How is research done in universities?

Question

I have searched for some time for the "insider scoop" on how academics operate when it comes to mathematical research (theoretical physics research would also be interesting, but mostly interested about math research).
I read "Letters to a young mathematician" which was a nice book but didn't talk much about the research process.
If I google "Research process", or "How is research done", I mainly get recommendations of steps of how to do independent research.
These step include brainstorming, choosing a research subject and looking through background information and articles.
But how are these done in an academic setting and in a group setting?
I understand usually while doing a PhD you would have an advisor which would help you with these things but you would mostly do them independently.

But I mean people who are employed as mathematics professors at a university, how do they go about it?
Do they do just this independent process? Do they cooperate together on the same research?
Mostly what is interesting to me is how a group of people can research something together, and if this is even attempted.

Answer 1

This really depends on your personality. When I was a graduate student, I worked a lot with my adviser who guided me through problems. Every time I had an idea, I would pass it by him to see how interesting it really was. After spending enough time on a particular problem, you start to get a feeling for how it can be extended or changed, and also what is difficult about that problem. It helps to get an outside perspective sometimes.

Now I work as a postdoc in a different field altogether. Though I still work on my mathematical research. For some problems, I am the only one who really knows anything about them, so I work alone. There are other problems that I would like to have more time to put to, but cannot for various reasons. For these problems I have several collaborators who also are interested in them. We meet from time to time to talk about what we have done, and to discuss future goals. This also helps me stay motivated about these problems.

There are people who insist on only working alone, and some are more successful than others. On the other hand, there are many people who don't do any work alone. There are highly successful mathematicians where almost all of their publications are collaborations.

There are always more problems than you have time to solve, and it helps to work with others.

Answer 2

I believe it is still the case (but less than in the past) that most mathematics research papers have only one author. So they were presumably done by the "independent research" method you wrote about. Even with two-person research, it is like the "independent" method, except there are two guys who bounce ideas back and forth (and in between meetings work on them alone) until they get results they like.

This is NOT like an experimental physics lab, where there is the Big Guy (or a Committee) that decides on what to do, and all the Little Guys carry it out.