Let me start by apologizing for the soft and lengthy nature of the question.

I am a third year graduate student (in India) working in complexity theory. Early this year, I developed an interest for topology owing to the problem in topological graph theory that I was working on. Since I didn't have much of a formal background in mathematics, I started studying up basic and graduate level math with the aim of being able to understand advanced topics in topology and geometry (low dimensional topology, differential geometry). While studying I felt that I was perhaps better cut out for doing pure math and was probably even more interested in topology and geometry, and started thinking about shifting to math.

Due to various reasons, I will not be shifting and will be continuing in theoretical CS. Now people who sympathize with my cause (including my professors) offer consolation by saying that I can work in topology and related areas once I have a PhD. This doesn't sound right to me. Immediately after a PhD, I would have to publish prolifically for securing and tenuring an academic position. Even when I'll finally be in a position to shift focus, I don't know how easy it will be to reach the research frontier of some area in geometry/topology. I should also add that my PhD will most probably end up being around applications of coding theory in complexity, which has little/no overlap with topology/geometry.

The way I see it, for any area (be it in math or CS) there seems to be an active community which works on it. I don't see how one can easily "enter" into the community and find collaborators, especially when coming from a different field.

So, let me come to the questions now. Firstly, should I have any hope for being able to do the "shift" at all? If yes, how can I keep in touch with the subject while not compromising on my complexity research? Or should I start looking at this stuff only when I am done with PhD? Any help/advice would be very useful.

isa gamble, but it is far from impossible. – Mikael Vejdemo-Johansson Nov 25 '12 at 15:43