I am second year PhD student in pure mathematics in Europe. My PhD advisor is a well-known and respected professor who has done and still does great maths.
During the first year and a half, I have been working in a quite hot topic (say some Annals papers in the last years) which I have liked and where I have been able to produce some (not very interesting) results. My advisor produced some theorems in this topic previously, but he is not an expert on it. Recently, he recommended that we move on since he doesn't know of any further interesting and approachable problems in this area. Nonetheless, he seems to have a good (or at least not bad) opinion of me.
The problem is that the new topic he is proposing is not very fashionable. Many years ago this was a very thriving topic where he produced top results. In fact, he probably is the leading expert on this. But in the last years there have been few articles on the topic, where few is an understatement. Also, this new problem is very far from the original one. Moreover, the PhD student that worked on this has left the graduate program (which of course might only be a coincidence).
What options do I have? I am scared I won't be able to find good postdocs if I devote too much time to this. Also, he is not forcing me to work on this. But I guess that if I refuse his problem, then I should come up with some problem to study, something which I find very hard to do.
EDIT: Thanks everyone for your answers! I will meet again with my advisor and ask him further questions about the problem he is proposing: whether its feasible, why no one has tried to solve it before (maybe its too hard, requires very new techniques, almost nobody has the necessary tools, is too technical, ...), whether he thinks it is interesting for the (mathematical) community, ... But right now I think I might give it a try at least. I'll keep reading the literature on my current problem though.
- I am not planning to change advisor as I am very happy with him both as a person and as a mathematician.
- When I am talking about this hot topic in mathematics, it is not that hot. What I mean is that it has some activity and some (very) good results have appeared in the last years. Almost all these results come from a small set of authors though (which does not include my advisor).
- By "forgotten" topic I mean a topic with few papers in ArXiv and few citations. Nonetheless, many of the techniques used to study this topic are alive and relevant in neighbouring fields.