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math2.0-setup : WpMU and BuddyPress

Last time we took a clean mac os x 10.5.6 installation and upgraded it to a MySQL and Apache-PHP+ server. This hour we’ll turn it into a math2.0-test environment. That is, we will install WordPressMU (the ‘multiple user’ version of WordPress which can host 10 or 100 or thousands of blogs on your computer). Then, we’ll turn this potential into a FaceBook-like social network.

Probably it is best to test this just on your ‘localhost’ before going worldwide. Therefore, I’ll describe here the test-environment-version (the changes to make for going www I’ll describe later, but, they are minor). Here’s the first problem : WordPressMu doesn’t recognize ‘localhost’ as a valid domain, so we’ll have to use something like ‘localhost.localdomain’ and tell our server to recognize this new address.

Open TextWrangler, under File/Open File by Name type /etc/hosts and add (again you’ll be asked to unlock the file as it is owned by ‘root’ and you’ll have to provide your sudo-password when you want to save it) to the end of that file (it’s short) the line

127.0.0.1 localhost.localdomain

Save it.Your file should now look like this



You can verify this by opening Safari and pointing it to http://localhost.localdomain. You should now see the default Apache-page (no need to restart the WebServer). Next, within TextWrangler do again a File/Open File by Name and type /etc/apache2/httpd.conf and under Search/Go to Line… say 211. The highlighted line reads

AllowOverride None

change it to (again the root and sudo routine as before)

AllowOverride All

and save it. Restart the WebServer (that is, open SystemPreferences go to Sharing, unmark and remark again after a tiny delay WebSharing). Okay, I think we are set


WordPress MU :

We need a database to store everything. If your database-root password set last time is myRootPassword then do the following (change to whatever it really is). Open Terminal and type to the prompt

mysql -u root -p

and provide MyRootPassword when asked. Now we have to tell MySQL to create a database, set a database-user and password (for safety reasons it better be a new user and password, but as this is merely a test-environment…). So, to the mysql-prompt we type the following string of commands

mysql> CREATE DATABASE mywordpressMU;
Query OK, 1 row affected (0.00 sec)

mysql> GRANT ALL PRIVILEGES ON mywordpressMU.* TO root@localhost IDENTIFIED BY “MyRootPassword”;
Query OK, 0 rows affected (0.00 sec)

mysql> FLUSH PRIVILEGES;
Query OK, 0 rows affected (0.01 sec)

mysql> SET PASSWORD FOR root@localhost = OLD_PASSWORD(“MyRootPassword”);
Query OK, 0 rows affected (0.01 sec)

mysql> EXIT
Bye

The OLD_PASSWORD-command seems odd, but is needed as WordPress doesn’t like the password-structure of MySQL 5. If you forget this, you’ll get database-errors.

Open Safari and download the latest WordPressMU here. Open two Finder windows, one pointing to YourHome/Downloads (in which a new folder “wordpress-mu” is created and another one pointing to “Macintosh HD/Library/WebServer/Documents. You may as well drop the content of this Documents-directory into the Trash (the file test.php we created last time excluded).

Select everything in the first-window wordpress-mu directory and drag all of it to the second-window Documents directory.



Open Terminal and type these to the prompt

$ cd /Library/WebServer
$ sudo chmod 777 Documents
Password: (fill in your sudo password)
$ cd Documents
$ sudo chmod 777 wp-content

Point your Safari to http://localhost.localdomain/index.php and you should get a WordPress mu install page. Make sure to choose the Sub-directories option (instead of the default sub-Domain option) and fill out the required info



You should get a success-page giving you the first password to login as admin. (In case you get a database-error, remove the wp-config.php file, redo the OLD_PASSWORD command given above and repeat the install. Everything should work!). Do this, go to the Users-tab and edit your admin-account to change the password (at the bottom) to something you can remember easily.

Remember to change the directory permissions again to 755. That is, open Terminal and do

$ cd /Library/WebServer
$ sudo chmod 755 Documents
Password: (fill in your sudo password)
$ cd Documents
$ sudo chmod 755 wp-content

We now have a working WordPressMU, you can create new users and they can start new blogs, but there isn’t yet any interaction between these users and your site looks, well…



Let’s spice it up a bit with

BuddyPress :

BuddyPress is a set of WordPress MU specific plugins, each plugin adding a distinct new feature. BuddyPress contains all the features you’d expect from WordPress but aims to let members socially interact.”

We start by getting the BuddyPress Combo Download (say the .zip file) which will create a buddypress-combo directory in YourHome/Downloads. Open this directory and as before drag its entire content to /Library/WebServer/Documents/wp-content/mu-plugins.

In the buddypress-theme directory there are two subdirectories which need to be put elsewhere. The subdirectory buddypress-home must be dragged to the wp-content/themes directory (containing at the moment only the classic, default and home theme) and the subdirectory member-themes must be dragged to the wp-content directory



Log in again as admin in your WpMU via http://localhost.localdomain/wp-admin/ under Site Admin/Themes activate the BuddyPress Home Theme and press the ‘Update Themes’ button.



Similarly, in Site Admin/Options mark under ‘Allow new registrations’ the option ‘Enables. Blogs and user accounts can be created’ and update these options via the button at the bottom. Finally, under Appearance click on the BuddyPress Home Theme and activate it (top right).

Now, visit your site and change it to your liking via adding widgets. For example, add ‘Welcome’, ‘Recent Blog Posts’ and ‘Site Wide Activity’ to the left column, ‘Who’s Online’ and ‘Members’ to the center column and ‘Groups’ and ‘Meta’ to the right column.

Next, create new users (via Site Admin/Users and not via signup as this is just an offline test-version and signup sends out activating emails…), create groups, blogs and posts, let users befriend one another and send wires, etc. etc. all the things people do in a web2.0 environment.

One further comment : if you want to have avatars uploaded you’ll have to open Terminal and type the following :

$ cd /Library/WebServer/Documents
$ sudo mkdir avatars
$ sudo chown _www:admin avatars

Finally, if your iMac is a proper web-server (that is, has its own URL) you can take your math2.0-network worldwide repeating the above procedure with obvious modifications (that is, replacing localhost.localdomain by the URL of your machine). In order to get the signup/email system going you may need to install the Swift-SMTP-Mailer plugin and feed it your outgoing mail-server (also you’ll have to enable plugins in the SiteAdmin/Options).

An embryonal version of your site may then look like this one



While this may already be good enough for the rest of the world, mathematicians talk LaTeX to each other, so we’d better include LaTeX-support (and perhaps also some Wiki-support). This we will do another time…

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can -oids save group-theory 101?

Two questions from my last group-theory 101 exam:

(a) : What are the Jordan-Holder components of the Abelian group $\mathbb{Z}/20 \mathbb{Z} $?

(b) : Determine the number of order 7 elements in a simple group of order 168.

Give these to any group of working mathematicians, and, I guess all of them will solve (a), whereas the number of correct solutions to (b) will be (substantially) smaller.

Guess what? All(!) my students solved (b) correctly, whereas almost none of them had anything sensible to say about (a). A partial explanation is that they had more drill-exercises applying the Sylow-theorems than ones concerning the Jordan-Holder theorem.

A more fundamental explanation is that (b) has to do with sub-structures whereas (a) concerns quotients. Over the years I’ve tried numerous methods to convey the quotient-idea : putting things in bags, dividing a big group-table into smaller squares, additional lessons on relations, counting modulo numbers … No method appears to have an effect, lasting until the examination.

At the moment I’m seriously considering to rewrite the entire course, ditching quotients and using them only in disguise via groupoids. Before you start bombarding me with comments, I’m well aware of the problems inherent in this approach.

Before you do groupoids, students have to know some basic category theory. But that’s ok with me. Since last year it has been decided that I should sacrifice the first three weeks of the course telling students the basics of sets, maps and relations. After this, the formal definition of a category will appear more natural to them than the definition of a group, not? Besides, most puzzle-problems I use to introduce groups are actually examples of groupoids…

But then, what are the main theorems on finite groupoids? Well, I can see the groupoid cardinality result, giving you in one stroke Lagrange’s theorem as well as the orbit-counting method. From this one can then prove the remaining classical group-results such as Cauchy and the Sylows, but perhaps there are more elegant approaches?

Have you seen a first-year group-theory course starting off with groupoids? Do you know an elegant way to prove a classical group-result using groupoids?

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math2.0-setup : mysql and php+

Last time I wondered whether a set-up like WordPress.com meets FaceBook with add-ons (such as wiki- and latex-support) might be a usable environment for people working in a specific arXiv-topic.

I’ve used WordPressMU and BuddyPress to create such an embryonal environment. At first I thought I’d extend it a bit before going online but I fail to have the energy right now so I might as well make the link available. If you are into math.QA and/or math.RA you are invited to join the experiment. But, please use this site gently as I’ll have to drop it otherwise.

I’ve no desire to maintain this site indefinitely but would welcome others to set up something similar. For this reason I’ll write a couple of posts explaining how you can build it yourself when you’d have a free afternoon and a spare Mac around. Each post should not take you longer than 1 hour. Today, we’ll provide the boring but essential basics : we must get a MySQL-server and a WebServer running. Next time, I’ll take you through the WordPressMU (MU for multi-users) and BuddyPress installation. After that, we’ll add extra functionality.

We will start from a vanilla 10.5.6 installation. We will often need to edit files, so we’d better grab a good, free  texteditor : TextWrangler, drag it to Applications and place it in the Dock. We’ll also type in commands so we want the TerminalApp (to be found in Applcations/Utilities) in the Dock. SystemPreferences and Safari are already in the Dock and as we will need these tools a lot we might rearrange the Dock to look like



From left to right : the Finder, Terminal, Safari, TextWrangler and System Preferences. From now on we will mean by ‘Open …’ that you click on the ‘…’ icon. In the end we want our computer to become a web-server, so we don’t want it to go to sleep. Open SystemPreferences and look for the ‘Energy Saver’-icon, click on the ‘Show details’ button and set the ‘Put the computer to sleep when it is inactive for:’ to Never and unmark the ‘Put the hard disk(s) to sleep when possible’ at the bottom.

We will need to start or stop the WebServer so here’s how that’s done : open SystemPreferences and look for the ‘Sharing’-icon. Marking the ‘Web Sharing’ option is equivalent to starting your webserver (you can verify this by opening Safari and pointing it to http://localhost/ and you should see the default Apache-screen), unmarking it stops the webserver (check this by repeating the previous, now you should get a ‘Safari can’t connect to the server’ message).

All of this was probably trivial to you so let’s do something a bit more advanced : setting up a database-server. OSX doesn’t come with MySql, so we need to download and install it.

MySQL :

Get the latest version : choose the Mac OS X 10.5 (x86)-package and download it (they ask you to register but you can bypass this by clicking on the ‘No thanks, just take me to the downloads’-link). It is a 55.3 Mb file so this may take a couple of minutes. If all goes well this window should pop-up



Click on the mysql-5.0.67-osx10.5-x86.pkg icon and follow the instruction (defaults suffice, you’ll be asked to give your sudo password and in all it will take less than a minute). Repeat this procedure with MySQLStartupItem.pkg. Done!

To verify it, Open Terminal and type this to the prompt

sudo /Library/StartupItems/MySQLCOM/MySQLCOM start

You’ll get a scary warning message but type in your sudo-password and the Mysql-server will start. You can access it by typing

/usr/local/mysql/bin/mysql

and type exit to the mysql-prompt to leave.
In all, your interaction with the terminal should look something like this



Clearly, you do not want to type all of this every time, so we will add the mysql-location to our ‘PATH’. To do this, open TextWrangler and add this line to the blank document

export PATH=$PATH:/usr/local/mysql/bin

and save the file as .profile in your home-directory (the one with the ‘House’-icon, usually given your name). You will get a warning that .-files are reserved but go ahead anyway by clicking the use . – button). Now, open Terminal and type this

source ~./profile
echo $PATH

if all went well you should now see the mysql-location at the end of your path. From now on you’ll only have to type

mysql

to the terminal-prompt to open MySql. At the moment the root-user of your mysql has no password which isnt safe so we’d better set one. Open terminal and type

mysqladmin -u root password NEWPASSWORD

where, of course, you replace NEWPASSWORD with your choice (use only letters and numbers). From now on you can access your mysql-server by opening Terminal and typing

mysql -u root -p

and giving your password. Okay, so we’ve established our first goal, we have a working Mysql. Take a break if you need one.

better PHP

Mac 10.5 comes equipped with php5 but unfortunately it isn’t quite up to what we need. So, we need to install a better one and tell the mysql-server and the webserver to use the new one instead of the standard one.

Open Safari and grab the better php-version by going to

http://www2.entropy.ch/download/php5-5.2.5-6-beta.tar.gz

It is a 85.2 Mb file, so have a bit of patience. The file gets unzipped automatically and downloaded in the Downloads-directory. Open Finder and go there. At the moment your Downloads-directory will look like



Doubleclick on the php5-5.2.5-6-beta.tar file and a new directory will be created called php5. We will now move this directory and lay some symbolic links. Open a new Terminal window and type the following commands (and provide your sudo-password when asked)

cd Downloads
sudo mv php5 /usr/local
sudo ln -sf /usr/local/php5/entropy-php.conf /etc/apache2/other/+entropy-php.conf
sudo mkdir /var/mysql
sudo ln -s /tmp/mysql.sock /var/mysql/mysql.sock

Next, we have to tell the webserver to use this new php-version instead of the old one. This information is contained in the apache-configuration file : httpd.conf. Open TextWrangler and under ‘File’ choose the option ‘Open File by Name’. Type /etc/apache2/httpd.conf in the field that appears. The file will now appear in the main window. Under ‘Search’ choose the ‘Go to Line’ option and fill in 114 and hit the Go To button. The follwing line should now be highlighted

#LoadModule php5_module libexec/apache2/libphp5.so

immediately under it add the following line (TextWrangler will tell you that the file is owned by root and ask you whether you want to open it, click yes and make the changes)

LoadModule php5_module local/php5/libphp5.so

(observe that line 114 is commented out, that is, starts with a #, whereas your added line is not).
Save the file (Textwrangler will ask you to provide the sudo-password).

Next, we will have to tell php to communicate with the mysql-server. Again, open TextWrangler, under ‘File’ choose ‘Open File by Name’ and type in /usr/local/php5/lib/php.ini-recommended. When the file appears, under ‘Search’ choose ‘Go to Line’ and type in 810. It will read

mysql.default_socket =

Change it as follows (that is, add to it)

mysql.default_socket = /var/mysql/mysql.sock

and now choose under ‘File’ the ‘Save as…’ option. In the window change php.ini-recommended to php.ini and click Save. Done!

Testing…

Restart your webserver. Recall that this means: open SystemPreferences, choose ‘Sharing’, unmark ‘Web Server’, wait 5 seconds and then mark it again.

Open TextWranger, make a new Text document containing just one line (remove the white space between the ?-signs and the brackets) :

< ?php phpinfo() ? >

Choose ‘File’ and ‘Save as…’ and in the window that appears navigate to YourHardDisk/Library/WebServer/Documents, name the file ‘test.php’ and click the ‘Save’ button



Finally, open Safari and point it to http://localhost/test.php. Cross your fingers and if you get a screen like the one below treat yourself to something nice!



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yet another math2.0 proposal

At present, some interesting experiments are going on exploring the potential of web 2.0 for mathematical research, that is, setting up a usable math 2.0 – environment.

The starting point is that math 2.0 should be something like blogs+extras. Most mathematicians are not that interested in the latest ICT-tools, but at least they are slowly getting used to reading blogs, so we should stick to this medium and try to enhance it for online-research.

Michael Nielsen has written a couple of posts on this : an after-dinner talk about doing science online aiming at a mathematics audience, building on an essay on the future of science.

Both posts were influential to Tim Gowers‘ dream of massively collaborative mathematics. He took an interesting problem, laid down a set of 12 rules-of-conduct and invited everyone to contribute. The project is still gaining momentum and Terry Tao is also posting about it on his blog.

Michael Nielsen compared Gowers’ approach to long established practice in the open-source software community.

Another interesting experiment is nLab, a knowledge-wiki set up by the reader-community of the n-category cafe. They describe it as : “In other words: this place is like the library, or alchemist’s laboratory, in the back room of the n-Category Café. You come here to work and go there to chat.
We are hoping to create here a space for presentation and archival storage of collaborative work of encyclopedic, didactic, expositional, but also original nature. This will include, but not be limited to, the subjects being discussed every day in the n-Café.”

Both experiments are working great, aided by the authority-status of the blogger, resp. the popularity of the blog, within the research topic. But, what about topics failing to have a blogger or blog of similar status? Should we all drop our current research-interest and convert to either combinatorics or higher-categories?

History taught us in case of failing authority we’d better settle for ‘manageable anarchy’. So, here’s my math2.0-anarchy-allowed-proposal :

  • per research-topic (say, an arXiv-topic) we’ll set up a seperate online-reasearch-environment
  • anyone interested in that topic is allowed to register and fill-out a profile linking to her list of publications, describe his research interests, her ongoing projects and other trivia
  • some may want to start a blog within the environment or join an already existing one, and should be allowed to do so
  • some may opt just to read blog posts and occasionally comment, and again, should be allowed to do so
  • some may want to set up a research-group to solve a specific problem. they may choose to do this in the open, or as a covert-operation, taking on new members only by invitation
  • some may use the environment mainly for networking or chatting-up with their friends
  • some research-groups may want to start a group-blog or knowledge-wiki to archive their finds
  • surely we’ll be not discussing math in ASCII but in latex
  • anyone will be able to follow specific sub-projects via RSS-feeds
  • anyone can see site-wide activity online, see who’s currently there and chat if they feel the need
  • anyone can do whatever sensible web2.0-thing there is I forgot by age and hence by ignorance

If this seems like a tall order to satisfy, a bit of research will show that we live at the fortunate moment in time when all the basic ingredient are there, freely available, to do just that!

Over the last weeks I’ve wasted too many hours googling for help, reading-up different fora to get it all working, but … somehow succeeded. Here’s a screen-shot of my very-own NSN (for : noncommutative-social-network) :



Please allow me a few more days to tidy things up and then I’ll make the link available so that anyone interested can experiment with it.

But then, I’ve no desire to spend my days web-mastering such a site. Perhaps some of you would like to take this on, provided you’d get it on a silver plate? (that is, without having to spend too much time setting it up).

So. I’ll run a series of posts explaining how to “set-up your own math2.0 environment”. I’m not aiming at the internet-savvy ones (they’ll probably do it a lot more efficiently), but at people like myself, who are interested to investigate web-based possibilities, but need to be told where to find the very basics, such as the location of their httpd.conf file or their php.ini and such.

I’ve zeroed my MacBookPro, re-installed OSX 10.5 from scratch, upgraded it to current 10.5.6 but no extras (say, vanilla 10.5.6). And I’ll guide you from there, in all gory details, with plenty of screen-shots as I would have liked to find them when I tried to set this up.

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On2 : extending Lenstra’s list

We have seen that John Conway defined a nim-addition and nim-multiplication on the ordinal numbers in such a way that the subfield $[\omega^{\omega^{\omega}}] \simeq \overline{\mathbb{F}}_2 $ is the algebraic closure of the field on two elements. We’ve also seen how to do actual calculations in that field provided we can determine the mystery elements $\alpha_p $, which are the smallest ordinals not being a p-th power of ordinals lesser than $[\omega^{\omega^{k-1}}] $ if $p $ is the $k+1 $-th prime number.

Hendrik Lenstra came up with an effective method to compute these elements $\alpha_p $ requiring a few computations in certain finite fields. I’ll give a rundown of his method and refer to his 1977-paper “On the algebraic closure of two” for full details.

For any ordinal $\alpha < \omega^{\omega^{\omega}} $ define its degree $d(\alpha) $ to be the degree of minimal polynomial for $\alpha $ over $\mathbb{F}_2 = [2] $ and for each prime number $p $ let $f(p) $ be the smallest number $h $ such that $p $ is a divisor of $2^h-1 $ (clearly $f(p) $ is a divisor of $p-1 $).

In the previous post we have already defined ordinals $\kappa_{p^k}=[\omega^{\omega^{k-1}.p^{n-1}}] $ for prime-power indices, but we now need to extend this definition to allow for all indices. So. let $h $ be a natural number, $p $ the smallest prime number dividing $h $ and $q $ the highest power of $p $ dividing $h $. Let $g=[h/q] $, then Lenstra defines

$\kappa_h = \begin{cases} \kappa_q~\text{if q divides}~d(\kappa_q)~\text{ and} \\ \kappa_g + \kappa_q = [\kappa_g + \kappa_q]~\text{otherwise} \end{cases} $

With these notations, the main result asserts the existence of natural numbers $m,m’ $ such that

$\alpha_p = [\kappa_{f(p)} + m] = [\kappa_{f(p)}] + m’ $

Now, assume by induction that we have already determined the mystery numbers $\alpha_r $ for all odd primes $r < p $, then by teh argument of last time we can effectively compute in the field $[\kappa_p] $. In particular, we can compute for every element its multiplicative order $ord(\beta) $ and therefore also its degree $d(\beta) $ which has to be the smallest number $h $ such that $ord(\beta) $ divides $[2^h-1] $.

Then, by the main result we only have to determine the smallest number m such that $\beta = [\kappa_{f(p)} +m] $ is not a p-th power in $\kappa_p $ which is equivalent to the condition that

$\beta^{(2^{d(\beta)}-1)/p} \not= 1 $ if $p $ divides $[2^{d(\beta)}-1] $

All these conditions can be verified within suitable finite fields and hence are effective. In this manner, Lenstra could extend Conway’s calculations (probably using a home-made finite field program running on a slow 1977 machine) :

[tex]\begin{array}{c|c|c} p & f(p) & \alpha_p \\ \hline 3 & 2 & [2] \\ 5 & 4 & [4] \\ 7 & 3 & [\omega]+1 \\ 11 & 10 & [\omega^{\omega}]+1 \\ 13 & 12 & [\omega]+4 \\ 17 & 8 & [16] \\ 19 & 18 & [\omega^3]+4 \\ 23 & 11 & [\omega^{\omega^3}]+1 \\ 29 & 28 & [\omega^{\omega^2}]+4 \\ 31 & 5 & [\omega^{\omega}]+1 \\ 37 & 36 & [\omega^3]+4 \\ 41 & 20 & [\omega^{\omega}]+1 \\ 43 & 14 & [\omega^{\omega^2}]+ 1 \end{array}[/tex]

Right, so let’s try the case $p=47 $. To begin, $f(47)=23 $ whence we have to determine the smallest field containg $\kappa_{23} $. By induction (Lenstra’s tabel) we know already that

$\kappa_{23}^{23} = \kappa_{11} + 1 = [\omega^{\omega^3}]+1 $ and $\kappa_{11}^{11} = \kappa_5 + 1 = [\omega^{\omega}]+1 $ and $\kappa_5^5=[4] $

Because the smallest field containg $4 $ is $[16]=\mathbb{F}_{2^4} $ we have that $\mathbb{F}_2(4,\kappa_5,\kappa_{11}) \simeq \mathbb{F}_{2^{220}} $. We can construct this finite field, together with a generator $a $ of its multiplicative group in Sage via


sage: f1.< a >=GF(2^220)

In this field we have to pinpoint the elements $4,\kappa_5 $ and $\kappa_{11} $. As $4 $ has order $15 $ in $\mathbb{F}_{2^4} $ we know that $\kappa_5 $ has order $75 $. Hence we can take $\kappa_5 = a^{(2^{220}-1)/75} $ and then $4=\kappa_5^5 $.

If we denote $\kappa_5 $ by x5 we can obtain $\kappa_{11} $ as x11 by the following sage-commands


sage: c=x5+1

sage: x11=c.nth_root(11)

It takes about 7 minutes to find x11 on a 2.4 GHz MacBook. Next, we have to set up the field extension determined by $\kappa_{23} $ (which we will call x in sage). This is done as follows


sage: p1.=PolynomialRing(f1)

sage: f=x^23-x11-1

sage: F2=f1.extension(f,'u')

The MacBook needed 8 minutes to set up this field which is isomorphic to $\mathbb{F}_{2^{5060}} $. The relevant number is therefore $n=\frac{2^{5060}-1}{47} $ which is the gruesome

34648162040462867047623719793206539850507437636617898959901744136581<br/>
259144476645069239839415478030722644334257066691823210120548345667203443<br/>
317743531975748823386990680394012962375061822291120459167399032726669613
<br/>
442804392429947890878007964213600720766879334103454250982141991553270171

938532417844211304203805934829097913753132491802446697429102630902307815

301045433019807776921086247690468136447620036910689177286910624860871748

150613285530830034500671245400628768674394130880959338197158054296625733

206509650361461537510912269982522844517989399782602216622257291361930850

885916974186835958466930689748400561295128553674118498999873244045842040

080195019701984054428846798610542372150816780493166669821114184374697446

637066566831036116390063418916814141753876530004881539570659100352197393

997895251223633176404672792711603439161147155163219282934597310848529360

118189507461132290706604796116111868096099527077437183219418195396666836

014856037176421475300935193266597196833361131333604528218621261753883518

667866835204501888103795022437662796445008236823338104580840186181111557

498232520943552183185687638366809541685702608288630073248626226874916669

186372183233071573318563658579214650042598011275864591248749957431967297

975078011358342282941831582626985121760847852546207377440873367589369439

085660784239080183415569559585998884824991911321095149718147110882474280

968166266224151511519773175933506503369761671964823112231808283557885030

984081329986188655169245595411930535264918359325712373064120338963742590

76555755141425

Remains ‘only’ to take x,x+1,etc. to the n-th power and verify which is the first to be unequal to 1. For this it is best to implement the usual powering trick (via digital expression of the exponent) in the field F2, something like


sage: def power(e,n):
...: le=n.bits()
...: v=n.digits()
...: mn=F2(e)
...: out=F2(1)
...: i=0
...: while i< le :
...: if v[i]==1 : out=F2(out_mn)
...: m=F2(mn_mn)
...: mn=F2(m)
...: i=i+1
...: return(out)
...:

then it takes about 20 seconds to verify that power(x,n)=1 but that power(x+1,n) is NOT! That is, we just checked that $\alpha_{47}=\kappa_{11}+1 $.

It turns out that 47 is the hardest nut to crack, the following primes are easier. Here’s the data (if I didn’t make mistakes…)

[tex]\begin{array}{c|c|c} p & f(p) & \alpha_p \\ \hline 47 & 23 & [\omega^{\omega^{7}}]+1 \\ 53 & 52 & [\omega^{\omega^4}]+1 \\ 59 & 58 & [\omega^{\omega^8}]+1 \\ 61 & 60 & [\omega^{\omega}]+[\omega] \\ 67 & 66 & [\omega^{\omega^3}]+[\omega] \end{array}[/tex]

It seems that Magma is substantially better at finite field arithmetic, so if you are lucky enough to have it you’ll have no problem finding $\alpha_p $ for all primes less than 100 by the end of the day. If you do, please drop a comment with the results…

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On2 : Conway’s nim-arithmetics

Last time we did recall Cantor’s addition and multiplication on ordinal numbers. Note that we can identify an ordinal number $\alpha $ with (the order type of) the set of all strictly smaller ordinals, that is, $\alpha = { \alpha’~:~\alpha’ < \alpha } $. Given two ordinals $\alpha $ and $\beta $ we will denote their Cantor-sums and products as $[ \alpha + \beta] $ and $[\alpha . \beta] $.

The reason for these square brackets is that John Conway constructed a well behaved nim-addition and nim-multiplication on all ordinals $\mathbf{On}_2 $ by imposing the ‘simplest’ rules which make $\mathbf{On}_2 $ into a field. By this we mean that, in order to define the addition $\alpha + \beta $ we must have constructed before all sums $\alpha’ + \beta $ and $\alpha + \beta’ $ with $\alpha’ < \alpha $ and $\beta’ < \beta $. If + is going to be a well-defined addition on $\mathbf{On}_2 $ clearly $\alpha + \beta $ cannot be equal to one of these previously constructed sums and the ‘simplicity rule’ asserts that we should take $\alpha+\beta $ the least ordinal different from all these sums $\alpha’+\beta $ and $\alpha+\beta’ $. In symbols, we define

$\alpha+ \beta = \mathbf{mex} { \alpha’+\beta,\alpha+ \beta’~|~\alpha’ < \alpha, \beta’ < \beta } $

where $\mathbf{mex} $ stands for ‘minimal excluded value’. If you’d ever played the game of Nim you will recognize this as the Nim-addition, at least when $\alpha $ and $\beta $ are finite ordinals (that is, natural numbers) (to nim-add two numbers n and m write them out in binary digits and add without carrying). Alternatively, the nim-sum n+m can be found applying the following two rules :

  • the nim-sum of a number of distinct 2-powers is their ordinary sum (e.g. $8+4+1=13 $, and,
  • the nim-sum of two equal numbers is 0.

So, all we have to do is to write numbers n and m as sums of two powers, scratch equal terms and add normally. For example, $13+7=(8+4+1)+(4+2+1)=8+2=10 $ (of course this is just digital sum without carry in disguise).

Here’s the beginning of the nim-addition table on ordinals. For example, to define $13+7 $ we have to look at all values in the first 7 entries of the row of 13 (that is, ${ 13,12,15,14,9,8,11 } $) and the first 13 entries in the column of 7 (that is, ${ 7,6,5,4,3,2,1,0,15,14,13,12,11 } $) and find the first number not included in these two sets (which is indeed $10 $).

In fact, the above two rules allow us to compute the nim-sum of any two ordinals. Recall from last time that every ordinal can be written uniquely as as a finite sum of (ordinal) 2-powers :
$\alpha = [2^{\alpha_0} + 2^{\alpha_1} + \ldots + 2^{\alpha_k}] $, so to determine the nim-sum $\alpha+\beta $ we write both ordinals as sums of ordinal 2-powers, delete powers appearing twice and take the Cantor ordinal sum of the remaining sum.

Nim-multiplication of ordinals is a bit more complicated. Here’s the definition as a minimal excluded value

$\alpha.\beta = \mathbf{mex} { \alpha’.\beta + \alpha.\beta’ – \alpha’.\beta’ } $

for all $\alpha’ < \alpha, \beta’ < \beta $. The rationale behind this being that both $\alpha-\alpha’ $ and $\beta – \beta’ $ are non-zero elements, so if $\mathbf{On}_2 $ is going to be a field under nim-multiplication, their product should be non-zero (and hence strictly greater than 0), that is, $~(\alpha-\alpha’).(\beta-\beta’) > 0 $. Rewriting this we get $\alpha.\beta > \alpha’.\beta+\alpha.\beta’-\alpha’.\beta’ $ and again the ‘simplicity rule’ asserts that $\alpha.\beta $ should be the least ordinal satisfying all these inequalities, leading to the $\mathbf{mex} $-definition above. The table gives the beginning of the nim-multiplication table for ordinals. For finite ordinals n and m there is a simple 2 line procedure to compute their nim-product, similar to the addition-rules mentioned before :

  • the nim-product of a number of distinct Fermat 2-powers (that is, numbers of the form $2^{2^n} $) is their ordinary product (for example, $16.4.2=128 $), and,
  • the square of a Fermat 2-power is its sesquimultiple (that is, the number obtained by multiplying with $1\frac{1}{2} $ in the ordinary sense). That is, $2^2=3,4^2=6,16^2=24,… $

Using these rules, associativity and distributivity and our addition rules it is now easy to work out the nim-multiplication $n.m $ : write out n and m as sums of (multiplications by 2-powers) of Fermat 2-powers and apply the rules. Here’s an example

$5.9=(4+1).(4.2+1)=4^2.2+4.2+4+1=6.2+8+4+1=(4+2).2+13=4.2+2^2+13=8+3+13=6 $

Clearly, we’d love to have a similar procedure to calculate the nim-product $\alpha.\beta $ of arbitrary ordinals, or at least those smaller than $\omega^{\omega^{\omega}} $ (recall that Conway proved that this ordinal is isomorphic to the algebraic closure $\overline{\mathbb{F}}_2 $ of the field of two elements). From now on we restrict to such ‘small’ ordinals and we introduce the following special elements :

$\kappa_{2^n} = [2^{2^{n-1}}] $ (these are the Fermat 2-powers) and for all primes $p > 2 $ we define
$\kappa_{p^n} = [\omega^{\omega^{k-1}.p^{n-1}}] $ where $k $ is the number of primes strictly smaller than $p $ (that is, for p=3 we have k=1, for p=5, k=2 etc.).

Again by associativity and distributivity we will be able to multiply two ordinals $< \omega^{\omega^{\omega}} $ if we know how to multiply a product

$[\omega^{\alpha}.2^{n_0}].[\omega^{\beta}.2^{m_0}] $ with $\alpha,\beta < [\omega^{\omega}] $ and $n_0,m_0 \in \mathbb{N} $.

Now, $\alpha $ can be written uniquely as $[\omega^t.n_t+\omega^{t-1}.n_{t-1}+\ldots+\omega.n_2 + n_1] $ with t and all $n_i $ natural numbers. Write each $n_k $ in base $p $ where $p $ is the $k+1 $-th prime number, that is, we have for $n_0,n_1,\ldots,n_t $ an expression

$n_k=[\sum_j p^j.m(j,k)] $ with $0 \leq m(j,k) < p $

The point of all this is that any of the special elements we want to multiply can be written as a unique expression as a decreasing product

$[\omega^{\alpha}.2^{n_0}] = [ \prod_q \kappa_q^m(q) ] $

where $q $ runs over all prime powers. The crucial fact now is that for this decreasing product we have a rule similar to addition of 2-powers, that is Conway-products coincide with the Cantor-products

$[ \prod_q \kappa_q^m(q) ] = \prod_q \kappa_q^m(q) $

But then, using associativity and commutativity of the Conway-product we can ‘nearly’ describe all products $[\omega^{\alpha}.2^{n_0}].[\omega^{\beta}.2^{m_0}] $. The remaining problem being that it may happen that for some q we will end up with an exponent $m(q)+m(q’)>p $. But this can be solved if we know how to take p-powers. The rules for this are as follows

$~(\kappa_{2^n})^2 = \kappa_{2^n} + \prod_{1 \leq i < n} \kappa_{2^i} $, for 2-powers, and,

$~(\kappa_{p^n})^p = \kappa_{p^{n-1}} $ for a prime $p > 2 $ and for $n \geq 2 $, and finally

$~(\kappa_p)^p = \alpha_p $ for a prime $p > 2 $, where $\alpha_p $ is the smallest ordinal $< \kappa_p $ which cannot be written as a p-power $\beta^p $ with $\beta < \kappa_p $. Summarizing : if we will be able to find these mysterious elements $\alpha_p $ for all prime numbers p, we are able to multiply in $[\omega^{\omega^{\omega}}]=\overline{\mathbb{F}}_2 $.

Let us determine the first one. We have that $\kappa_3 = \omega $ so we are looking for the smallest natural number $n < \omega $ which cannot be written in num-multiplication as $n=m^3 $ for $m < \omega $ (that is, also $m $ a natural number). Clearly $1=1^3 $ but what about 2? Can 2 be a third root of a natural number wrt. nim-multiplication? From the tabel above we see that 2 has order 3 whence its cube root must be an element of order 9. Now, the only finite ordinals that are subfields of $\mathbf{On}_2 $ are precisely the Fermat 2-powers, so if there is a finite cube root of 2, it must be contained in one of the finite fields $[2^{2^n}] $ (of which the mutiplicative group has order $2^{2^n}-1 $ and one easily shows that 9 cannot be a divisor of any of the numbers $2^{2^n}-1 $, that is, 2 doesn’t have a finte 3-th root in nim! Phrased differently, we found our first mystery number $\alpha_3 = 2 $. That is, we have the marvelous identity in nim-arithmetic

$\omega^3 = 2 $

Okay, so what is $\alpha_5 $? Well, we have $\kappa_5 = [\omega^{\omega}] $ and we have to look for the smallest ordinal which cannot be written as a 5-th root. By inspection of the finite nim-table we see that 1,2 and 3 have 5-th roots in $\omega $ but 4 does not! The reason being that 4 has order 15 (check in the finite field [16]) and 25 cannot divide any number of the form $2^{2^n}-1 $. That is, $\alpha_5=4 $ giving another crazy nim-identity

$~(\omega^{\omega})^5 = 4 $

And, surprises continue to pop up… Conway showed that $\alpha_7 = \omega+1 $ giving the nim-identity $~(\omega^{\omega^2})^7 = \omega+1 $. The proof of this already uses some clever finite field arguments. Because 7 doesn’t divide any number $2^{2^n}-1 $, none of the finite subfields $[2^{2^n}] $ contains a 7-th root of unity, so the 7-power map is injective whence surjective, so all finite ordinal have finite 7-th roots! That is, $\alpha_7 \geq \omega $. Because $\omega $ lies in a cubic extension of the finite field [4], the field generated by $\omega $ has 64 elements and so its multiplicative group is cyclic of order 63 and as $\omega $ has order 9, it must be a 7-th power in this field. But, as the only 7th powers in that field are precisely the powers of $\omega $ and by inspection $\omega+1 $ is not a 7-th power in that field (and hence also not in any field extension obtained by adjoining square, cube and fifth roots) so $\alpha_7=\omega +1 $.

Conway did stop at $\alpha_7 $ but I’ve always been intrigued by that one line in ONAG p.61 : “Hendrik Lenstra has computed $\alpha_p $ for $p \leq 43 $”. Next time we will see how Lenstra managed to do this and we will use sage to extend his list a bit further, including the first open case : $\alpha_{47}= \omega^{\omega^7}+1 $.

For an enjoyable video on all of this, see Conway’s MSRI lecture on Infinite Games. The nim-arithmetic part is towards the end of the lecture but watching the whole video is a genuine treat!

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On2 : transfinite number hacking

In ONAG, John Conway proves that the symmetric version of his recursive definition of addition and multiplcation on the surreal numbers make the class On of all Cantor’s ordinal numbers into an algebraically closed Field of characteristic two : On2 (pronounced ‘Onto’), and, in particular, he identifies a subfield
with the algebraic closure of the field of two elements. What makes all of this somewhat confusing is that Cantor had already defined a (badly behaving) addition, multiplication and exponentiation on ordinal numbers.

Over the last week I’ve been playing a bit with sage to prove a few exotic identities involving ordinal numbers. Here’s one of them ($\omega $ is the first infinite ordinal number, that is, $\omega={ 0,1,2,\ldots } $),

$~(\omega^{\omega^{13}})^{47} = \omega^{\omega^7} + 1 $

answering a question in Hendrik Lenstra’s paper Nim multiplication.

However, it will take us a couple of posts before we get there. Let’s begin by trying to explain what brought this on. On september 24th 2008 there was a meeting, intended for a general public, called a la rencontre des dechiffeurs, celebrating the 50th birthday of the IHES.

One of the speakers was Alain Connes and the official title of his talk was “L’ange de la géométrie, le diable de l’algèbre et le corps à un élément” (the angel of geometry, the devil of algebra and the field with one element). Instead, he talked about a seemingly trivial problem : what is the algebraic closure of $\mathbb{F}_2 $, the field with two elements? My only information about the actual content of the talk comes from the following YouTube-blurb

Alain argues that we do not have a satisfactory description of $\overline{\mathbb{F}}_2 $, the algebraic closure of $\mathbb{F}_2 $. Naturally, it is the union (or rather, limit) of all finite fields $\mathbb{F}_{2^n} $, but, there are too many non-canonical choices to make here.

Recall that $\mathbb{F}_{2^k} $ is a subfield of $\mathbb{F}_{2^l} $ if and only if $k $ is a divisor of $l $ and so we would have to take the direct limit over the integers with respect to the divisibility relation… Of course, we can replace this by an increasing sequence of a selection of cofinal fields such as

$\mathbb{F}_{2^{1!}} \subset \mathbb{F}_{2^{2!}} \subset \mathbb{F}_{2^{3!}} \subset \ldots $

But then, there are several such suitable sequences! Another ambiguity comes from the description of $\mathbb{F}_{2^n} $. Clearly it is of the form $\mathbb{F}_2[x]/(f(x)) $ where $f(x) $ is a monic irreducible polynomial of degree $n $, but again, there are several such polynomials. An attempt to make a canonical choice of polynomial is to take the ‘first’ suitable one with respect to some natural ordering on the polynomials. This leads to the so called Conway polynomials.

Conway polynomials for the prime $2 $ have only been determined up to degree 400-something, so in the increasing sequence above we would already be stuck at the sixth term $\mathbb{F}_{2^{6!}} $…

So, what Alain Connes sets as a problem is to find another, more canonical, description of $\overline{\mathbb{F}}_2 $. The problem is not without real-life interest as most finite fields appearing in cryptography or coding theory are subfields of $\overline{\mathbb{F}}_2 $.

(My guess is that Alain originally wanted to talk about the action of the Galois group on the roots of unity, which would be the corresponding problem over the field with one element and would explain the title of the talk, but decided against it. If anyone knows what ‘coupling-problem’ he is referring to, please drop a comment.)

Surely, Connes is aware of the fact that there exists a nice canonical recursive construction of $\overline{\mathbb{F}}_2 $ due to John Conway, using Georg Cantor’s ordinal numbers.

In fact, in chapter 6 of his book On Numbers And Games, John Conway proves that the symmetric version of his recursive definition of addition and multiplcation on the surreal numbers make the class $\mathbf{On} $ of all Cantor’s ordinal numbers into an algebraically closed Field of characteristic two : $\mathbf{On}_2 $ (pronounced ‘Onto’), and, in particular, he identifies a subfield

$\overline{\mathbb{F}}_2 \simeq [ \omega^{\omega^{\omega}} ] $

with the algebraic closure of $\mathbb{F}_2 $. What makes all of this somewhat confusing is that Cantor had already defined a (badly behaving) addition, multiplication and exponentiation on ordinal numbers. To distinguish between the Cantor/Conway arithmetics, Conway (and later Lenstra) adopt the convention that any expression between square brackets refers to Cantor-arithmetic and un-squared ones to Conway’s. So, in the description of the algebraic closure just given $[ \omega^{\omega^{\omega}} ] $ is the ordinal defined by Cantor-exponentiation, whereas the exotic identity we started out with refers to Conway’s arithmetic on ordinal numbers.

Let’s recall briefly Cantor’s ordinal arithmetic. An ordinal number $\alpha $ is the order-type of a totally ordered set, that is, if there is an order preserving bijection between two totally ordered sets then they have the same ordinal number (or you might view $\alpha $ itself as a totally ordered set, namely the set of all strictly smaller ordinal numbers, so e.g. $0= \emptyset,1= { 0 },2={ 0,1 },\ldots $).

For two ordinals $\alpha $ and $\beta $, the addition $[\alpha + \beta ] $ is the order-type of the totally ordered set $\alpha \sqcup \beta $ (the disjoint union) ordered compatible with the total orders in $\alpha $ and $\beta $ and such that every element of $\beta $ is strictly greater than any element from $\alpha $. Observe that this definition depends on the order of the two factors. For example,$ [1 + \omega] = \omega $ as there is an order preserving bijection ${ \tilde{0},0,1,2,\ldots } \rightarrow { 0,1,2,3,\ldots } $ by $\tilde{0} \mapsto 0,n \mapsto n+1 $. However, $\omega \not= [\omega + 1] $ as there can be no order preserving bijection ${ 0,1,2,\ldots } \rightarrow { 0,1,2,\ldots,0_{max} } $ as the first set has no maximal element whereas the second one does. So, Cantor’s addition has the bad property that it may be that $[\alpha + \beta] \not= [\beta + \alpha] $.

The Cantor-multiplication $ \alpha . \beta $ is the order-type of the product-set $\alpha \times \beta $ ordered via the last differing coordinate. Again, this product has the bad property that it may happen that $[\alpha . \beta] \not= [\beta . \alpha] $ (for example $[2 . \omega ] \not=[ \omega . 2 ] $). Finally, the exponential $\beta^{\alpha} $ is the order type of the set of all maps $f~:~\alpha \rightarrow \beta $ such that $f(a) \not=0 $ for only finitely many $a \in \alpha $, and ordered via the last differing function-value.

Cantor’s arithmetic allows normal-forms for ordinal numbers. More precisely, with respect to any ordinal number $\gamma \geq 2 $, every ordinal number $\alpha \geq 1 $ has a unique expression as

$\alpha = [ \gamma^{\alpha_0}.\eta_0 + \gamma^{\alpha_1}.\eta_1 + \ldots + \gamma^{\alpha_m}.\eta_m] $

for some natural number $m $ and such that $\alpha \geq \alpha_0 > \alpha_1 > \ldots > \alpha_m \geq 0 $ and all $1 \leq \eta_i < \gamma $. In particular, taking the special cases $\gamma = 2 $ and $\gamma = \omega $, we have the following two canonical forms for any ordinal number $\alpha $

$[ 2^{\alpha_0} + 2^{\alpha_1} + \ldots + 2^{\alpha_m}] = \alpha = [ \omega^{\beta_0}.n_0 + \omega^{\beta_1}.n_1 + \ldots + \omega^{\beta_k}.n_k] $

with $m,k,n_i $ natural numbers and $\alpha \geq \alpha_0 > \alpha_1 > \ldots > \alpha_m \geq 0 $ and $\alpha \geq \beta_0 > \beta_1 > \ldots > \beta_k \geq 0 $. Both canonical forms will be important when we consider the (better behaved) Conway-arithmetic on $\mathbf{On}_2 $, next time.

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best of 2008 (2) : big theorems

Charles Siegel of Rigorous Trivialities ran a great series on big theorems.

The series started january 10th 2008 with a post on Bezout’s theorem, followed by posts on Chow’s lemma, Serre duality, Riemann-Roch, Bertini, Nakayama’s lemma, Groebner bases, Hurwitz to end just before christmas with a post on Kontsevich’s formula.

Also at other blogs, 2008 was the year of series of long posts containing substantial pure mathematics.

Out of many, just two examples : Chris Schommer-Pries ran a three part series on TQFTs via planar algebras starting here, at the secret blogging seminar.
And, Peter Woit of Not Even Wrong has an ungoing series of posts called Notes on BRST, starting here. At the moment he is at episode nine.

It suffices to have a quick look at the length of any of these posts, to see that a great deal of work was put into these series (and numerous similar ones, elsewhere). Is this amount of time well spend? Or, should we focus on shorter, easier digestible math-posts?

What got me thinking was this merciless comment Charles got after a great series of posts leading up to Kontsevich’s formula :

“Perhaps you should make a New Years commitment to not be so obscurantist, like John Armstrong, and instead promote the public understanding of math!”

Well, if this doesn’t put you off blogging for a while, what will?

So, are we really writing the wrong sort of posts? Do math-blog readers only want short, flashy, easy reading posts these days? Or, is anyone out there taking notice of the hard work it takes to write such a technical post, let alone a series of them?

At first I was rather pessimistic about the probable answer to all these questions, but, fortunately we have Google Analytics to quantify things a bit.

Clearly I can only rely on the statistics for my own site, so I’ll treat the case of a recent post here : Mumford’s treasure map which tried to explain the notion of a generic point and how one might depict an affine scheme.

Here’s some of the Google Analytics data :



The yellow function gives the number of pageviews for that post, the value ranges between 0 and 600 (the number to the right of the picture). In total this post was viewed 2470 times, up till now.

The blue function tells the average time a visitor spend reading that post, the numbers range between 0 and 8 minutes (the times to the left of the picture). On average the time-on-page was 2.24 minutes, so in all people spend well over 92 hours reading this one post! This seems like a good return for the time it took me to write it…

Some other things can be learned from this data. Whereas the number of page-views has two peaks early on (one the day it was posted, the second one when Peter Woit linked to it) and is now steadily decreasing, the time-on-page for the later visitors is substantially longer than the early readers.

Some of this may be explained (see comment below) by returning visits. Here is a more detailed picture (orange = new visits, green=returning visits, blue=’total’ whatever this means).



All in all good news : there is indeed a market for longer technical math-posts and people (eventually) take time to read the post in detail.

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best of 2008 (1) : wiskundemeisjes

Of course, excellent math-blogs exist in every language imaginable, but my linguistic limitations restrict me to the ones written in English, French, German and … Dutch. Here a few links to Dutch (or rather, Flemish) math-blogs, in order of proximity :
Stijn Symens blog, Rudy Penne’s wiskunde is sexy (math is sexy), Koen Vervloesem’s QED.

My favorite one is wiskundemeisjes (‘math-chicks’ or ‘math-girls’), written by Ionica Smeets and Jeanine Daems, two reasearchers at Leiden University. Every month they have a post called “the favorite (living) mathematician of …” in which they ask someone to nominate and introduce his/her favorite colleague mathematician. Here some examples : Roger Penrose chooses Michael Atiyah, Robbert Dijkgraaf chooses Maxim Kontsevich, Frans Oort chooses David Mumford, Gunther Cornelissen chooses Yuri I. Manin, Hendrik Lenstra chooses Bjorn Poonen, etc. the full list is here or here. This series deserves a wider audience. Perhaps Ionica and Jeanine might consider translating some of these posts?

I’m certain their English is far better than mine, so here’s a feeble attempt to translate the one post in their series they consider a complete failure (it isn’t even listed in the category). Two reasons for me to do so : it features Matilde Marcolli (one of my own favorite living mathematicians) and Matilde expresses here very clearly my own take on popular-math books/blogs.

The original post was written by Ionica and was called Weg met de ‘favoriete wiskundige van…’ :

“This week I did spend much of my time at the Fifth European Mathematical Congress in Amsterdam. Several mathematicians suggested I should have a chat with Matilde Marcolli, one of the plenary speakers. It seemed like a nice idea to ask her about her favorite (still living) mathematician, for our series.

Marcolli explained why she couldn’t answer this question : she has favorite mathematical ideas, but it doesn’t interest her one bit who discovered or proved them. And, there are mathematicians she likes, but that’s because she finds them interesting as human beings, independent of their mathematical achievements.

In addition, she thinks it’s a mistake to focus science too much on the persons. Scientific ideas should play the main role, not the scientists themselves. To her it is important to remember that many results are the combined effort of several people, that science doesn’t evolve around personalities and that scientific ideas are accessible to anyone.

Marcolli also dislikes the current trend in popular science writing: “I am completely unable to read popular-scientific books. As soon as they start telling anecdotes and stories, I throw away the book. I don’t care about their lives, I care about the real stuff.”

She’d love to read a popular science-book containing only ideas. She regrets that most of these books restrict to story-telling, but fail to disseminate the scientific ideas.”

Ionica then goes on to defend her own approach to science-popularization :

“… Probably, people will not know much about Galois-theory by reading about his turbulent life. Still, I can imagine people to become interested in ‘the real stuff’ after reading his biography, and, in this manner they will read some mathematics they wouldn’t have known to exist otherwise. But, Marcolli got me thinking, for it is true that almost all popular science-books focus on anecdotes rather than science itself. Is this wrong? For instance, do you want to see more mathematics here? I’m curious to hear your opinion on this.”

Even though my own approach is somewhat different, Ionica and Jeanine you’re doing an excellent job: “houden zo!”

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5 years blogging

Here’s a 5 move game from $\mathbb{C} $, the complex numbers game, annotated by Hendrik Lenstra in Nim multiplication.

$\begin{matrix} & \text{White} & \text{Black} \\ 1. & 3-2i & { 3_{\mathbb{R}} } \\ 2. & 3_{\mathbb{R}} & (22/7)_{\mathbb{Q}} \\ 3. & (-44_{\mathbb{Z}},-14_{\mathbb{Z}})? & { -44_{\mathbb{Z}} } \\ 4. & -44_{\mathbb{Z}} & ( 0_{\mathbb{N}},44_{\mathbb{N}} )! \\ 5. & \text{Resigns} & \\ \end{matrix} $

He writes : “The following 5 comments will make the rules clear.

1 : White selected a complex numbers. Black knows that $\mathbb{C} = \mathbb{R} \times \mathbb{R} $ by $a+bi = (a,b) $, and remembers Kuratowski’s definition of an ordered pair: $~(x,y) = { { x }, { x,y } } $. Thus black must choose an element of ${ { 3_{\mathbb{R}} }, { 3_{\mathbb{R}},-2_{\mathbb{R}} } } $. The index $\mathbb{R} $ here, and later $\mathbb{Q},\mathbb{Z} $ and $\mathbb{N} $, serve to distinguish between real numbers, rational numbers, integers and natural numbers usually denoted by the same symbol. Black’s move leaves White a minimum of choice, but it is not the best one.

2 : White has no choice. The Dedekind definition of $\mathbb{R} $ which the players agreed upon identifies a real number with the set of all strictly larger rational numbers; so Black’s move is legal.

3 : A rational number is an equivalence class of pairs of integers $~(a,b) $ with $b \not= 0 $; here $~(a,b) $ represents the rational number $a/b $. The question mark denotes that White’s move is a bad one.

4 : The pair $~(a,b) $ of natural numbers represents the integer $a-b $. Black’s move is the only winning one.

5 : White resigns, since he can choose between ${ 0_{\mathbb{N}} } $ and ${ 0_{\mathbb{N}},44_{\mathbb{N}} } $. In both cases Black will reply by $0_{\mathbb{N}} $, which is the empty set” (and so wins because White has no move left).

These rules make it clear what we mean by the natural numbers $\mathbb{N} $ game, the $\mathbb{Z} $-game and the $\mathbb{Q} $ and $\mathbb{R} $ games. A sum of games is defined as usual (players are allowed to move in exactly one of the component games).

Here’s a 5 term exercise from Lenstra’s paper : Determine the unique winning move in the game $\mathbb{N} + \mathbb{Z} + \mathbb{Q} + \mathbb{R} + \mathbb{C} $

It will take you less than 5 minutes to solve this riddle. Some of the other ‘exercises’ in Lenstra’s paper may take you a lot longer, if not forever…

Exactly 5 years ago I wrote : “As it is probably better to run years behind than to stand eternally still, I’ll try out how much of a blogger I am in 2004.”

5 months ago this became : “from january 1st 2009, I’ll be moving out of here. I will leave the neverendingbooks-site intact for some time to come, so there is no need for you to start archiving it en masse, yet.”

5 minutes before the deadline, this will be my last post….

of 2008

less entropy in 2009!

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