My background: I have two master degrees in pure math and applied CS. I am interested in working in both areas, and there are certain topics in each area (extensions of my -completely different- two master theses) that I am interested in working in, so there does not exist the possibility of combining them in any way: math its 100% proofs and nothing else; computer science it's code a prototype, computing the some accuracy metrics, writing more code.

However, my working style is to switch between these two field every 9 month, since I quickly get bored, if I only work in one of the fields.

I recently received an offer for a PhD at a top European university (think Cambridge/Oxford/ETH... ). Like any funded place, this one is tied to a single topic from one of those two fields and some compromises have to be made to harmonize my topic choice with the problems my supervisor likes to work on.

1) Do you know of any strategies that I could use to keep switching topics, so to preserve my passion for both fields?
("Utopically", I would get accepted into a second PhD, where I would work on a topic from the other field and then keep rotating every 0.5 - 1 years between them, i.e. as soon as I achieved a milestone in field and start to feel boredom creeping on, go to the other field. But no sane supervisor would accept that, I think. And the program structure would.)

2) Are there ways to convince my supervisor to let maybe 1.5 take year-long time off? That would already help a lot I think. How should I best go about this, without giving a bad first impression ("you haven't even started, and already you think about going away").

3) Or should I just stick with it and try to finish as fast as possible with the PhD and then try to move to the other field for postdoc (supposing I could get a postdoc position in say, math, if I did my PhD in CS, and only published 2 smaller articles in math in the meantime)?

4) If there absolutely no way to switch between topics for my PhD, I'm unsure which field would be the best to go for for my PhD: Pure math (doing proofs) or CS (coding proof-of-concepts in machine learning and related areas)?

I assume many will say things like "this is a PhD! You have to learn how to stick with one topic, because this is life and life can be hard". But working on multiple problems at once is a tried and tested way to keep the fun in the research (as done perhaps most famously by Perelman: "he liked to work on several problems at once" quote). Alas, if I knew them, I would not post here; I'm really at a loss for what the best step would be to preserve my love for doing research, while not ending up unemployed and keeping the door open for a career in academia.

edit: While the current answer provides very good general feedback, I would still like to know more (very) concrete information about 1)-4).

closed as primarily opinion-based by Buzz, scaaahu, Scientist, user3209815, Flyto Aug 7 '18 at 12:44

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.

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    You seem to be under the impression that mathematics and computer science are disjoint. Some of the most influential computer scientists were renowned mathematicians. Think Turing, Godel. You should also keep in mind that a Ph.D. is about learning how to do research. Once you have one, there isn't really a point in doing a second. – koverman47 Aug 6 '18 at 17:04
  • @koverman47, what you say is true, but the examples you give come from a time when CS didn't actually exist. Much of CS grew out of math and, as you say, there is still a connection on the theoretical side. Some modern topics also depend strongly on a mathematics background and some vital CS topics like security and privacy may depend even more on math in the future. – Buffy Aug 6 '18 at 17:14
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    @Buffy My 'examples' were more anecdotal. It certainly depends on the subfield of CS you are interested in but there are still plenty of areas in CS that are heavily reliant on advanced mathematics – koverman47 Aug 6 '18 at 17:17
  • @koverman47 I'm doing the PhD in a narrow subfield of these fields (where my topics of interest lie). And these are as disjoint as they can get. In math its 100% proofs and nothing else. In computer science it's a much more experimental approach: Writing code, computing the some accuracy metrics, writing more code. And only maybe, at the end, a short proof at the end. – MyCatsHat Aug 6 '18 at 17:17
  • @koverman47, the opposite is also true of course. Some proofs in mathematics have needed heavy use of computing. – Buffy Aug 6 '18 at 17:19

Generally speaking PhD studies are all about specialization. One dives very deeply into a narrow field. This is especially true of math and can also be true of CS, depending on subfield.

There can be exceptions and some synthetic work is possible, but it depends a great deal on your advisor and what he or she is comfortable with. If you can find an advisor at the intersection you should be able to pursue a joint approach.

After the PhD you are much freer to pursue your own goals not constricted by the concerns of your advisor. I would suggest, pretty strongly, that you explore your wishes with a doctoral supervisor and take their advice. If the advice is to pursue the one field for now, it isn't a decision you need to live with for all time. But the advice will likely be good advice.

Doing a second doctorate will probably be superfluous. You will have deep research experience from the first. If it is in math, much of what you learned will have some application to thinking about some of what is important in CS.

Doing two doctorates simultaneously seems masochistic to me, especially as you will have two advisors each with different, possibly conflicting, expectations and each putting constraints on your time.

Think specialize now. Think widening interests for the future. It will better match the expectations of those around you and make life easier.

But you will also need ways to not get bored, given your description of the problem. It might even be that while focusing deeply on one of the fields you can 'play about' in the other for a short periods just as a stress reducer as needed.

  • No synthetic approaches are possible; the topics are too different. I've added an edit to my question, to point this out. – MyCatsHat Aug 6 '18 at 17:18
  • @MyCatsHat, understood. Maybe my 'play about' idea could help. There is no need to be narrow even if you are diving into a narrow subfield. My diversion might by cycling. Yours might be programming (or whatever). Though an active body aids an active mind. – Buffy Aug 6 '18 at 17:21
  • Regarding "a second doctorate will probably be superfluous": I think already the first doctorate degree is superfluous, since I have published papers in both fields already. (Of course I don't have as extensive experience as someone who has completed a PhD, but I think if I could just be working for one intensive year with a professor I would have enough experience; no need for a formal PhD. But it is necessary, since it often is a formal requirement for postdocs. – MyCatsHat Aug 6 '18 at 17:23
  • The "play about" idea is good. But how to do that in practice, if I will have a strict timeline with various deadlines, my supervisor expects me to publish with him etc.? I couldn't easily say to my supervisor "hey, I'll take a break now for a week -whilst you keep paying me- since I have this cool idea in a totally different field that I'd like to try out", right? Or maybe I could, if I frame it right, but I don't know how to do that... – MyCatsHat Aug 6 '18 at 17:25
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    @MyCatsHat. You don't need "education: at all to learn how to think or write - though it helps most people. But if you want a career in Academia, it is nearly impossible without a doctoral level education formally certified, stamped, sealed, etc. Nevermind postdocs. Your goal should go beyond that. – Buffy Aug 6 '18 at 17:26

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