I've noticed that some algebra / number theory / geometry students seem to come to a dead-end in mathematics and tell me, "yea, no more math for me, no way, can't wait to be done with it", while the ones who specialized in analysis and PDEs, who don't have a future in analysis or PDE theory, seem to fall back nicely on theoretical CS, data science, machine learning, optimization, and other numerics-type of mathematics.

Is focusing on algebra during one's math PhD significantly riskier and generally not advisable?


closed as off-topic by Brian Borchers, Massimo Ortolano, Brian Tompsett - 汤莱恩, Stephan Kolassa, Wrzlprmft May 11 '16 at 20:04

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  • 3
    The labels/distinctions in the hypotheses of the question are not really accurate... That is, not only do those broad "areas" not classify mathematics, but also the "numerical/not" aspect is manifest/not throughout mathematics. Also, describing "theoretical CS, data science, ..." as "numerical" is comparably dubious. Maybe you should not take your acquaintances' descriptions as being reliable accounts of how things work. – paul garrett May 11 '16 at 18:25
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    the sense of the question seems to be firmly based in a fairly naive conception of mathematics and its applications outside of academic math itself. E.g., it is not the case that "algebra/number-theory" makes much sense, since, among other reasons, number theory involves many, many things that are "not algebra" (by any standard), ... and modern crypto and coding are significant instances of "applications". But, yes, there are traditions in which "applied math" is literally a synonym for PDE and maybe numerical solution thereof ... even when not applied... :) ... – paul garrett May 11 '16 at 21:37
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    ... [cont'd] One could perhaps reasonably argue that most applications of mathematics to the external-to-academic world prior to 1940 were about "mechanics", and, thus, PDE. But cryptography, and error-correcting codes in the last several decades change that. The advent of category-theoretic ("algebra"?) ideas in computer science in recent years? The REAL point is that the academic and other job markets are currently tougher than at some other times. The relative "risk" of any general direction is entirely swamped by the quality of one's thesis work, one's advisor's sensibilities, and so on. – paul garrett May 11 '16 at 21:41
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    ... [cont'd] It may be true that some "summer research programs for undergrads" are misleading because of their pose as having very low prerequisites, and then kids think that they're already on top of the game... and call it "algebra" or something else... thus corrupting the sense of that label, which already was vague. But, yes, then, if the true sense of the label is "low-prereq, no-heavy-lifting", well, sure, that's not good preparation for much of anything, and ... disillusionment that one is not, in fact, already expert? Tsk. Much noise in opinions of scarcely-informed people... – paul garrett May 11 '16 at 21:44
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    If the OP is suggesting that people who like and then don't like algebra/number theory leave the mathematical sciences entirely while people who like and then don't like analysis switch to some other branch of the mathematical sciences: nope, I haven't seen any such trend (and presumably I have a larger sample-size than the OP). Bottom line: if you want to plan for a non-academic post-PhD career, plan for it in a positive way -- learn skills now that would be valuable in such a career. Don't choose your academic specialty based on what a few people who decide not to do that go on to do! – Pete L. Clark May 13 '16 at 5:07