Kontsevich conjectures to be isomorphic to the Grothendieck-Teichmuller group. See section 4 of his paper Operads and motives in deformation quantzation for more details. It also appears in the renormalization results of Alain Connes and Dirk Kreimer. A very readable introduction to this is given by Alain Connes himself in Symmetries Galoisiennes et renormalisation . Perhaps the latest news on Cartier’s dream of a cosmic Galois group is the paper by Alain Connes and Matilde Marcolli posted last month on the arXiv : Renormalization and motivic Galois theory . A good web-page on all of this, including references, can be found here. OK! I asked to get side-tracked by comments so now that there is one I’d better deal with it at once. So, is there any relation between the non-commutative (algebraic) geometry based on formally smooth algebras and the non-commutative _differential_ geometry advocated by Alain Connes? Short answers to this question might be (a) None whatsoever! (b) Morally they are the same! and (c) Their objectives are quite different! As this only adds to the confusion, let me try to explain each point separately after issuing a _disclaimer_ that I am by no means an expert in Connes’ NOG neither in$C^* $-algebras. All I know is based on sitting in some lectures by Alain Connes, trying at several times to make sense of his terribly written book and indeed by reading the Landi notes in utter desperation. (a) _None whatsoever!_ : Connes’ approach via spectral triples is modelled such that one gets (suitable) ordinary (that is, commutative) manifolds into this framework. The obvious algebraic counterpart for this would be a statement to the effect that the affine coordinate ring$\mathbb{C}[X] $of a (suitable) smooth affine variety X would be formally smooth. Now you’re in for a first shock : the only affine smooth varieties for which this holds are either _points_ or _curves_! Not much of a geometry huh? In fact, that is the reason why I prefer to call formally smooth algebras just _qurves_ … (b) _Morally they are the same_ : If you ever want to get some differential geometry done, you’d better have a connection on the tangent bundle! Now, Alain Connes extended the notion of a connection to the non-commutative world (see for example _the_ book) and if you take the algebraic equivalent of it and ask for which algebras possess such a connection, you get _precisely_ the formally smooth algebras (see section 8 of the Cuntz-Quillen paper “Algebra extensions and nonsingularity” Journal AMS Vol 8 (1991). Besides there is a class of$C^* $-algebras which are formally smooth algebras : the AF-algebras which also feature prominently in the Landi notes (although they are virtually never affine, that is, finitely generated as an algebra). (c) _Their objectives are quite different!_ : Connes’ formalism aims to define a length function on a non-commutative manifold associated to a$C^* $-algebra. Non-commutative geometry based on formally smooth algebras has no interest in defining some sort of space associated to the algebra. The major importance of formally smooth algebras (as advocated by Maxim Kontsevich is that such an algebra A can be seen as a _machine_ producing an infinite family of ordinary commutative manifolds via its _representation varieties_$\mathbf{rep}_n~A $which are manifolds equipped with a$GL_n $-action. Non-commutative functions and diifferential forms defined at the level of the formally smooth algebra A do determine similar$GL_n $-invariant features on _all_ of these representation varieties at once. Here’s a part of yesterday’s post by bitch ph.d. : But first of all I have to figure out what the hell I’m going to teach my graduate students this semester, and really more to the point, what I am not going to bother to try to cram into this class just because it’s my first graduate class and I’m feeling like teaching everything I know in one semester is a realistic and desireable possibility. Yes! Here it all is! Everything I have ever learned! Thank you, and goodnight! Ah, the perpetual motion machine of last-minute course planning, driven by ambition and sloth!. I’ve had similar experiences, even with undergraduate courses (in Belgium there is no fixed curriculum so the person teaching the course is responsible for its contents). If you compare the stuff I hoped to teach when I started out with the courses I’ll be giving in a few weeks, you would be more than disappointed. The first time I taught _differential geometry 1_ (a third year course) I did include in the syllabus everything needed to culminate in an outline of Donaldson’s result on exotic structures on$\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!