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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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Joseph O'Rourke
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    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. – user74230 Nov 29 '14 at 23:25
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    I feel that this question is too broad and unfocused, as well as being too "opinion-based". – Andy Putman Nov 29 '14 at 23:36
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    @AndyPutman: It is indeed both broad and opinon-based. Which does not mean we cannot learn from responses. – Joseph O'Rourke Nov 29 '14 at 23:38
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    BSD wins for me, but I am biased because it is something that I am working on. However, my biases aside, the problem is currently producing lots of advances in mathematics (e.g. progress in p-adic L-functions and Iwasawa theory by Urban et al, and all of the work by Bhargava surrounding the solution to BSD "on average"). – Johannas Nov 29 '14 at 23:41
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    Grothendieck's standard conjectures? – Venkataramana Nov 30 '14 at 01:26
  • An MO reference re @Venkataramana: "Has there been any progress on the standard conjectures on algebraic cycles?." – Joseph O'Rourke Nov 30 '14 at 02:18
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    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.) – Henry Cohn Nov 30 '14 at 02:40
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    My memory is that in algebraic geometry, Grothendieck hoped for a program to develop quite a lot of machinery in order to prove some outstanding (Weil?) conjectures. When Deligne proved them without all the machinery Grothendieck hoped would be developed, Grothendieck, according to the story, was quite upset: the point wasn't the result, but the rest of the theory. I bring this up to support the idea that "open conjectures that would advance mathematics" might make such an advance because they seem to need dramatically new machinery or ideas, which would then be available for other uses. – Theo Johnson-Freyd Nov 30 '14 at 03:20
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    @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). – user74230 Nov 30 '14 at 07:46
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    Meta post: http://meta.mathoverflow.net/questions/2000/should-we-reopen . – Emil Jeřábek Nov 30 '14 at 23:28
  • @user74230 Good to know! As I hope was clear from my comment, I am not an expert. – Theo Johnson-Freyd Dec 01 '14 at 05:43
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    What about a contradiction in Peano Arithmetics or First Order Logic? However, it's hard to argue about consequences (at all) - any way out would certainly be a true advance. – Andreas Thom Dec 01 '14 at 17:45
  • What about specializing to applications? My life wouldn't change if the Riemann hypothesis is proven/disproven, and knowing P=/!=NP also not necessarily would have THAT impact with computers. – Hauke Reddmann Dec 01 '14 at 17:55
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    @AndreasThom: Peano arithmetic and first order logic are consistent. That’s a (fairly easy) theorem, not a conjecture, notwithstanding a host of popular misconceptions about Gödel’s theorems. – Emil Jeřábek Dec 01 '14 at 20:27
  • @EmilJeřábek Does Voevodsky agree meanwhile? :-) –  Dec 01 '14 at 20:42
  • @EmilJeřábek: You don't use the verb "are" in a mathematically sound way. Kleene on Gentzen's work "To what extent the Gentzen proof can be accepted as securing classical number theory [...] is in the present state of affairs a matter for individual judgement [...]." – Andreas Thom Dec 01 '14 at 20:47
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    @AndreasThom: I’m using the word in a perfectly sound technical mathematical way, I’m not talking about Gentzen’s proof, and even if I did, there is a huge difference between “securing classical number theory”, and just proving something as a theorem. Kleene’s issue is with philosophical implications of Gentzen’s proof for the purposes of Hilbert’s program, not with the mathematics involved in the proof. I’m not going to comment on the infamous Voevodsky’s talk, which is a distillation of popular misconceptions about Gödel’s theorems. – Emil Jeřábek Dec 01 '14 at 20:57
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    Actually, @AndreasThom: please read Timothy Chow’s answer to the question linked by quid, it explains it quite well. If you want a quote: "In the case of Con(PA), the aforementioned "normal conditions" for removing its "open problem" status have been met, and in fact exceeded." – Emil Jeřábek Dec 01 '14 at 21:08
  • @EmilJeřábek: I did not claim consistency cannot be proved. My point was just that the existence of a concrete contradiction is still possible (and I thought "are" was to dispute this.) – Andreas Thom Dec 01 '14 at 21:08
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    In many cases, it will depend on how the conjecture is resolved (not just the answer, but the method of proof). Fermat's last theorem and the 3-dimensional Poincaré conjecture are both examples where the result proved was more important than the most famous conjecture. In the case of FLT, if a direct proof had been found before the development of modern number theory, that might have actually slowed down our understanding of mathematics. – Colin Reid Dec 01 '14 at 21:43
  • Please, everyone, vote up the comment above about the meta post so that people see it. – Scott Morrison Dec 01 '14 at 22:03
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    @AndreasThom: It’s possible exactly as much as it is possible that the reals $0$ and $1$ are equal, or that there are only finitely many primes, or that there exists a nonconstant polynomial over $\mathbb C$ without a root or ... (you get the idea). Are any of these (equivalent) statements “conjectures/problems whose resolution would significantly advance mathematics”? Matter of opinion, but IMHO, no. For one thing, they already have been mathematically resolved. – Emil Jeřábek Dec 01 '14 at 22:33
  • @EmilJeřábek: Your way to argue applies just as well to naive set theory, before Russell's paradox. Our vision was advanced considerably by the need to rethink how we set up mathematics in a way that avoids the paradox. You can "mathematically solve" these problems ten times, but a concrete contradiction would force you to start again from scratch. Call that progress/advancement or not. Anyhow, it seems unlikely to happen and my initial comment was not entirely serious anyway. – Andreas Thom Dec 01 '14 at 22:55
  • Cantor’s set theory was a huge leap into an uncharted territory full of new unfamiliar concepts and pitfalls. This is not comparable to the proof of consistency of PA using the satisfaction predicate for the ring of integers, which only employs simple standard techniques that are used on daily basis all over mathematics (the most complicated being the construction of a sequence of objects ${a_n:n\in\mathbb N}$ by induction on $n$.) – Emil Jeřábek Dec 01 '14 at 23:05
  • @EmilJeřábek: I am sure you can prove consistency of Cantor's set theory pretty much the same way as consistency of PA, using only "daily basis" naive set/model theory. And this exactly shows how useful such a proof can be (in any situation) when it comes to the question of existence of a contradiction - but you know that. Your point can merely be that this seems rather unlikely to happen (given all our positive experience with basic reasoning) and I agree (but not based on any mathematical reasoning/argument, rather some sort of belief/platonism.). – Andreas Thom Dec 02 '14 at 08:29
  • Oh really? Show me your proof of consistency of Cantor's theory, I'm all ears. – Emil Jeřábek Dec 02 '14 at 10:52
  • On second thought, the problem seems to be that you are constantly confusing theory and metatheory. It's probably unhelpful that the result we are talking about involves the word "consistency", which has the curious psychological effect on many mathematicians to treat it as special, and pretend they suddenly became die-hard finitists. So let's call it the Peano representation theorem instead. Now, just like many other theorems, it has two aspects. (1) It is a mathematical problem; and this is the aspect this question asks about. The problem has been solved some time ago, and it's on the ... – Emil Jeřábek Dec 02 '14 at 12:12
  • ... level of elementary algebra. (2) The theorem can be interpreted as saying something about reality outside mathematics. This is a philosophical problem, which is not what this question asks about. As such, it is subject to endless debate whether applying the theorem to reality in this way makes sense in the first place, and if so, whether the reality validates the theorem (if not, mathematics may be in serious trouble), and so on. There is nothing mathematics can do to "resolve" this debate, as it is neither mathematical nor conclusively resolvable by its very nature. The fact that ... – Emil Jeřábek Dec 02 '14 at 12:13
  • ... the Peano representation theorem happens to talk about the consistency of some formal system is irrelevant, all these issues arise for any result with extra-mathematical applications. – Emil Jeřábek Dec 02 '14 at 12:13
  • My initial problem did not involve the word "consistency", I was talking about the existence of a concrete contradiction (if you want, we can call that a meta-question about mathematics). It seems that this is not a mathematical problem in your sense, and your mathematical formulation does not answer the legitimate meta-question. A proof in ZF that PA is consistent is only meaningful if ZF is consistent. In particular, it cannot have any meaning if PA is inconsistent. But certainly, this meta-question is a problem about mathematics with potential to advance our understanding of it. – Andreas Thom Dec 02 '14 at 13:39
  • A proof of anything in ZF is only meaningful if ZF is consistent. The consistency of PA is not any different in this respect from any other theorem. – Emil Jeřábek Dec 02 '14 at 16:39
  • Right, I didn't say it was any different. I understand what you mean but I don't understand why you doubt the legitimacy of the problem and the potential progress in case of a solution. Anyhow, I guess we can end this discussion. – Andreas Thom Dec 02 '14 at 22:12

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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.

Paul Siegel
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  • 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 – Yemon Choi Nov 30 '14 at 00:43
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    The Baum–Connes conjecture in Wikipedia: "The conjecture sets up a correspondence between different areas of mathematics." – Joseph O'Rourke Nov 30 '14 at 00:45
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    @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. – Paul Siegel Nov 30 '14 at 01:10
  • @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 :) – Yemon Choi Nov 30 '14 at 01:34
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    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 ? – Joël Dec 08 '14 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.

Charles Matthews
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How about the Millennium Prize Problems

Gerald Edgar
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