Being a successful research mathematician, although I am not one, sounds nowadays like more about figuring out the market for your skills among multitudes of opaque research projects so I imagine following your gut you might just get into a good post if you're lucky.
We are truly in a time where there are plenty of (bad, unsubstantial) ideas out there, i reckon it takes a good instinct to be a good mathematician (or any other thing that does not rely on public opinion).
This is a philosophical viewpoint but in some sense all of them are lucky.
Louis de Branges claimed for years to have a proof of Riemann Hypothesis. No one really agreed with his conclusion. He is certainly a first rate mathematician and had he came up with a proof that others accepted he’d be lauded as a great mathematician.
Abhyankar claimed that no one has ever really understood Hironaka’s resolution of singularities paper. Hironaka is a Field’s Medalist and Abhyankar a great mathematician in his own right. He was never able to find a simpler proof for resolution of singularities. If he had gotten lucky then he would have.
The point is, that luck plays a role in terms of whether or not the right idea pops into your head. How many brilliant people labored over problems that simply have no solution and thus aren’t considered one of the greats? Newton wrote more about alchemy than math or physics. He’s not considered a great chemist. One thing the greats have in common is spending a great deal of time thinking about problems. That increases the probability of coming up with a brilliant insight.
Of course, it is not all luck. You do have to have good intuition. Here’s a quote by Chaitin:
Gödel's incompleteness theorem tells us that within mathematics there are statements that are unknowable, or undecidable. Omega tells us that there are in fact infinitely many such statements: whether any one of the infinitely many bits of Omega is a 0 or a 1 is something we cannot deduce from any mathematical theory. More precisely, any maths theory enables us to determine at most finitely many bits of Omega.
In some sense we get a survivorship bias when talking about the greats. They happened to work on a problem that was solvable. I suggest there are many more equally brilliant people who didn’t get lucky and thus are unknown.
Successful research mathematicians do consistently good work on many problems. While there may be one or two highlights that really boost a career, few people are one-hit wonders that stumble upon a solvable problem.
Most are one hit wonders. Most research mathematicians end up having at most 1 or 2 Ph.D. students. Most publish minor extensions of known results. The greats have an insight and end up doing a lot in that one minor area. Very few people have more than a couple of truly remarkable theorems.
