Timeline for Tools for the Langlands Program?
Current License: CC BY-SA 2.5
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 23, 2010 at 21:43 | answer | added | Greg Kuperberg | timeline score: 7 | |
| Feb 16, 2010 at 4:34 | comment | added | Emerton | The aspect of things that Zavosh is referring to relates to functoriality (see my answer). The fundamental lemma resolves a certain aspect of this, called endoscopy. Most instances of functoriality are not instances of endoscopy. (See Langlands's "Beyond endoscopy" papers.) As I discuss in my answer, the reciprocity conjecture (relating automorphic forms and motives) is yet another kettle of fish. | |
| Feb 16, 2010 at 3:02 | answer | added | Emerton | timeline score: 33 | |
| Feb 16, 2010 at 1:03 | vote | accept | Ben | ||
| Feb 16, 2010 at 1:03 | |||||
| Feb 15, 2010 at 23:36 | answer | added | Kevin Buzzard | timeline score: 44 | |
| Feb 15, 2010 at 23:28 | comment | added | Ben | @Zavosh: Thanks! That's precisely the type of response I was looking for. If I remember correctly, Langlands discusses (in an article I cannot recall) the scope and limitations of the trace formula as well. I was also curious about the proposed connections between autmorophic forms and motives, etc. Lots of wonderful math to study! | |
| Feb 15, 2010 at 22:53 | comment | added | Zavosh | Here is my vague, limited and non-geometric understanding. There is not a black box theory whose existence would prove the conjectures of Langlands as there was with the Weil conjectures. However, one general strategy is to use the Arthur-Selberg trace formula on two different reductive groups, match up the geometric sides of the formula (as much as possible), and then use the spectral sides to relate the automorphic forms of the groups. There are several technical difficulties in getting this to work in general, with a major problem having been the Fundamental Lemma, finally resolved by Ngo. | |
| Feb 15, 2010 at 21:56 | comment | added | Harry Gindi | If you check out either of the links I gave you, you'll see just how much stuff is actually being used. | |
| Feb 15, 2010 at 21:55 | comment | added | Harry Gindi | No, I mean, it's a really really huge program. I think that any satisfying answer to this question either doesn't exist or could take tens if not hundreds of pages. | |
| Feb 15, 2010 at 21:49 | comment | added | Ben | @fpqc: I understand, generally, what the Langlands program is about, but what I don't get, in a certain respect, are the difficulties are with it. Do we need some sort of cohomology theory? Or do we need to understand a certain mathematical object better? What type of machinery are we lacking? | |
| Feb 15, 2010 at 21:39 | comment | added | Harry Gindi | math.uchicago.edu/~mitya/langlands.html and math.utexas.edu/users/benzvi/Langlands.html | |
| Feb 15, 2010 at 21:37 | comment | added | Harry Gindi | This question is too broad in my opinion. I'm sure there are good papers giving an overview of the Langlands program. I know that Mitya Boyarchenko and David Ben Zvi have some stuff about geometric Langlands that might help you get up to speed. | |
| Feb 15, 2010 at 21:32 | history | asked | Ben | CC BY-SA 2.5 |