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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!

5 Comments

beyond the blog

For starters, apologies for flooding your RSS-aggregators a couple of days ago. Ive been copying my posts at F_un mathematics and have cross-posted them here. I will continue to do so as I prefer to search just one blog instead of two to find stuff. Besides, it’s unclear how long the F_un site will survive. Javier will be moving from MPI to London later this month, and is uncertain on the implications this will have for his research. Other people who told they’d like to post at F_un haven’t done so far… and I see little point in continuing a singleton-‘group blog’.

Over at the secret blogging seminar there is an interesting series on TQFTs via planar algebras by Chris Schommer-Pries. They also had a few nice words on the design of the F_un-site (though their commenters prefer a ‘traditional’ blog-layout). I think these days most people read blogs via their RSS-feeds, so are ignorant about the actual layout of a blog until they want to follow up a story that interests them. Besides, the main point of using the open book wordpress theme, which is a so called ‘magazine’-theme, was to try to get more structure in the blog (such as : indicating the intented audience for a post, organizing posts wrt. the papers mentioned etc.). Still, such themes are designed for news-sites having new content every hour/day, something we cannot say of the F_un-site…

Also at the n-category cafe they are thinking aloud on how to improve the blog-medium for mathematics-research. See the discussion following David Corfield’s beyond the blog post. Often, the comment-thread of an n-cafe post is a better read than the actual post, but the blog-concept is not very good at picking-out interesting comments. That’s why they are trying to set up a wiki-like thing with pointers to such interesting discussions. It’s still early days but they’ve started the nLab (powered by instiki) and describe it as “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”. Surely an interesting experiment to follow.

Finally, a link to images des mathematiques which is a news-site-style blog on mathematics run by the CNRS (the French NSF). They give their posts ‘colours’, indicating the intented public, surely a simple idea we can all implement that will make math-blogs a lot more useful. They also have repeating topics, such as ‘the object of the month’, portraits of mathematicians etc. Perhaps an idea to follow-up by other math-societies.

If you have ideas to improve the structure and usability of math-blogs, please share them!

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This week at F_un Mathematics (1)

In case you haven’t noticed it yet : I’m not living here anymore.

My blogging is (at least for the moment) transfered to the F_un Mathematics blog which some prefer to call the “ceci n’est pas un corps”-blog, which is very fine with me.

Javier gave a talk at MPI on Soule’s approach to algebraic geometry over the elusive field with one element $\mathbb{F}_1 $ and wrote two posts about it The skeleton of Soule’s F_un geometry and Gadgets a la Soule. The rough idea being that a variety over the field with one element only acquires flesh after a base extension to $\mathbb{Z} $ and to cyclotomic integers.

I did some posts on a related (but conceptually somewhat easier) approach due to Alain Connes and Katia Consani. I’ve tried to explain their construction at the level of (mature) undergraduate students. So far, there are three posts part1, part2 and part3. Probably there is one more session to come in which I will explain why they need functors to graded sets.

In the weeks to come we plan to post about applications of this F_un-geometry to noncommutative geometry (the Bost-Connes system) and Grothendieck’s anabelian geometry (the theory of dessins d’enfant). I’ll try to leave a short account of the main posts here, but clearly you are invited to feed your feedreader this.

Perhaps I’ll return here for a week mid november to do some old-fashioned vacation blogging. I have to admit I did underestimate Numeo.fr. Rumours have it that our place is connected wirelessly to the web…

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