In the past, practical applications have motivated the development of mathematical theories, which then became the subject of study in pure mathematics, where mathematics is developed primarily for its own sake. Thus, the activity of applied mathematics is vitally connected with research in pure mathematics.
I wonder which problems in pure mathematics could be tackled once an algorithm solving NP-complete problems is found. The list of instances of NP-complete problems is long, but are they any other, let's say stand-alone problems in pure mathematics that could be solved?
Sure, we could get more numerical data to tighten some bounds, but that's not what I'm after...
All of them. It's possible to verify if a proof is valid in polynomial time, and it's possible to check if the last step of a proof is the theorem under consideration in polynomial time, so just encode "valid proof of length N" as a satisfiability problem.
Unless the theorem is unprovable, which is really just another theorem to check. A tractable SAT solver would be ridiculously powerful.
I'll try to be more explicit. A proof is a string of characters. You could ask "what is the length of your proof?" and get a response "1300 characters long". In academic papers only the outlines to proofs tend to be published, but there are proofs where every single step can be checked.
Consider an algorithm $\text{ValidProof}(P, T)$ that returns true iff $P$ is a valid proof of $T$. We already know constructively how to implement $\text{ValidProof}$ in polynomial time.
If $P=NP$, then for any algorithm $Q(x)$ which runs in polynomial time, one can solve the problem "is there $x$ of length $N$ such that $Q(x)$ is true?". You can also determine what that $x$ is. That's the entire point of the $NP$ complexity class.
Since $Q(x)$ can be $\text{ValidProof}(x, T)$, $P=NP$ would allow anyone to just solve for $x$, the proof.