Talk:Fundamental lemma (Langlands program)

Needs much work. From a talk of Laumon it seems that local field is to be taken seriously: reduction step to the geometric case of formal power series fields over finite fields. Also reduction to the Lie algebra setting. And then a serious translation into geometry, and the Hitchin fibration. The proof seems to have taken five years to check. A full article here would be quite substantial. Charles Matthews (talk) 22:40, 21 December 2009 (UTC)


 * It is beyond me, but if Time thinks it was one of the "Top 10 Scientific Discoveries of 2009" then it needs some sort of article. Feel free to improve it. --Rumping (talk) 01:02, 22 December 2009 (UTC)


 * It's all about good PR. I don't think that Time is qualified to make that judgment. For starters, how can a 2004 (or 2008) proof of a technical result in one of the most specialized areas of mathematics, conjectured by Landlands in the late 1970s, be billed as a scientific discovery of 2009? In any case, the potential world readership for this topic is limited to a few dozen people working in automorphic forms and wikipedia seems a wrong medium to present a survey on it. Arcfrk (talk) 06:51, 27 December 2009 (UTC)


 * You may be correct about PR, but I came looking after the Time list, found a redlink and started the article. No doubt others came looking too. It is notable. --Rumping (talk) 19:21, 29 December 2009 (UTC)


 * It's an interesting slice of the history of modern mathematics, at the least: the number of mathematicians contributing to the proof of one "lemma" is perhaps an all-time record. I don't have any concerns about notability: Time has added &epsilon;, for a mathematician, but &epsilon; > 0 as usual. Charles Matthews (talk) 19:29, 29 December 2009 (UTC)

"Needs much work" is true, but it is also a vast understatement. If no one is wiling to go to the trouble of even stating in English what this conjecture means -- instead sticking readers with a bunch of symbolic mumbo-jumbo -- then this article would be better not existing. A bunch of undefined symbols should not be confused with a clear statement of Langlands's Fundamental lemma.

One more thing -- the phrase "fundamental lemma" is used in thousands of different contexts in mathematics, so this article should be renamed something more specific, like "Langlands's fundamental lemma".Daqu (talk) 18:39, 7 January 2010 (UTC)


 * I am not so sure about "thousands of different contexts": perhaps, I am too close to the field, but that is the only entity called "the fundamental lemma" I've ever encountered in mathematics. Perhaps, you are thinking of various fundamental theorems? Arcfrk (talk) 16:41, 8 January 2010 (UTC)


 * Well, now that the symbols have been removed, I wonder if the article is really improved at all. There is the problem with overstated criticism, and deletion of valid material, that the purposes of the encyclopedia are actually not served. Further, you should note that the title "fundamental lemma of Langlands and Shelstad" was there before, and was changed. Altogther a more constructive approach would be welcome. Charles Matthews (talk) 16:33, 8 January 2010 (UTC)


 * I take full responsibility for removing the "statement", as the symbols in it carried no meaning. If you look over one of the references, say Rapoport's laudatio or Dat's talk at Bourbaki seminar, no symbols even begin to appear until a lot of explanation has been given, perhaps, more than what is appropriate for wikipedia, and the statement, even in a special case, comes at the very end. Concerning the title, as far as I am aware, everyone in automorphic forms calls it simply the "fundamental lemma", with attribution to Langlands. That's why I changed the title to "fundamental lemma (automorphic forms)", with the parenthetical clause providing the context in accordance with wikipedia's conventions. What is the source of "Fundamental lemma of Langlands and Shelstad"? Arcfrk (talk) 16:59, 8 January 2010 (UTC)


 * I don't think the title matters so much: I was just pointing out that we have vehement opinions on both sides. As for the formula, there is a problem I pointed out on WT:WPM: the glossary of Lie algebras ducks the definitions of "regular element". So the basis for a concise explanation isn't there - it isn't interesting to explain technical mathematics by means of redlinks (or links to pages that don't have definitions). But there is no reason that there can't be an explanation by means of some special case(s). If someone comes along who will do that, then they can get the formula from the history. I've added quite a lot to the article, while knowing nothing about the content (I have some idea of the context). I don't intend to work on this more, at least at present. Charles Matthews (talk) 21:06, 8 January 2010 (UTC)

Okay, "fundamental lemma" according to Google can apply to Calculus of Variations, Sieve Theory, Advanced Calculus, Statistics (the Neyman-Pearson fundamental lemma), Interpolation Theory, Semigroup Theory, one about normal coordinates for spaces of paths, one for Dirchlet's theory for arithmetic progressions, one for weak limit directions of plane sets, one about the limit of a sum, one about multiple trigonometric sums, one for Homological Algebra, one about Information Theory, one in Functional Analysis,. . . and -- oh,yeah -- the Langlands one. (And that's just from the first few pages of the 60,000 or so Google hits.)

Also: I was not advocating the removal of the symbolic version, only for an explanation of it. I do agree that no symbols is better than meaningless ones, but best of all would be an explanation of their meaning.Daqu (talk) 01:00, 17 January 2010 (UTC)

Shelstad
I see her role in the conjectures has been downgraded. Charles Matthews (talk) 12:23, 16 February 2010 (UTC)

Move
I'd like to suggest moving this article to something like "Fundamental lemma (Langlands program)". The fundamental lemma is a local statement about orbital integrals on reductive groups, whereas automorphic forms are global objects to which, through much theory, the fundamental lemma eventually applies. And I'm not sure anything more specific than "Fundamental lemma (Langlands program)" (e.g. "Fundamental lemma (endoscopy)" or "Fundamental lemma (stable trace formula)") would be any better. RobHar (talk) 15:27, 25 March 2010 (UTC)


 * Seems OK to me. Charles Matthews (talk) 16:23, 25 March 2010 (UTC)

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