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