Do you know interesting consequences of (standard) conjectures in complexity theory in other fields of mathematics (i.e. outside of theoretical computer science)?I would prefer answers where:the complexity theory conjecture is as general and standard as possible; I am ok with consequences of the hardness of specific problems too, but it would be nice if the problems are widely believed to be hard (or at least have been studied in more than a couple of papers)the implication is a statement that is not known to be true unconditionally, or other known proofs are considerably more difficultthe more surprising the connection the better; in particular, the implication should not be a statement explicitly about algorithmsIf pigs could fly, horses would sing type of connections are ok, too, as long as the flying pigs come from complexity theory, and the singing horses from some field of math outside of computer science.This question is in some sense the converse of a question we had about surprising uses of mathematics in computer science. Dick Lipton had a blog post exactly along these lines: he writes about consequences of the conjecture that factoring has large circuit complexity. The consequences are that certain diophantine equations have no solutions, a kind of statement that can very hard to prove unconditionally. The post is based on work with Dan Boneh, but I cannot locate a paper. EDIT: As Josh Grochow notes in the comments, his question about applications of TCS to classical math is closely related. My question is, on one hand, more permissive, because I do not insist on the classical math restriction. I think the more important difference is that I insist on a proven implication from a complexity conjecture to a statement in a field of math outside TCS. Most of the answers to Josh's question are not of this type, but instead give techniques and concepts useful in classical math that were developed or inspired by TCS. Nevertheless, at least one answer to Josh's question is a perfect answer to my question: Michael Freedman's paper which is motivated by a question identical to mine, and proves a theorem in knot theory, conditional on $\mathsf{P}^{\#P} \ne \mathsf{NP}$. He argues the theorem seems out of reach of current techniques in knot theory. By Toda's theorem, if $\mathsf{P}^{\#P} = \mathsf{NP}$ then the polynomial hierarchy collapses, so the assumption is quite plausible. I am interested in other similar results.
Mathematical implications of complexity theory conjectures outside TCS
cc.complexity theory;conditional results
Here's another example from graph theory. The graph minor theorem tells us that, for every class $\mathcal{G}$ of undirected graphs that is closed under minors, there is a finite obstruction set $\mathcal{Obs(G)}$ such that a graph is in $\mathcal{G}$ if and only if it does not contain a graph in $\mathcal{Obs(G)}$ as a minor. However, the graph minor theorem is inherently nonconstructive and does not tell us anything about how big these obstruction sets are, i.e., how many graphs it contains for a particular choice of $\mathcal{G}$. In Too Many Minor Order Obstructions, Michael J. Dinneen showed that under a plausible complexity-theoretic conjecture, the sizes of several of such obstruction sets can be shown to be large. For example, consider the parameterized class $\mathcal{G}_k$ of graphs of genus at most $k$. As $k$ increases, we can expect the obstruction sets $\mathcal{Obs}(\mathcal{G}_k)$ to become more and more complicated, but how much so? Dinneen showed that if the polynomial hierarchy does not collapse to its third level then there is no polynomial $p$ such that the number of obstructions in $\mathcal{Obs}(\mathcal{G}_k)$ is bounded by $p(k)$. Since the number of minor obstructions for having genus zero (i.e. being planar) is just two ($\mathcal{Obs}(\mathcal{G}_0) = \{K_5, K_{3,3}\}$), this superpolynomial growth is not immediately obvious (although I believe it can be proven unconditionally). The nice thing about Dinneen's result is that it applies to the sizes of obstruction sets corresponding to any parameterized set of minor ideals $\mathcal{G}_k$ for which deciding the smallest $k$ for which $G \in \mathcal{G}_k$ is NP-hard; in all of such parameterized minor ideals the obstruction set sizes must grow superpolynomially.  