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I am doing my PhD in algebraic graph theory, for not much more reason than that was what was available. However, I love deep structure and theory in mathematics, and I do not particularly want to be a graph theorist for the rest of my life.

I have heard of mathematicians changing from more theoretical subjects to broad ones like combinatorics, but not the other way round, and am concerned that this is because the time it takes to learn the requisite theory for deeper subjects is hard to come by after grad school.

Has anyone done this, or known of someone who has? Is it at all likely or possible I will be able to get a post-doc position in a subject only marginally connected to my thesis? I have been using small amounts of algebraic number theory, and would ideally like to go on and specialise in something similar.

Any advice much appreciated!

EDIT (June '12) - I came across this old question of mine, and now feel rather silly for having asked it at all. For the record, and for anyone who might be having similar doubts to me: a year or so on I have realised that, as was mentioned below, I should not pigeonhole myself, and that the most interesting problems are often those that lead to other seemingly disparate subfields. Perhaps more to the point, I actually can't now think of a subject I'd rather do than algebraic graph theory! If you find yourself in the position I was in, I advise you to just look for those problems you find most interesting in your "chosen" field - they are bound to lead to other areas. And anyway, surely all that matters is that you are interested and inspired?! In my own research I have used number theory, galois theory, group theory, and even a bit of probability. I might even opine that graph theory is one of the subjects most likely to appear in intra-disciplinary work, which seems to be forming an ever higher proportion of mathematical research.

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On the more theoretical side, there's been very interesting work about about analogies of graphs with Riemann surfaces. That might pave the way into algebraic geometry. I'll let people who are more informed than me to link to papers on the topic. –  James D. Taylor Jul 10 '11 at 15:37
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Lots depends on your specific situation. But generic advise (for a US research university) would be: wait until you are tenured before changing area. –  Gerald Edgar Jul 10 '11 at 16:47
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gowers's answer here (to what is almost the same question) mathoverflow.net/questions/12684/switching-research-fields seems about as concise and spot-on as could possibly be hoped for. Find something in the target field that relates to something you know about in your field of expertise, and let that lead your way. –  Cam McLeman Jul 10 '11 at 17:16
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Here are 3 cases I came across in recent years: from harmonic analysis to bio-informatics, from algebraic geometry to applied cryptography (via elliptic curves of course!), from logic to incidence geometry and group theory; the latter person moved from the very pure to the pure. Be careful w.r.t. funding: you will not get funded in your old subject if you state clearly that you want to change, and you will not get funded in your new subject unless you prove a meaningful result. –  Alain Valette Jul 10 '11 at 17:29
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Any shift in research can be done, if done carefully and strategically. The trick is to keep as active as you can in your current area, while working on the new area but telling as few people about your new area as possible. Only after you have some new substantial results in the new area should you try to announce your new research specialty. But you should also make sure you talk as much as you can to good people in your new area, so you have a clear understanding of what's considered significant and important and what's not. –  Deane Yang Jul 12 '11 at 2:19

5 Answers 5

up vote 37 down vote accepted

Speaking as someone whose thesis was also in algebraic graph theory but who has later gone on to do research in other areas, I would say that it is definitely possible to switch fields. The main skills you need are management skills: the ability to manage your own time so that you can spend some time learning a new field while also producing something that others will value, and the ability to manage other people's view of your work. In this regard, I believe that Deane Yang's comment is right on the money. You probably can't afford to drop your initial specialty abruptly and produce no results while you retrain yourself. But if you manage things carefully then anything is possible, regardless of whether you have tenure or are switching to a field that requires a lot of background study.

There are some things you can do to help you solve these management problems. If you can find overlap between your current field and your new field, that will obviously help. Mathematics is so interconnected that this is usually not too hard; in the specific case of algebraic graph theory versus algebraic number theory, the first topic to come to mind is the Ihara zeta function of a graph, but I'm sure there are others.

In my case, I decided it would help to switch from academics to industry/government. I personally found it easier in industry/government to spend x% of my time producing results that pleased my employers and spending the remaining (100-x)% of my time training myself in a new area. Your mileage may vary, of course; in my case, I found that teaching drained me of too much of my energy, but this is not true of everybody.

One last comment I have is that if you take this route, then you will need the ability to maintain a clear sense of your own identity and goals and not be unduly swayed by other people's categorizations. For example, when I switched out of academics, many others regarded me as "leaving mathematics." In fact I was only leaving academics and not leaving mathematics, and it was important for me to ignore other people's view of the matter. As another example, people will want to pigeonhole you as a "something-ist" (and it seems you have been influenced by this point of view, since you use the phrase "being a graph theorist for the rest of my life"); you should resist this pigeonholing, and instead think of yourself just as someone with certain abilities and interests. Thinking of yourself as either a graph theorist or a number theorist is unnecessarily limiting. (Of course you may need to bill yourself as one or the other for the purposes of managing other people's view of you while you're making a transition, but you should not necessarily believe your own propaganda.)

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Dear Timothy, thank you for sharing your experiences as someone who has actually done it. –  Hailong Dao Jul 12 '11 at 16:16
    
Thanks Timothy...this is very useful advice. –  Adam Jul 14 '11 at 14:38

When I see an applicant for a postdoctoral or tenure-track position who has switched/is switching fields, I ask: 'why'? You'll face this question, and will need to be clear to your interlocutor about your reasons. I read CVs for evidence of mathematical strength. In young researchers, I also look for evidence of intellectual independence from the PhD/postdoc supervisors and their research program. A well-conceived move of fields is a positive here.

Consider the following:

(1) Candidate A is interested in spectral theory and microlocal analysis, and moves into numerical analysis and fast integral equation techniques because a specific problem demanded a computational approach. Candidate A then begins to contribute to NA, and takes care to publish substantial papers in reputed journals. During an interview Candidate A can precisely describe the motivating problem, and why they needed to move into numerical analysis.

(2) Candidate B starts off in microlocal analysis, and switches to math biology with no apparent link. The publications in math bio don't signal deep engagement with the new field, and aren't in the better journals. During an interview Candidate B is not quite clear about why they moved, but funding and jobs come up often.

Speaking only for myself, Candidate A's move is viewed as intellectually courageous and nimble, whilst Candidate B appears cynical. No one in field Y wants to hear that a candidate moved from field X to Y because of the money. Instead, they want to hear why the candidate finds Y an appealing, natural field to work in.

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Some thoughts from a fellow Vietnamese coffee lover (warning: I am rarely able to answer these types of questions, so certainly this is from a non-expert).

Are you sure you do not want to be a graph theorist? Recently I started learning a bit about graphs, and it looks like a beautiful subject with connections to a lot of other areas, and no lack of mysterious structures. I think all of us have doubts about our chosen fields at some points. At times I was very frustrated with what I study, especially when applying for jobs! But now I love every minute doing it. So may be just hanging there long enough would help.

But if you really have to change, I think doing it gradually and beginning with something close to what you are doing already (as others have already said) is a good advice. From outside, it looks to me that algebraic number theory takes a huge amount of background study, so make sure you have some sound financial and emotional support.

One more thing: it is perhaps not best for you to advertise to everyone that you want to switch fields. After all, you may still need a job, and it may come from people who are impressed more with your graph theory than algebraic number theory.

Best of luck!

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And I am sorry if my answer is not very encouraging. But it is my honest thoughts. –  Hailong Dao Jul 12 '11 at 2:46
    
No problem, I appreciate the advice. –  Adam Jul 12 '11 at 5:55

I don't have the necessary reputation to comment, so I'll leave this as an answer. You asked if we knew of anyone who has made this switch, so I'll chime in: Several well-known mathematicians have done this, and surely many more not-so-well-known mathematicians have as well. The first one that comes to mind is Barry Mazur, who started off working in algebraic topology before switching to number theory. Serre did something similar.

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+1. But let me add, with no disrespect at all towards Adam or you: some people can do basically everything. –  quid Jul 12 '11 at 14:47
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@quid: I was thinking the exact same thing when I wrote this answer! And no disrespect taken: I am certainly not one of those people! It is definitely easier for some people to transition between fields, but I suppose the existence of any such people at all should be encouraging to people like Adam. –  Jeff H Jul 12 '11 at 14:50
    
Jeff H, thank you for the reply; we are thinking along the same lines; and I (thus) find the information you give useful. –  quid Jul 12 '11 at 15:04

Adam you can also study the topology of graphs.

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