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Some time ago Andrew Wiles proved fermat's last theorem. The four colour theorem has been proved and Kepler's Conjecture has been proved. But what is the most important mathematical proof yet to be devised?

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closed as not constructive by Asaf Karagila, Micah, Pedro Tamaroff, Austin Mohr, Did Sep 22 '12 at 21:58

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What does 'important' mean here? Practical applications? Historical significance? Perceived difficulty? Philosophical implications? Etc. –  Clive Newstead Sep 22 '12 at 21:04
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I don't think FTL was important. Famous, certainly. –  MJD Sep 22 '12 at 21:24
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The phrasing "…left" suggests that there was some original supply of unsolved problems that is being used up. That is not the case. –  MJD Sep 22 '12 at 21:26
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I do not like the fact that all it takes are five fast-fingered conservatives to shut down questions that can generate good answers. –  zyx Sep 22 '12 at 21:55
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@zyx: You should probably delete this comment. –  Asaf Karagila Sep 22 '12 at 22:06

2 Answers 2

up vote 3 down vote accepted

See the list in the Millennium Prize Problems (right pane) here: http://www.claymath.org/millennium/

I have always wondered how those could be ranked, but that is likely a matter of whom you are talking to.

HTH ~W

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"HTH" is "hope this helps", but it is usually deployed sarcastically. –  MJD Sep 22 '12 at 21:24
    
Hope This Helps, and I did not intend any sarcasm, my apologies if it came across that way (I never even realized that). ~A –  Amzoti Sep 22 '12 at 21:31

Mathematics has progressed from specific problems to long-range research programs. It is more the latter that are considered important.

There is the century-long project of understanding the parallelism between number theory and geometry and especially its manifestation in zeta and L functions. In some sense this includes everything to do with the Riemann hypothesis, the Langlands programs, special values of L-functions, motives, trace formulas, arithmetic/Arakelov geometry, complex geometry, and as though that stream of words were not enough, a large fraction of everything else in pure mathematics.

This does not necessarily have the practical or philosophical implications of P/NP or any strong connection to physics, but both in the enormous amount of structure already discovered, and the larger amount believed to be discoverable, this is a pure mathematicians' problem par excellence.

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