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I notice that many pure math PhD students, doing things like algebraic geometry or representation theory, drop out and enter industry and become software developers. The ones who finish their pure math PhDs also drop out and do things such as blogging, tutoring, etc. These students are pretty "top of the heap" too, attending the top pure math PhD programs like at UC Berkeley and UCLA.

Then I notice that many applied math PhD students typically finish, and also get post-doc offers. These PhD students are typically doing numerical methods for the solutions of partial differential equations.

Why does this happen? Do pure math PhD tracks often lead students to dead ends? Is it too hard, even for the brightest PhD candidates at the best programs?

In numerical PDEs, it seems that just a little progress leads to a PhD degree, e.g. solving numerically a PDE in one space dimension. (I know firsthand of someone getting their applied math PhD from an Ivy League program for solving a PDE in 1D.)

I have friends in both pure math and applied math PhD programs and have noticed this trend for many years now.

closed as primarily opinion-based by Morgan Rodgers, Jon Custer, Richard Erickson, Brian Borchers, StrongBad Jul 24 at 19:59

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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    Do you have the definitive figures for all institutions or just two of your friends left from one dept compared to one from the other? – Solar Mike Jul 15 at 7:26
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    I somewhat agree with you that numerical results are often just incremental, but please do not be fooled by PDE dimensions. There are plenty of equations, where even an 1D-result can rightfully be considered a major breakthrough, though they might look deceptively similar to others, where completely solving them in n dimensions is just a textbook problem. – mlk Jul 15 at 9:40
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    I voted to close and think this question is highly misleading in its premise. You give no evidence that any of what you wrote is true, nor does much of it even ring true, for example I know of no algebra PhDs who left academia to do blogging and/or tutoring. So, we can have an opinion-based discussion and hear some random anecdotes, but I suspect at the end of the day we’d be none the wiser (and possibly we’d be less wise for having a false impression that we know something that isn’t even true). – Dan Romik Jul 15 at 16:00
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    ... However I think there may be some legitimate, on-topic questions here, e.g., about whether the job prospects for academic jobs are better in applied math than in certain areas of pure math. Some people may be able to dig up statistics about this. It would be good if you could edit the question to make it more focused on such things. – Dan Romik Jul 15 at 16:05
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    Isn't a PDE in one dimension an ODE? – Michael Seifert Jul 16 at 14:36
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Only my personal experience, but I hope it will help you a little bit:

I am almost in the exact situation you describe in your question. I finished my PhD in Algebra (coding theory, lot's of linear algebra and representation theory) and am now working as a software developer. As a fun fact, this wasn't always planned. When I started my PhD, I was still dreaming about getting a PostDoc and ending up as a tenured prof at a prestigious university.

For me, personally, there were three reasons to take this step:

  1. The PhD was quite hard and taxing on my mental health. Having a rather theoretic topic, you sit at your desk all day and try to come up with an idea. There is often no way to track progress for a long time, as you either have a clever idea or you don't, you can't set up a step by step plan to see if you are doing ok for such things. There is also always the fear that you simply won't have the right idea before your time/funding runs out. Due to that reason, I decided already some time before finishing it that I will run from academia without looking back as soon as I am done.
  2. Closely related to the first reason, I wanted to do something more applied. I found a job where I can use my skills to really help people/the climate/... (our company develops AI solutions for agriculture, allowing farmers to better use resources and minimizing the need for pesticides).
  3. I still plan to become a professor, one day. But after that experience during the PhD, I'm aiming for a position that is more focused on teaching and applied research (e.g. with industrial partners). For this, at least here in Germany, you need some years of industry experience.
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    I have seen similar stories from people in all areas of mathematics, not just algebra. – Randall Jul 15 at 17:51
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    It’s amazing how similar your situation was to mine. I would add to point one, that it also gets incredibly lonely doing research in a highly theoretical field. There were only a handful of folks I could converse with about stuff I was thinking about all day. I even started talking to myself a lot.... this wasn’t the fulfillment I had imagined entering grad school. – Joseph Wood Jul 15 at 18:22
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    #1 is a very good point. I found out that while I'm pretty good at solving problems, I'm lousy at coming up with problems that are worth solving. So it's much easier if I work for people who'll say "Hey, we need this problem solved, and we'll give you a bunch of money to do it." – jamesqf Jul 15 at 18:33
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    @jamesqf: I have a bunch of problems that need solved! Sadly, I'm also blessed with a minimal amount of The Root Of All Evil. So, not only do my problems not get solved, but I also don't get to be Evil. (*sigh*) :-( – Bob Jarvis Jul 16 at 16:24
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    @Dirk Do you think it is because time has changed? Many people who are professors now got their PhDs before the internet boom and the subsequent explosion in tech. There are almost impossibly many problems dealing with technology (in domains such as robotics, artificial intelligence, computer architecture, network protocols......)that are equally as intellectually challenging, and often times financially rewarding in an increasingly expensive world. Maybe the PhD that primarily deals with theoretical research has ran out of its uses. – Carlos - the Mongoose - Danger Jul 16 at 20:25
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Well, it's been more than a couple years for me since grad school, but I can add my personal experience to what is appreciably a rather subjective question.

Of my group of students at a pure math department, with a few world renowned old timers, I would not say algebra was considered the harder path. On the contrary geometry and analysis students, such as those studying general relativity or geometric analysis, such as stochastic differential equations over manifolds, were generally seen as having more brass. Ironically geometry and PDEs are closer to what's studied in "applied math" but this was a pure math department and numerical analysis was not studied.

The students I knew doing algebraic subjects, such as number theory or topology were more successful. One of whom got a post-doc at MIT, doing cetegorical homotopy theory and such.

But I think your point of view applies to the difference between pure math and applied math in general. Of some 15-20 of my grad school friends, most completed their PhDs and eventually worked for Google or in cyber-security or for IBM, such as myself, after perhaps teaching math for 3 or 4 years. I think 5 have settled down as math professors.

The simple answer to your question is just that pure math is very hard. I cannot speak to applied math training, I was not much exposed to it, but from what I understand proving an unknown statement or theorem is not necessarily required, whereas it generally is for a pure math thesis. It is occasionally said in academics that obtaining a pure math degree is the biggest challenge among PhDs. As the decades pass the bar is set higher and higher because math is a progress field and it is being completed. A similar sentiment is often expressed regarding physics and other hard sciences.

So, whether or not a student completes a thesis, being a mathematician is onerous work and requires an emotional commitment to continue at any level.

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    The low-hanging fruit was picked ages ago, and new candidates have to keep climbing higher and higher to find unpicked apples. – RonJohn Jul 15 at 19:28
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    @RonJohn ... or pick from from a novel new fruit tree. – chux Jul 17 at 4:48
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    @chux but you've got to find the new fruit tree... – RonJohn Jul 17 at 12:18
  • @RonJohn And all the low-hanging fruit trees were found ages ago! – knzhou Jul 18 at 23:22
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This is also something that is very true in statistics as well (applied vs theoretical).

A theoretical math PhD is (obviously) extremely difficult and most of the topics in it tend not be directly beneficial towards future employment in anything but more work in abstract theoretical math. While many people think this is what they want to do at the beginning of the degree, the reality of the work is often far worse or different than what they are prepared for.

Now this is hardly an uncommon situation in many of the degrees out there but there's a catch for people in degrees like pure math that isn't true (as an example) for someone getting their PhD in history or literature: the skillset of a pure math PhD student is extremely lucrative. When you can either continue to bang your head against the wall on incredibly difficult and esoteric topics, or get an offer the next day for a 100k + paying job in tech or banking, I don't think it's much of a surprise a lot choose the latter. Often times they have recruiters seeking them out telling and them as much.

The difference for applied PhD's is that the end product is even more lucrative/valued and opens doors that someone with a masters degree isn't qualified for. If you complete a PhD in something like cryptography or applied AI/ML, you will have career opportunities in Google, Microsoft, blah blah blah, as they are specifically seeking out people who are experts in those topics. Whereas with pure math PhD's they will still seek you out, but only because people with those degrees also have a skillset they value, not because they care about your thesis on << ultra abstract pure math topic here >>.

Of course there are exceptions, but I believe this is the general reason you see this phenomena.

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    Yes. Students may have much student debt and tech/banking companies offer a salary that alleviates the burden dramatically, vs continuing in academia. – qwr Jul 16 at 21:05
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    @qwr (also adressed to eps) Hi, I'm a soon to be math graduate student in mathematics. I have heard the claim before that even pure math PhD's are valued in industry. I believe you, but I am somewhat confsed about this. I do not see how studying some abstract math topic makes you competitive for a high level job in industry. If you don't mind, could you give some general examples of positions that pure math PhD's would be competitive for? Thanks. – Ovi Jul 18 at 14:39
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    @Ovi even if the research topic itself you've done in your PhD isn't interesting for employers, it may still a) involve tools that are just as important for another, applied subject, and by using those tools in the pure setting you get lots of valuable experience with them b) proove that you're in general be able to do novel research. Somebody who's been successful in a difficult (and “boring”?) pure-maths project will with some likelihood also be successful in an applied project, even if the concrete topic has little to do with it. – leftaroundabout Jul 18 at 14:53
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    @Ovi personally I've seen some pure math people go into quantitative finance or statistics, both of which can offer six digit salaries relatively quickly. It's not directly related to pure math but companies value the problem-solving and technical abilities. – qwr Jul 18 at 15:53
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    @Ovi higher level math students almost always tend to be very good at abstract problem solving in general, and also tend to have taken enough programming courses to be fairly skilled in that area. Even if you've never seen a machine learning algo I would guess most people with that background could figure out what is going on rather quickly, and analytics / data science in general benefits from being able to think abstractly and approach problems from different angles. – eps Jul 23 at 18:02
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First, it is not accurate to use "algebra" to refer to broad swaths of mathematics, any more than it is accurate to refer to "pure math", in fact. One immediate objection is already partly related to the issue of the question, namely, these are labels very often used by outsiders to refer to "not what they themselves do", and/or reflections of their own lack of awareness of mathematics outside their immediate competence. (It is not a moral failing to not be a universal scholar, but it starts to become some sort of problem when one loses sight of the skewing that one's own ignorance may engender.)

More significantly, I really don't think that there is any coherent "subject" that is "algebra", for example. And in any case number theory is not a subset of "algebra", nor is algebraic geometry, nor "representation theory" (despite the weird and technically inaccurate wikipedia entry for it), nor... logic? set theory?

Is there "pure math"? Well, depends what you mean, obviously. A very common usage of the phrase is by people who have a limited understanding of mathematics, and use the phrase to refer to things they don't understand, and don't see the point of. Also, some people who are "pure mathematicians" give themselves this label as some sort of idealistic thing. BUT in reality the best mathematics tends to be relevant to lots of things. Yes, sometimes people get involved in "deep background", and due to non-trivial difficulties do not manage, in the lifetimes, to return to resolve the original questions that sent them off on their inquiries.

My 40 years of observation of people working on PhD's in math at good places does not indicate that "pure math" people have a harder time either finishing or getting jobs... except for the subset of those people for whom the label does truly refer to something they've discovered they don't care about.

And, again, the potential relevance of any part of mathematics to things outside of mathematics proper is huge. Category theory to computer science!?! Stochastic differential equations to finance? Elliptic curves to cryptography!?! More elementarily: quaternions and 3D games (not to mention aerospace?!?)

To recap: the framing of the question is unsurprisingly naive, but, also, inaccurate in the current reality. Not so surprising that beginners are not aware. Also, skewed language not only reflects misunderstanding or ignorance of reality, but can limit one's vocabulary so as to create difficulties in having coherent discussions about reality.

(It may be worth noting that in the U.S. some of the "research experiences for undergrads" are (perhaps necessarily) so artificial as to be pretty silly. Maybe fun, fun to be with other enthusiastic kids, but often quite misleading about what genuine contemporary mathematics is, and what "research" in it would mean. But/and it often seems to happen that the people slide into an apparent enthusiasm for an alleged part of mathematics that they give some naive name... and, often, become disillusioned when the part of actual mathematics that has that name is not at all what they thought they'd bargained for.)

EDIT: in light of several (generally understandable comments):

Yes, it's not easy to complete a PhD, and it's not easy to get a job, and in all cases one can easily feel that one has no hope to be any sort of heroic contributor. But may that last bit is asking too much for most of us, in any case.

In my years of observation: the primary determiners of success or failure in completion of the PhD itself, and in getting an academic job, are the advisor ... and, second, the student's attitude.

Seriously, for completion of the degree, itself, it's not that any kind of math is any easier than any other, unless by mischance standards are lowered. And it is not easy to gauge the latter. That is, often, novelty is a valuable thing... even if it doesn't pay off... and serious novelty is truly harder to achieve in topics that have been around for 200 years.

Some less constructive "criticisms" of my earlier remarks seem to hinge on the allegedly obvious esotericn-ness and irrelevance of sheaves... or something... etc. Ok, I have to disagree with this, and claim that such math is old news, and has proven its value in understanding basic things. The possible fact that novices do not understand/appreciate its value... while a significant fact... only makes discussion more difficult.

My own general subject of interest, number theory (and its applications), nicely shows the vacuity of attempted partitioning of "math" into "analysis, algebra, ..." (though, yes, not so many years ago my own university's math dept had a supremely idiotic categorization as the basis for a hiring plan, etc.)

E.g., one of my recent PhD students (Kim K-L) solved a differential equation in automorphic forms that expresses four-loop (if I remember correctly) graviton interations in string theory. Simplistic classification is...?

What I do observe is true, again, is that novices' naive ideas about "what subjects are" leads them often to naive decisions, causing them to lose enthusiasm when the reality catches up... Perhaps a significant difference between "applied math" (maybe the questioner means "modeling"???) and (then???) "proving theorems" is that the bait-and-switch on theorem-proving might be perceived as far worse than in engineering-oriented "applied math".

The amazing thing about (good) math is that it is not only relevant but decisive in so many human endeavors. Whatever one's specialty, if one is _good_at_it_, one will have both specific and abstract mathematical chops, but/and demonstrated resilience to certain sorts of scientific/intellectual adversity. Crazily-enough, not all STEM disciplines teach that.

In brief: not, it's the person and their advisor, not the topic.

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    I think I somewhat get what you are trying to achieve, but maybe it would be easier to bring across the point if you switched it to be positive (i.e., describe what current maths research (the type the OP calls "pure") is, not just say that it isn't what OP thinks it is). Also, maybe there is some simple misunderstanding going on; I have no trouble understanding what "applied math" is, and can easily see myself calling everything else "non-applied" or "pure" math. This may be naive, but still valid in the context of the question. "Algebra" seems just an example here. – AnoE Jul 17 at 9:16
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    Rubbish. Of course there's a subject called modern algebra. It doesn't matter if it's old news. One of the biggest "divides" in math is still whether you trend toward algebraic or analytic tools. And there are departments that specialize in either "pure" or "applied" math. They have a notion of what they mean by saying they specialize. This is all obvious. Why obfuscate? – B Custer Jul 17 at 22:32
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    @paulgarrett well, don't you think that proof techniques developed for homotopy theory or algebraic geometry (sheaf theory etc) is pretty well diverged from those of ordinary old geometry (ie. like John Milnor books) or simple functional analysis? In my experience people that read the one seldom want a conversation with people who read the other. – B Custer Jul 17 at 23:50
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    This answer is just not satisfying -- it avoids discussing an issue that many real students in the field have seen for themselves, possibly out of some misplaced sense of required humility. "Are cars really more dangerous than planes?" "First, it is not accurate to use "cars" to refer to broad swaths of vehicles. Cars are not inherently better than planes, and many people can drive both. Planes even have wheels, just like cars do. In fact, some small planes can even function as cars, and what will you say when flying cars appear? The framing of the question is, unsurprisingly, naive." – knzhou Jul 18 at 12:20
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    Operations research, mathematical modeling, and aerodynamics are applied. The stuff on ncatlab is pure. There may be other subfields that are edge cases, but no mathematics department in the world has my examples swapped. Which is why it’s bewildering to see so many pure mathematicians show up in the comments denying there is a distinction. – knzhou Jul 19 at 1:24