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This is something I have been thinking about for some time. What I study is pretty much on the edge between applied and pure math. Specifically, I am interested in stochastic dynamics including SPDEs. Thus I have been investing a lot of time into stochastic analysis, really enjoy the subject, and am looking into some new ideas in my "spare time". I say my spare time because right now I am really productive on the more applied side where I am looking at stochastic dynamics in developmental biology and numerical methods for stochastic systems (I have a few publications I will be able to push out before advancement).

There is some trade-off here. My adviser strongly thinks I should drop all other activities because I am already building a solid foundation in the applied math / systems biology direction, and going 100% in it would make me have a higher chance of being a top candidate in the field come graduation. However, I kind of think that incorporating more "theoretical" stochastic analysis into my studies/research will lead to stronger results further down the line, but in the short term will inevitably be a hindrance. If I am thrown in the same pool as other applied mathematicians (who tend to publish more than pure math) and judged in the same manner for productivity, it seems that the extra time it takes for pure math results would most likely be detrimental to my chances of getting a post-doc.

I assume I am not the only person in academia who has faced this kind of decision. Is this something to come back to as a post-doc, or tackle now? How did you handle it? How did your colleagues handle it? How did it turn out? Do you think you made the right decision?

I do want to emphasize that right now I am doing quite well, and I think that matters a lot in this decision. However, since I will have a good amount of publications pre-advancement (based off of my own ideas), have fellowships, and will have a masters in both math and bio: does it make sense to take the time to diversify my personal portfolio? [Currently I am thinking of doing like 75% applied 25% pure, trying to spend less time optimizing code and using the down time as theory time.]

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up vote 8 down vote accepted

(Sorry; though I'm a mathematician, I have no professional familiarity with stochastic anything.)

"Even in mathematics," most postdocs are given because a particular faculty member wants to work with you or is at least interested in hearing what you are working on. Can you identify mathematicians who are working on both the pure and applied sides of SPDE? For them, your aspirations at breadth will be viewed as a positive. If there are no tenure track faculty members doing this, then trying to do this as a PhD student looks, from a purely strategic perspective, rather risky.

In my experience -- though, beware, my mathematics is as pure as the driven snow -- graduate school is too early to "show breadth" in an advantageous way. Up until relatively recently, the most common number of pure papers a strong graduating pure math PhD student had was zero. (This was my situation, and I graduated from a top-three department.) Instead they had a letter from their advisor talking about how important their thesis work was (going to be). Things have changed a bit, but even now in pure math one great theorem is worth ten very good theorems. My feeling is that, except for a targeted audience as above, the applied math faculty is going to judge you for the applied math you've done, and the pure math faculty is going to judge you for the pure math you've done. These accomplishments will not be added together: rather, you need to be sufficiently impressive on either scale separately. If you're doing 25% pure math, then unless you have an amazing project it will be hard to compete with those who are doing 100% pure math.

I have to say though that I have some misgivings in answering your question from a strategic viewpoint. I would really like to say that PhD students in math should study what interests them the most, and not worry about "selling themselves" until later in their career. Unfortunately in the current job market I think it is fair to warn you that you may be taking a hit for this -- but nevertheless it may well be what you want to do. Doing what you want to do is a powerful, wonderful thing and should not be underestimated.

Finally, I wonder if you can escape the "all other things being equal" aspect of your question. There really is not a "fixed amount of productivity" possessed by every math PhD student: on the contrary, I have observed an almost limitless fungibility there. Maybe you could stay in your PhD program for an extra year. You would think that only weaker students do this...but you'd be wrong -- I've observed a positive correlation between both age of a newly minted PhD student and total time in school with the quality of their work. Maybe you could spend a semester or a year working with a faculty member who is more focused on the pure side of what you want to do: that could work out very positively. Maybe you could just vow to yourself to do the same amount of applied math you were doing before, do pure math on top and watch less TV. Or whatever. It's your life, and who the heck am I to tell you not to chase your dreams? Just chase them with your eyes open. (Hmm...)

Good luck!

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That's exactly what I needed to know. I guess a good way to go forward is to think of pure math / theory as fun, but is not likely to matter much in hiring decisions. I think learning the pure math parts, but only focus on applied papers/results unless something really clicks, at least until postdocs time. When I hit postdocs I will use the built-up knowledge to diversify. Thanks! – Chris Rackauckas Feb 2 at 6:28

If you are able to publish in both pure and applied math and keep a leg in both worlds, I would say that can give you a very real advantage later when you are looking for a faculty position. E.g., you may be perceived as an applied guy since that's where 80% of your papers are, but the pure math people would much prefer hiring an applied guy that they can see themselves having fun talking to than one they can't, so if a department is hiring in applied math in a certain year, a candidate who can appeal to both groups can get more faculty to coalesce in support of them. I have seen specific cases where that kind of breadth has made a difference in hiring decisions.

With that said, obviously there is a trade-off and an opportunity cost, and the time you will be spending developing the theory side will be time when you might have written an additional paper supporting your "main" line of research. In my opinion, each math researcher needs to consider for themselves how much diversity is optimal for them. Personally, I'm someone who favors more breadth and diversity (in my case, not in the sense of doing both pure and applied work but doing work in several areas of pure math), and have been able to build a good career doing that. But it's not something I'd recommend to everyone to try to do.

Putting aside career concerns and the fact that there are only 24 hours in a day, in a utopian sense I think the best mathematicians (and the ones I personally respect the most) are the ones who have as broad of a background as possible, since such people are often able to draw surprising connections between unrelated areas that other, more narrow, people can't. Mathematics has a surprising unity to it, and it's often amazing how an idea in one area can inform developments in another seemingly completely unrelated area. But, as you yourself clearly recognize, it would be unwise to consider only such utopian, philosophical aspects of the question without considering the realities of the academic world and the limited time window one gets in which to launch one's math career. As Pete Clark said in his excellent answer - good luck!

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