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This is an impossibly broad question, and makes the unwarranted assumption that Mathematics is a uniform field. It might make more sense to ask the same question restricted to, say, Mathematical Logic, or Differential Geometry, or Number Theory. Nevertheless, I think there may be a sense in which certain fields of mathematics are (temporarily) blocked by major unresolved conjectures, and others are not. In which case, a breakthrough in a "blocked" field might advance Mathematics—broadly construed—the most.

For example, prior to 2002, I would have ventured: the Poincaré conjecture & Thurston's geometrization conjecture. But since Perelman's resolution, I see (in my limited vision) no equivalent major conjecture outstanding in Differential Geometry. Similarly, one might might view Fermat's last theorem as a key blockage within Number Theory before the 1995 Wiles-Taylor resolution; or twin primes before Zhang's 2013 breakthrough and subsequent polymath's improvements.

My parochial, biased viewpoint is that resolution of the P=NP question, tomorrow, would represent the greatest advance in Mathematics. But perhaps others think that, e.g., resolution of the Riemann hypothesis would represent the larger advance? Of course, how a question is resolved makes a huge difference: A narrow proof is less an advance than a broad reconceptualization that reshapes the landscape.

Q. I welcome responses specific to narrow fields, but also remarks on Mathematics (if such exists!) as a whole—from which we may all learn from one another.

Editor's note: there is a meta thread about this post at Should we reopen "The resolution of which conjecture/problem would-advance mathematics the most?". Please read that before casting open/close votes.

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    $\begingroup$ Fermat's Last Theorem was not blocking progress, since it was an isolated (albeit historically important) puzzle whose statement was disconnected from the main developments of algebraic number theory. However, its reduction to modularity of Galois representations (a problem that, expressed in multiple ways, was a key blockage) by Ribet did provide inspiration for Wiles to work on the problem and the techniques invented to thereby prove modularity theorems indeed opened the floodgates to a tremendous number of subsequent advances. $\endgroup$
    – user74230
    Commented Nov 29, 2014 at 23:25
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    $\begingroup$ I feel that this question is too broad and unfocused, as well as being too "opinion-based". $\endgroup$
    – Andy Putman
    Commented Nov 29, 2014 at 23:36
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    $\begingroup$ One somewhat objective way to answer this question is by asking about the known consequences. In that case it's probably hard to beat the Riemann hypothesis and P vs. NP, but deducing consequences directly from a single statement is a pretty limited notion of advancing mathematics. The deeper question is what the proof techniques could tell us if we had them, but that's really not easy to predict. (For example, before the elementary proof of the prime number theory, people imagined it would lead to a revolution in number theory, but that turned out not to happen.) $\endgroup$
    – Henry Cohn
    Commented Nov 30, 2014 at 2:40
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    $\begingroup$ @TheoJohnson-Freyd: Deligne's later Weil II provides tremendously powerful results on the cohomology of $\ell$-adic sheaves going way beyond anything that could be extracted from the standard conjectures. There's so much more to the role of $\ell$-adic cohomology in mathematics than for motives, and I've always wondered if Grothendieck ever re-evaluated his opinion after Weil II was produced (though he had left the scene by then). $\endgroup$
    – user74230
    Commented Nov 30, 2014 at 7:46
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    $\begingroup$ Meta post: meta.mathoverflow.net/questions/2000/should-we-reopen . $\endgroup$
    – Emil Jeřábek
    Commented Nov 30, 2014 at 23:28

3 Answers 3

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I am going to submit the Baum-Connes conjecture because probably nobody else will and I believe its importance is quite understated. The conjecture asserts that the assembly map from the K-homology of the appropriate classifying space of a group $G$ to the K-theory of the reduced group C*-algebra of $G$ is an isomorphism for every group $G$. A proof would immediately imply two old conjectures in completely different areas of mathematics:

  • Injectivity implies the Novikov conjecture in high-dimensional topology
  • Surjectivity implies the Kadison-Kaplansky conjecture in analysis

Aside from that, the conjecture is deeply linked to a variety of other open problems in differential geometry and topology, for instance in the theory of positive scalar curvature invariants and eta invariants. A bit more speculatively, the conjecture suggests a number of deep dualities in representation theory which could shed light on open problems in that area as well; see, for instance, Aubert-Baum-Plyman's recent work on local Langlands.

Finally, unlike with many conjectures in mathematics, a counter-example would in some ways be more exciting than a proof. Some experts think that $SL(\mathbb{Z},3)$ is a counter-example to surjectivity, but proving it is well beyond the reach of current techniques. Counter-examples to injectivity might require completely new ways to construct discrete groups, and would undoubtedly have applications to geometric group theory.

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  • $\begingroup$ I'm not disagreeing with your claims of importance, but I am amused to hear that nowadays its importance is "quite understated" -- when I was a PhD student it seemed to be extremely fashionable $\endgroup$
    – Yemon Choi
    Commented Nov 30, 2014 at 0:43
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    $\begingroup$ The Baum–Connes conjecture in Wikipedia: "The conjecture sets up a correspondence between different areas of mathematics." $\endgroup$
    – Joseph O'Rourke
    Commented Nov 30, 2014 at 0:45
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    $\begingroup$ @YemonChoi Its importance is adequately (or perhaps excessively) stated among operator algebraists, but having found myself in a department without any I was surprised to learn that almost nobody has heard of it. $\endgroup$
    – Paul Siegel
    Commented Nov 30, 2014 at 1:10
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    $\begingroup$ @PaulSiegel In my case it wasn't operator algebraists per se, it was geometric group theorists, or at least the operator algebraists wanting to get in on the GGT action :) $\endgroup$
    – Yemon Choi
    Commented Nov 30, 2014 at 1:34
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    $\begingroup$ I am not an expert but there are counter-examples to the conjecture "with coefficients" since about 20 years (due to V. Lafforgue and others). Is the conjecture without coefficients almost as useful than the conjecture in general ? $\endgroup$
    – Joël
    Commented Dec 8, 2014 at 15:55
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I conjecture the question to be premature ...

I would nominate the Riemann Hypothesis, since it is clear that something occurs that we fundamentally don't understand. But folding other things in with RH, such as the Artin Conjecture, is known to be a good idea (since Weil). And the good behaviour of L-functions should be expressed by a "geometric" principle. Such a thing, if convincingly formulated, would have a strong claim.

There is presumably another major conjecture to do with how K-theory would rule geometry. Algebra rules, subordinating geometry then analysis. Such a clutch of conjectures would delineate the reach of structure, at least into the heart of the mathematical heritage of the 19th century.

Anyway, top-down questions provoke top-down answers, by suggesting "incremental" progress is beside the point. But it isn't, of course.

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How about the Millennium Prize Problems

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    $\begingroup$ Could you narrow down to one of the MP problems? $\endgroup$
    – Yemon Choi
    Commented Nov 30, 2014 at 19:12
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    $\begingroup$ @JosephO'Rourke I don't think the list order is indicative of anything. After all, P vs NP is "número tres" at claymath.org/millennium-problems . $\endgroup$
    – S. Carnahan ♦
    Commented Dec 1, 2014 at 3:23
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    $\begingroup$ @Yemon: the question is "on hold" as "opinion based", and you want me to choose one of the problems? $\endgroup$
    – Gerald Edgar
    Commented Dec 1, 2014 at 14:52
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    $\begingroup$ The question is open again. But even if it weren't, I think it makes sense to choose one, as the question seems to call for some sort of ordered list. $\endgroup$
    – Todd Trimble
    Commented Dec 1, 2014 at 18:34
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    $\begingroup$ In my opinion you really should not vote to close while having answered; this holds in general, and especially for this type of question. $\endgroup$
    – user9072
    Commented Dec 1, 2014 at 22:16

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