There are lots of statements that have been conditionally proved on the assumption that the Riemann Hypothesis is true.
What other conjectures have a large number of proven consequences?
10 Answers
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Set theory is of course completely saturated with this feature, since the independence phenomenon means that a huge proportion of the most interesting natural set-theoretic questions turn out to be independent of the basic ZFC axioms. Thus, most of the interesting work in set theory is about the relations beteween these various independent statements. They typically have the form of implications assuming the truth of a hypothesis not known to be true (and often, known in some sense not to be provably true), and therefore are instances of what you requested. The status of these various hypotheses as conjectures, however, to use the word you use, has given rise to vigorous philosophical debate in the foundations of mathematics and set theory, as to whether or not they have definite truth values and how we could come to know them.
Examples of such hypothesis that are used in this way would include all of the main set-theoretic hypotheses known to be independent. This list would run to several hundred natural statements, but let me list just a few:
- The Continuum Hypothesis (also the Generalized Continuum Hypothesis)
- The negation of the Continuum Hypothesis
- Martin's Axiom
- More generally, other forcing axioms, such as PFA or MM
- Cardinal characteristics relations, such as b < d
- The entire large cardinal hierarchy
This last example is extremely important and a unifying instance of what you requested, for the large cardinal hierarchy is a tower of increasingly strong hypotheses, which we believe to be consistent, but haven't proved, and indeed, provably cannot prove, to be consistent, unless set theory itself is inconsistent. From any level of the large cardinal hierarchy, if consistent, we provably cannot prove the consistency of the higher levels.
So in this sense, the large cardinal hierarchy provides enormous iterated towers of your phenomenon.
This might seem at first to be a flaw. Why would we be interested in these large cardinals, if we cannot prove they exist, cannot prove that their existence is consistent, and indeed, can prove that we cannot prove they are consistent, assuming our basic axioms are consistent? The reason is that because of Goedel's incompeteness theorem, we know and expect to find such statements, that are not settled, even when we assume Con(ZFC) and more. Thus, we know there is hierarchy of consistency strength towering above us. The remarkable thing is that this tower turns out to be describable in terms of the very natural infinite combinatorics of large cardinals. These were notions, such as inaccessible, Ramsey and measurable cardinals, that arose from natural questions about infinite combinatorics, independently of any considerations of consistency strength.
Some of the most interesting uses of large cardinals have been equiconsistencies between large cardinals and other natural mathematical statements. For example, the impossibility of removing AC from the Vitali construction of a non-measurable set is exactly equiconsistent with the existence of an inaccessible cardinal. And the complete determinacy of infinite integer games (with not AC) is equiconsistent with the existence of infinitely many Woodin cardinals.
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$\begingroup$ Does the simple negation of the continuum hypothesis really have a lot of interesting consequences? $\endgroup$– bofCommented Aug 6, 2015 at 5:27
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$\begingroup$ Why didn't the axiom of constructibility make your list? $\endgroup$– bofCommented Aug 6, 2015 at 5:28
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1$\begingroup$ Of course, we can (in principle) prove large cardinals don't exist, like Reinhardts in ZFC. $\endgroup$– David Roberts ♦Commented Aug 6, 2015 at 13:56
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The standard conjectures (http://en.wikipedia.org/wiki/Standard_conjectures_on_algebraic_cycles) were pretty much designed to be used in this way (and then proved); but proofs are lacking, and some of the results now have non-conditional proofs. There are many related results in the theory of motives.
In number theory, Vandiver's conjecture (http://en.wikipedia.org/wiki/Vandiver%27s_conjecture) has begun to stand out, because of its connection with K-theory (which is another area in which there are large scale conjectures used in this way).
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$\begingroup$ Or even the Bombieri-Lang conjecture which says that if $K$ is a number field and $X/K$ is a variety of general type, then $X(K)$ is not Zariski dense in $K$. This is the special case of Vojta's conjecture (formulated earlier) applied to a variety of general type. Among it's numerous consequences is that among all curves $C/K$ of a fixed genus, there is an upper bound for $\#C(K)$ that is independent of $C$. (Due to Caparaso-Mazur-Harris.) $\endgroup$ Commented Dec 25, 2015 at 1:12
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Wiki also says:
A famous network of conditional proofs is the NP-complete class of complexity theory
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Resolution of singularities for algebraic varieties in positive characteristic is another example. Many statements in algebraic K-theory have been proven to follow from this conjecture.
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What is usually referred to as Lusztig's Conjecture in the modular representation theory of semisimple algebraic groups has been enormously influential, as seen in Jantzen's treatise Representations of Algebraic Groups. It is actually a series of closely related conjectures, from 1979 on, inspired by the (soon proved) Kazhdan-Lusztig Conjecture (1979) on the formal characters of the usually infinite dimensional simple highest weight modules for a complex semisimple Lie algebra: such a character can be written as a $\mathbb{Z}$-linear combination of the known formal characters of Verma modules whose coefficients are values at 1 of Kazhdan-Lusztig polynomials for the Iwahori-Hecke algebra of the Weyl group $W$. The original characteristic $p$ conjecture has a similar flavor, but with the affine Weyl group (whose translations are multiplied by $p$) replacing $W$ and with the essential proviso that $p$ be not too small. It is expected that the Coxeter number of $W$ will be a suitable lower bound, but so far the partial proofs by Andersen-Jantzen-Soergel, Fiebig, and Bezrukavnikov-Mirkovic do not achieve a reasonable bound.
If proved, the conjecture would combine with older results of Curtis and Steinberg to yield all modular irreducible characters of finite groups of Lie type in the defining characteristic (but still with the lower bound on $p$), as well as the formal characters and dimensions of all restricted representations of the Lie algebra of the given semisimple group. Andersen and others have formulated further consequences, in terms of the structure of Weyl modules, the extensions and cohomology of simple or Weyl modules, etc. (Adapted to general linear groups, there are also implications for modular characters of symmetric groups.) The later conjectures of Lusztig, proved for large enough $p$ in a preprint by Bezrukavnikov and Mirkovic, go further with the non-restricted Lie algebra representations as well in a unified geometric setting which promises further applications.
ADDED: I should point out that many special cases of the more general results which would follow from Lusztig's Conjecture have in fact been verified, but usually by computational or somewhat ad hoc methods. Plus the existing proofs of the conjecture itself for "large enough" primes, which don't seem improvable without new methods.
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2$\begingroup$ (now that this question has been bumped to the front page anyway) It might be worth pointing out that it has turned out that the bound on $p$ unfortunately cannot be just the Coxeter number, or for that matter linear (probably not even polynomial) in the Coxeter number by work of Williamson. $\endgroup$ Commented Aug 6, 2015 at 6:15
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$\begingroup$ @Tobias: A useful observation, though I've since come around to the viewpoint that Lusztig's "inverse" approach (without a lower bound on $p$) in his 1980 Advances in Math. paper will be more useful than his 1979 Santa Cruz conjecture which imitated the original Kazhdan-Lusztig conjecture closely. A key difference in characteristic $p$ is that the inverse K-L polynomials are not just the original polynomials in disguise. Also, it's likely that affine Weyl groups relative to higher powers of $p$ will be an essential tool. $\endgroup$ Commented Aug 6, 2015 at 13:13
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In computational complexity there are several conjectures which are stronger than $NP \ne P$ which have important consequences. To mention a few
1) The conjecture that factoring is computationally hard is the basis to much theoretical and practical cryptography.
2) More broadly the conjecture that one-way functions exist has many consequences.
3) The conjecture that the polynomial hierarchy ($PH$) does not collapse has many consequences.
4) Khot's unique game conjecture has many important consequences for hardness of approximation.
5) The exponential time hypothesis ($ETH$)and strong exponential time hypothesis ($SETH$) are strong form of $NP \ne P$ with important consequences.
6) There are stronger and stronger versions of the "derandomization" conjecture with many consequences.
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$\begingroup$ For instance: pseudorandom generators exist if and only if one-way functions do (thanks, Wikipedia!) $\endgroup$– David Roberts ♦Commented Aug 6, 2015 at 13:51
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Clicking on Toolbox-What Link's Here in the wikipedia article Conditional proof brings up Schinzel's hypothesis H which the article says is used to prove conditional results in diophantine geometry.
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In the representation theory of a reductive algebraic group $G$ in positive characteristic $p$, there is a conjecture known as the Humphreys-Verma conjecture, which states that an indecomposable injective module for a Frobenius kernel $G_r$ of $G$ should lift uniquely to a module for $G$. There is also a refinement of the conjecture, known as Donkin's tilting conjecture, which specifies which $G$-module this lift should be (an indecomposable tilting module with a specified highest weight).
Both conjectures are known to be true when $p\geq 2h-2$ where $h$ is the Coxeter number, and while it is not particularly common to see statements formulated conditional on either conjecture, the condition $p\geq 2h-2$ is exceedingly common, and quite often this condition could be replaced by an assumption that one or both of the above conjectures are true.
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Related to the answer about NP-complete problems, there are a number of theorems that state "either x is true, or P=NP." The most interesting of these in my opinion are hardness of approximation results. For example: "Given two graphs on $n$ vertices, one with max clique size $n^\alpha$ and one with max clique size $n^{1-\alpha}$, there is no polynomial time algorithm that determines which is which, or P=NP."
Most results like this are proven via the PCP Theorem, by showing that if you can approximate a result to a certain extent, you can then convert that into a proof of the statement.