There's a big difference between being one of the only ones in a field because it does not yet exist (i.e. you are founding the field) and because it is exhausted (i.e. all the big results are already proven).
It is generally not the case in any area of math that "all the big results are already proven".
True. But, well, for instance, all the finite simple groups are classified.
(Not in the sense of the NSA.)
We have a professor who is one of a few dozen active researchers in his field, and I have developed some algorithms for him. It is really trivial to publish in fields like that. And back in the 50s and 60s anybody could get an algorithm named after them.
Sorry to let you know, but most professors are in fields like that. That's why I left the subject. :-(
Really? Most of our CS professors are in much more well-established fields. Our department head is in search engines, my advisor is data mining, we have some GPU computing folks, and some database security guys, etc. Then the one guy in a highly esoteric field, where a "conference" involves a half dozen people meeting for a week.
I was in math.
I've heard that CS is better. But still has tendencies that way.
I understand this confusion, but it totally isn't how research is done. It's very much a community and social effort; it can be very lonely working all by yourself on something.
I don't understand the logic here. I've been told many times that the academic's dream is to find a field where he's the only active researcher. Fewer active researchers means fewer PhD positions, sure, but it also means more prominence for those who remain, less effort involved in finding publishable results, etc.
My go-to example of offensively low-hanging fruit is De Morgan's law(s), which I still can't believe were named after a person. They state, in plain english:
1. If it is not the case that a collection of claims are all true, then one or more of the claims is false.
2. If it is not the case that any of a collection of claims is true, then they are all false.
When you're the only active researcher in a field, you can have observations like that named after you too!