Mostly, your initial years as an independent researcher will be spent working on problems that you had worked on during your doctorate degree. So in most cases,

Mostly, your initial years as an independent researcher will be spent working on problems that you had worked on during your doctorate degree. So in most cases, the Ph. D advisor tends to have a lot of influence both in the topic and the style of your research. There are also many people who end up not working at all in actuality with their advisors, they end up working with fellow grad students etc. In none of the cases are they from a common pool as you suggest. I think there is a lot of "school" philosophy in mathematics. Most productive groups tend to work on the same topics : analysis, algebra, number theory, probability etc. in the case of Soviet Union, for example, but not many on combinatorics.