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Which results from the last 30 years, in any area of mathematics, do you think are the most important ones?

Specifically, which are the ones that will have more impact across all math and/or settle significant questions, in your opinion? It does not matter if they are big, lond-proofed results like a classification theorem, or little, elementary lemmas proved in a paragraph.

To keep this reasonable, I propose that every contributor posts at most 10 results.

Please, write one result per post, so that we can all vote properly and have a real ranking afterwards. Also, please don't just write the name, write a little description (hopefully, self-contained) and put some link if desirable.

(My personal aim with this question is to learn. I hope to be a bit better versed after studying your answers!)

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closed as off-topic by András Bátkai, Michael Albanese, Ben Linowitz, Stefan Kohl, Dirk Jan 18 '17 at 19:05

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question does not appear to be about research level mathematics within the scope defined in the help center." – András Bátkai, Michael Albanese, Ben Linowitz
If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ The Classification of Finite Simple Groups! $\endgroup$ – Nick Gill Jan 17 '17 at 11:26
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    $\begingroup$ The proof of the modularity theorem. $\endgroup$ – Sylvain JULIEN Jan 17 '17 at 11:32
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    $\begingroup$ This post has been "put on hold as primarily opinion-based". So now an educated opinion is a bad thing for us to have and share? $\endgroup$ – Jose Brox Jan 17 '17 at 13:32
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    $\begingroup$ @JoseBrox, I see your point which is entirely valid... But I must admit that one thing that gave me pause with this question is that I would, generally speaking, prefer to avoid "ranking" mathematical work if I can. The system in which we work -- prestigious journals, getting grants, getting jobs -- forces this upon us at times and, to be honest, I don't much care for it. I know, of course, that sometimes there are good reasons to discriminate between different pieces of mathematics on grounds of value... but writing lists of this sort of thing is so subjective that it bothers me a bit... $\endgroup$ – Nick Gill Jan 17 '17 at 15:09
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    $\begingroup$ @Jose , yes. On this forum, it is a bad thing to have and share an educated opinion. Primarily because this place is intended as a repository for specific questions and answers to serve as a useful resource into the future, not a place to conduct opinion polls of limited temporal scope. If you want to turn your encyclopedia into a personal scrapbook, that is up to you. This forum is the wrong place for your question. Gerhard "That's Why There's A Blogosphere" Paseman, 2017.01.17. $\endgroup$ – Gerhard Paseman Jan 17 '17 at 15:29
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Taniyama, Shimura and later Weil conjectured around 60 years ago that the L-function of an elliptic curve arises from a modular form. This conjecture was known to entail the last Fermat theorem after pioneering works of Hellegouarch, Frey and Ribet. In 1994, after a very long and technical work made in isolation and relying on Galois representations, Wiles managed to prove this conjecture in the semi-stable case, which was enough to establish the veracity of the now called Fermat-Wiles theorem. Based on this work, an article published in 1999 by Breuil, Conrad, Diamond and Taylor establish the result in full generality.

This theorem is a tiny part of the Langlands program, dating from 1967.

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The proof of the Milnor-Bloch-Kato conjecture by Vladimir Voevodsky (Fields medal 2002):

One has a norm residue isomorphism $$\mathrm{K}^\mathrm{M}_n(k)/\ell \to \mathrm{H}^n(k, \mu_\ell^{\otimes n})$$ from Milnor $K$-theory to étale cohomology, $\ell$ invertible in $k$.

This is translated into a statement of the comparison of the cohomology of motivic complexes with respect to different topologies.

For $\ell = 2$, this is the Milnor conjecture. Even for this, Voevodsky got the Fields medal.

For $n = 0$, this is obvious, for $n = 1$, it follows from Hilbert's Theorem 90. A corollary is that the Galois cohomology ring is generated by elements of degree $1$.

It implies the Beilinson-Lichtenbaum conjecture, a more general conjecture for varieties, not just fields.

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    $\begingroup$ amazon.com/gp/aw/d/1568810830 ...Monastyrsky's book Modern Mathematics in the Light of the Fields Medals [this comment, perhaps still useful, was made before an edit -- Todd Trimble]. $\endgroup$ – Steve Huntsman Jan 17 '17 at 13:02
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In discrete math:

Keevash's probabilisitic proof of Steiner's conjecture (On the Existence of Designs) is perhaps the biggest recent-ish result in combinatorics [due to the problem's ridiculously long pedigree].

But if you allow a little further back, I think Szemeredi's regularity lemma would have to take the cake for its outrageous applicability.

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Probably one of the major developments in homotopy theory is the solution to the Kervaire invariant one problem by Hill, Hopkins and Ravenel. It showcased the power of equivariant homotopy theory in solving a problem that a priori has nothing to do with it. In particular the idea to use the slice spectral sequence to get a handle on the homotopy fixed point spectral sequence of a certain spectrum is a real eye-opener.

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Perelman's proof of the Geometrization conjecture (see here, here and here) was the crowning achievement of decades of work. It was the most important of Thurston's conjectures about the topology of 3-manifolds.

Oh, and I nearly forgot to mention that the Poincare Conjecture is a consequence.

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Agol's proof of the Virtual Haken conjecture was a wonderful application of the tools developed by Wise and his coauthors in geometric group theory to 3-manifold topology. The Virtual Haken conjecture, which can be thought of as the topological classification of compact 3-manifolds, is a fundamental result.

Agol's theorem:

Every cubulable hyperbolic group is virtually special.

also has other dramatic consequences for hyperbolic groups. It's arguably the most important theorem proved in both algebra and topology in the last five years.

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    $\begingroup$ I have learned from this post that "cubulable" is a word. $\endgroup$ – Gerry Myerson Jan 18 '17 at 21:00
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I'm going for:

Gauss's Class Number Problem

solved for $m\le100$ by Watkins in 2004, in 'Class Numbers of Imaginary Quadratic Fields'.

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  • $\begingroup$ Why does this answer have been negatively received? The explanation will be instructive too! $\endgroup$ – Jose Brox Jan 23 '17 at 1:02
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    $\begingroup$ @JoseBrox See this blog post for a discussion of this paper, and a criticism of this answers negative status. $\endgroup$ – tracing Jan 23 '17 at 2:27
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Kahn--Markovic's proofs of the Surface Subgroup conjecture and the Ehrenpreis conjecture.

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