While I was visiting AA last Spring, we got into a discussion about old algebra junk (God knows why) that came around to Vietà. I told AA that AI had given a lecture about the cohomology of some Lie groups that brought in as a technique something called the Vietà map. AA was shocked that an old-junk player like that could be playing rock and roll, but there it is.
I think I had told AA I would look into it but have forgotten all about it till now. The link above is to a .ps file (AA, if you want, I'll convert it to .pdf and mail it to you) of Etingoff's notes of a talk by Varchenko about Knizhnik Zamolodchikov eqn. A new section starts on the second page that's intelligible. The first is the end of the proof of a previously stated theorem.
Sunday, October 22, 2006
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Hey, JJ, ps is fine now that have at my fingertips the amalgamated computing fury of Canuckia West. But the link seems to be non-existent? Yes, I would be left agape at the prospect of seeing the pre-harpsichord Viete playing R&R...... but, first, the link?
JJ, you need to learn how to make links -- once you really master the art of clicking on things.
That said (this is a bit like my preamble tirades on frenchy pinko-lefty ribve gauchisme befor replying to a Pepe post) -- yes, the Vieta map is fun. I learned to use that term from a little gray tome by Arnol'd. Tried a couple of bottle of Italian wine tonight, though (kind of light, no comparison to full-bodied frenchy stuff), so I won't try to find a link. But if you press me on that (and perhaps, tell me if there are any drinkable Italian wines -- all that I tried upm to now are rather watery), I'll tell you more.
Try again. I have changed the link slightly. If you still have trouble, go here and feel lucky.
Watery and Italian and you want French instead? You must still be reeling from the expenditure of $12 on cheap rose' or whatever that Baron Merdique stuff was.
And (as Groucho Marx would say) another thing, Prof Html, all you had to do to find the appropriate link was roll over the link I gave. All it was missing was an 'http://'.
Maybe you ought to go back to that rose', AI.
Ok, so googled the term and came across some relevant hyperplanic papers. I see that Vieta's Map bares the rough same resemblance to what Vieta actually did as Riemann's "element of distance" does to Pythagoras' work. .....It is actually refreshing to see, 400 years after the fact, a mathematician finally get so honored. It had seemed that, once Descartes and Fermat got their bagfuls of wreaths, that there was nothing left for the maestro of the Analytical Art.
As for wine, and spirits, just had a Canadian [Molson]. It is Coors Norte.
It's exactly the same thing. Here is the book AI was talking about of Arnol'd's.
Well, it is not "exactly the same thing". For instance, the paper I saw stated it as a map from C[n] to C[n], and then the product of the nonzero roots has the alternating sign propert, etc.... Now, Vieta had no conception of C, never mind C[n]. That is, although the 16th Century had seen such as Cardano and Bombelli make working with roots of negative numbers a fruitful "imaginary" operation, and Vieta made some much smaller use of these, he sees it simply as a "mysterious operation" allowing real results to be arrived, for real polynomials, by "passage through imaginaries" Heck, [almost all the time he restricts his work to roots as positive numbers, uncommonly negative, and rarely introduces imaginaries. Remember, his "Analytic Art" is still closely tied to geometric constructions, and in fact is the background to the work of Descartes and Fermat, and others, on what became "Analytic Geometry"] . In particular, there was no inkling of 'the complex plane", or C as a system of numbers, until the 18th Century. [Wallis had tried in the late 17th Century, but his geometric conception of these was very far from being the complex plane].
So, no, they are NOT exactly the same. As for Pythagoras, he had a measure of distance with his/his schools "Theorem", and as Euclid makes clear the Greeks developed the notion of a "straight line" [segment].as being "the shortest distance between two points". The generalization of this to curved surfaces, then to the "element of distance" is indeed a far generalization, yet the key idea, locally, is kept. And Vieta's little play with roots, generalized into the "Vieta map' of hyperplanicating mathematics, has undergone indeed a far generalization.
But, if you think Vieta's work is "exactly the same", then by all means read his many works ["The Analytic Art" is just his most famous, he was quite prolific], and show me it. I, at least, have not seen it.
By the way, to give you a sense of what you might be up against in trying to read the original Vieta, read Mazur's little piece, of 4 years ago or so, in the Math Intelligencer on Bombelli's delineation of the arithmetical properties of imaginaries.
AA, there's gotta be better lager than Molson up there in the PacNW?
There is, but the joint I was at was out of draft beers, out of Pale Ales, and everything half decent in bottles was about 3 dollars more than what I cared to dish out.....It was Molson, or it was Coors, or it was Bud. I had no choice, man!
I have got a copy of the Mazur article. It is hard to copy into this blog well. Goddamn Springer. Anyway, you have a 'number' of details wrong, but I will read the article in depth at my earliest convenience.
I will be in a casket about then if these days have been any indication.
Well AA, since you brought the context and economics to the fore, I have to say you made the best choice out of them all. I don't care to shell out mucho dinero for one beer, all the time, like JJ does. Then again, that's why we pay him the big bucks.
I can only drink one a night or I go to the hospital the next day, mft. You should be so lucky.
Details? On Vieta? On Bombelli? What details exactly? Or, JJ, are you suggesting that the hyperplanic paper I came acroos should have stated it was a map from quaternionic n space to quaternionic n space?
The uncertainty of it, egad, could drive a man to Molson
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