Comments on: Looking for F_un http://www.neverendingbooks.org/index.php/looking-for-f_un.html lieven le bruyn's blog Mon, 08 Mar 2010 18:45:28 +0100 http://wordpress.org/?v=2.9.1 hourly 1 By: javier http://www.neverendingbooks.org/index.php/looking-for-f_un.html/comment-page-1#comment-5289 javier Wed, 04 Jun 2008 13:25:01 +0000 http://www.neverendingbooks.org/?p=424#comment-5289 <p>Back to the point, just realized that the paper <em>Sur les analogues algebriques des groupes semi-simples complexes</em> by Jacques Tits, together with <em>Lectures on zeta functions and motives</em>, by Y. Manin, are available at Katia Consani's website: http://www.math.jhu.edu/~kc/</p> Back to the point, just realized that the paper Sur les analogues algebriques des groupes semi-simples complexes by Jacques Tits, together with Lectures on zeta functions and motives, by Y. Manin, are available at Katia Consani’s website: http://www.math.jhu.edu/~kc/

]]> By: javier http://www.neverendingbooks.org/index.php/looking-for-f_un.html/comment-page-1#comment-5288 javier Wed, 04 Jun 2008 13:20:12 +0000 http://www.neverendingbooks.org/?p=424#comment-5288 <p>I cannot say if the story about Faltings and Durov is true (Durov left MPI right before I arrived here), but in his lecture in Vanderbilt, right after defining the field with one element using Durov's approach, Eugene Ha showed how to construct <em>the field with no elements</em>. I wouldn't expect it to be of any use, though :-P</p> I cannot say if the story about Faltings and Durov is true (Durov left MPI right before I arrived here), but in his lecture in Vanderbilt, right after defining the field with one element using Durov’s approach, Eugene Ha showed how to construct the field with no elements. I wouldn’t expect it to be of any use, though :-P

]]> By: John Baez http://www.neverendingbooks.org/index.php/looking-for-f_un.html/comment-page-1#comment-5285 John Baez Wed, 04 Jun 2008 07:22:00 +0000 http://www.neverendingbooks.org/?p=424#comment-5285 <p>"I’d like to find out what the fuzz is all about..."</p> <p>Unless you're talking about fuzzy set theory, I think you mean "what the fuss is all about".</p> <p>But seriously: great post! Lots of mathematicians don't know about this "field with one element" problem - they're going to wake up too late and find mathematics transformed while they were snoozing! </p> <p>I hope you talk a bit about Durov's nice simple idea, even though you seem to be getting ready to explain the work of Connes and Marcolli, which I'm eager to hear more about.</p> <p>I heard that Faltings, after hearing a talk by Durov on the field with one element, said: "Very nice! Maybe next time you will tell us about the field with two elements!" </p> <p>Any truth to this?</p> “I’d like to find out what the fuzz is all about…”

Unless you’re talking about fuzzy set theory, I think you mean “what the fuss is all about”.

But seriously: great post! Lots of mathematicians don’t know about this “field with one element” problem – they’re going to wake up too late and find mathematics transformed while they were snoozing!

I hope you talk a bit about Durov’s nice simple idea, even though you seem to be getting ready to explain the work of Connes and Marcolli, which I’m eager to hear more about.

I heard that Faltings, after hearing a talk by Durov on the field with one element, said: “Very nice! Maybe next time you will tell us about the field with two elements!”

Any truth to this?

]]> By: Kea http://www.neverendingbooks.org/index.php/looking-for-f_un.html/comment-page-1#comment-5284 Kea Wed, 04 Jun 2008 05:34:22 +0000 http://www.neverendingbooks.org/?p=424#comment-5284 <p>But from a physical point of view one only cares about 'proving' RH (say by showing it to be independent of Set, or any 1-categorical system of axioms) because it just keeps getting in the way. The use of F1, however, is more directly relevant to something that might be called <i>cyclic n-multicategory</i> axioms, which should be recursive in n, starting at 1 or 0. In this context, saying that F1 does not exist is like saying that p=1 is not prime, which might be a valid point of view but is not quite correct.</p> But from a physical point of view one only cares about ‘proving’ RH (say by showing it to be independent of Set, or any 1-categorical system of axioms) because it just keeps getting in the way. The use of F1, however, is more directly relevant to something that might be called cyclic n-multicategory axioms, which should be recursive in n, starting at 1 or 0. In this context, saying that F1 does not exist is like saying that p=1 is not prime, which might be a valid point of view but is not quite correct.

]]> By: jb http://www.neverendingbooks.org/index.php/looking-for-f_un.html/comment-page-1#comment-5283 jb Wed, 04 Jun 2008 00:44:11 +0000 http://www.neverendingbooks.org/?p=424#comment-5283 <p>In the second sentence of fourth paragraph, you forgot to assume that your ring is nonzero!</p> <p>:)</p> In the second sentence of fourth paragraph, you forgot to assume that your ring is nonzero!

:)

]]> By: javier http://www.neverendingbooks.org/index.php/looking-for-f_un.html/comment-page-1#comment-5281 javier Tue, 03 Jun 2008 13:04:13 +0000 http://www.neverendingbooks.org/?p=424#comment-5281 <p>The "nonexistence" of [tex] F_1 [/tex] doesn't look like a big deal. After all, nobody want to stdy the properties of the field with one element, but the properties of <em>objects</em> (vector fields, varieties, schemes, or whatever) constructed over such field.</p> <p>Isn't that exactly the same philosophy underlying Representation Theory? You don't care about the algebra, but only about the category of modules over the algebra. If you take this point of view to the last extreme, it doesn't matter at all how the particular algebra is defined... and it shouldn't, since rep. theory wouldn't distinguish between Morita equivalent algebras.</p> <p>In the conference you mention, I found very interesting the talk by Lisa Carbone (her slides are linked somewhere in the post by Connes and Consani), and while reading some papers about the topic really liked the (never-published) one by Kapranov and Smirnov. Wasn't able to find the "original" stemming paper by Tits, so if anyone can provide a link to it, I'd be very grateful.</p> The “nonexistence” of [tex] F_1 [/tex] doesn’t look like a big deal. After all, nobody want to stdy the properties of the field with one element, but the properties of objects (vector fields, varieties, schemes, or whatever) constructed over such field.

Isn’t that exactly the same philosophy underlying Representation Theory? You don’t care about the algebra, but only about the category of modules over the algebra. If you take this point of view to the last extreme, it doesn’t matter at all how the particular algebra is defined… and it shouldn’t, since rep. theory wouldn’t distinguish between Morita equivalent algebras.

In the conference you mention, I found very interesting the talk by Lisa Carbone (her slides are linked somewhere in the post by Connes and Consani), and while reading some papers about the topic really liked the (never-published) one by Kapranov and Smirnov. Wasn’t able to find the “original” stemming paper by Tits, so if anyone can provide a link to it, I’d be very grateful.

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