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May 19, 2008

Ambiguity Theory

Posted by David Corfield

This paper – Ambiguity theory, old and new – is rather fun and would be good to understand thoroughly if we hope to get 2-Galois to do anything important. It’s by Yves André of the ENS, and refers to a comment made by Galois that he was working with a théorie de l’ambiguïté. Good to see Albert Lautman receiving a mention.

For those who want something less introductory, on the same day André has deposited Galois theory, motives and transcendental numbers. Lots there about Kontsevich and Zagier’s Periods, described in their article of that name in Mathematics Unlimited – 2001 and beyond, pages 771-808, unfortunately now no longer available on the Web.

Posted at May 19, 2008 1:48 PM UTC

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7 Comments & 1 Trackback

Re: Ambiguity Theory

Yes those two papers by Andre are very helpful to see the bigger picture and motivate algebro-geometric stuff.

As for the Periods paper of Kontsevich and Zagier parts are available on google books, it’s a shame the preprint isn’t online. This idea of finding algebraically investigable yet transcendental numbers is great indeed.

Posted by: tom on May 20, 2008 12:31 PM | Permalink | Reply to this

Link to “Periods”

Here’s a copy of Kontsevich and Zagier’s paper on periods.

The IHES preprint server seems to be having problems: if you click on any preprint prior to 2004, you get a technical error message of a type that users shouldn’t see. (I suspect this is related to the fact that the IHES have just revamped their entire site.) I don’t think Kontsevich and Zagier’s paper has been deliberately taken offline, which is why I feel OK about making a copy available.

Posted by: Tom Leinster on May 20, 2008 1:04 PM | Permalink | Reply to this

Re: Link to “Periods”

The Link to Kontsevich and Zagier’s Periods paper does not work. When I click on it nothing happens. Please send to me a link that does work. Thank you.

Bill Messing

Posted by: William Messing on September 29, 2008 11:23 PM | Permalink | Reply to this

Re: Link to “Periods”

If you click on the link with the right-hand mouse button, a little menu should appear. Selecting the appropriate item from the menu should allow you to save a copy of the paper. (For me, this item is labelled “Save Link As…” It might be slightly different on your browser.) A copy will then be saved to your computer, and you can view it in the usual way.

Posted by: Tom Leinster on September 30, 2008 12:09 AM | Permalink | Reply to this

Re: Link to “Periods”

There seems to be something odd about the PostScript file that I linked to. It’s readable by PostScript viewers, but not easily convertible into PDF, which means that some people can’t view it at all.

But I think I’ve fixed it, and you can get a readable pdf file here.

(I just tried to check the IHES site to make sure that my justification for hosting a copy of their paper, given above, still holds. But the whole IHES site is currently down.)

Posted by: Tom Leinster on September 30, 2008 3:19 AM | Permalink | Reply to this

Re: Ambiguity Theory

Here’s an attempted translation of a portion of Galois’ last letter, quoted in Yves Andre’s paper Ambiguity theory, old and new. It explains why ‘ambiguity theory’ is a perfect name for Galois theory:

My principal meditations for some time have been directed towards the application of the theory of ambiguity to transcendental analysis. It was a question of seeing a priori in a relation between quantities or transcendent functions, what exchanges one could make, which quantities one could substitute for the given quantities without the original relation ceasing to hold. That immediately made clear the impossibility of finding many expressions that one could look for. But I do not have time and my ideas are not yet well developed on this ground which is immense.

(This translation is a collaboration between James Dolan and Babelfish, neither of whom speak French. Improvements are welcome!)

Posted by: John Baez on May 26, 2008 8:55 PM | Permalink | Reply to this
Read the post Imperfections, Ambiguities and Physics
Weblog: The n-Category Café
Excerpt: Can Lautman's concept of the dialectical relationship between perfection and imperfection be realised in physics?
Tracked: August 30, 2008 10:04 AM

Information = Comprehension × Extension

The mother of all Galois correspondences may well be the supposed duality between extensions and intensions of concepts, a notion that incited C.S. Peirce to some degree of reflection, and out of which he developed his incipient theory of information to resolve the tension between ex- and in-.

Here’s one place where I’ve been studying the issue:

Information = Comprehension ×\times Extension

Posted by: Jon Awbrey on October 15, 2009 7:34 PM | Permalink | Reply to this

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