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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.

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The following seems very similar… Voting to close as duplicate. – user9072 Nov 25 '12 at 15:12
Why not shift your PhD to computational topology? – Kelly Davis Nov 25 '12 at 15:31
I thought of that. The issue with that is I don't have anyone to guide me or for that matter even discuss with in my institute, if I choose computational topology. Of course, one can argue that I can have collaborators outside the institute, but I am not sure how pragmatic that option is. Taking all factors into consideration, I took the call of working in coding theory. – Abhishek Bhrushundi Nov 25 '12 at 15:39
In particular, by stepping stone I mean to look for computational topologists who are interested in more complexity knowhow in their own workgroups, and use your participation in a postdoc in such a group as a way to bootstrap yourself into computational topology. After 2-3 years you'll be prolific in your new field instead. It is a gamble, but it is far from impossible. – Mikael Vejdemo-Johansson Nov 25 '12 at 15:43
@Todd Trimble: Regarding the duplicate, my point of view here is that the details of the situation (and giving individual advice) are anyway outside the possibilities of MO (and for this the background information is also not detailed enough), and for the generalities it's about the same. And, in a rough sense there is even a compatibility of the fields. – user9072 Nov 25 '12 at 17:18

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