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(Old question: How much of mathematics is or could be done by the 'geniuses'?)

A lot of important theorems or even theories end up being named after or otherwise attributed to one person or a small group of people. This is often fair, but taken as an overall trend it can give the (hopefully false!) impression that the most important mathematics is being done by a small minority of mathematicians. What I'm wondering is, how much does the opposite phenomenon occur, where it's very clear that a result is a large team effort and that no small group of authors deserves the lion's share of the credit? The most obvious example I can think of is the Classification of Finite Simple Groups, but I'm sure there are others from different areas of maths.

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This question seems too discussion-y to me. I think it is more appropiate for other fora. –  Alberto García-Raboso Sep 28 '10 at 15:55
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Exemplar for "subjective and argumentative" ;). Perhaps every mathematician should do it's best and not ponder what would happen without him ;). –  Martin Brandenburg Sep 28 '10 at 15:56
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On reflection, I agree. Perhaps I should instead have asked something more concrete, such as what are some examples of important theorems that can't be boiled down to the work of one or two big names. The classification of finite simple groups would be an obvious example here - it seems unlikely that it would ever have been proved in a world with very few mathematicians, even if they were all brilliant. –  Colin Reid Sep 28 '10 at 16:53
    
Now that the question has been completely rewritten, perhaps someone should start a meta thread to attract attention for unclosing. –  JBL Sep 28 '10 at 18:28
    
@JBL: done: tea.mathoverflow.net/discussion/687/… –  Tony Huynh Sep 28 '10 at 23:14
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3 Answers

Aside from the Geometrization Theorem/Poincare conjecture, probably the deepest theorem in low-dimensional topology in the last 10 years is the classification of hyperbolic structures on 3-manifolds with finitely generated fundamental group. Aside from the topological type, it turns out they they are classified by certain invariants "at infinity" (either Riemann surfaces at infinity or so-called "ending laminations"). The proof of this uses work of an enormous number of people : Agol, Alhfors, Bers, Brock, Calegari, Canary, Gabai, Namazi, Kleineidam, Kra, Lecuire, Marden, Maskit, Masur, Minsky, Mostow, Ohshika, Prasad, Rees, Souto, Sullivan, Thurston, and probably people I'm forgetting.

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Most theorems in Mathematics build upon the work of many people. Maybe a good recent example is the proof of Serre's conjecture by Khare and Winterberger. They contributed a decisive, fundamental piece of the puzzle, but it depends on the work of e.g. Serre, Tate, Ribet, Mazur, Taylor, Wiles, Kisin, Dieulefait and many others.

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One possible answer (and hopefully the source of many future answers) is Tim Gowers' polymath experiment. In short, the idea of the project is to test if massively collaborative mathematics over the internet is possible.

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