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I don't know what kind of job I want, but I know I like pure math. It seems people study pure math and then, in the end, realise there is nothing out there for them (jobwise). I don't want this to happen to me (and many other students as well I guess). This question was particularly interesting to read :

How do I sell out with abstract algebra?

Next year, I get to choose between a mix of applied and pure math courses. Given that I have no idea of what I want to do after my degree :

Should I go « all-in » and study as much pure math as possible, basically follow my heart ?

Should I take a mix between both but reducing my chances to study more interesting math (from my point of view) but it is a safer option

Or should I sell out already and take as many applied courses as possible ?

I have studied physics two years along with math and really did not enjoy it so I am a bit scared of taking applied courses.

If you could give an example of a course that made an impact (positive or negative) on your mathematical career, that would be nice !

Thanks a lot !

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Choose wisely... –  Asaf Karagila Nov 20 '13 at 23:39
@Hypem : it is extremely difficult to make a living doing pure math. There is much more demand for people who can make a computer do things than for people who can prove theorems. If you have any doubts, you should learn some programming. A bachelor's degree in mathematics, even with perfect grades, is a near-useless job credential. People will dispute this, but if you press them for counterexamples, they will probably cite examples of people who studied math and something more marketable. As an experiment, you can try entering "mathematician" into monster.com (are they still around?) –  Stefan Smith Nov 20 '13 at 23:57
... and "software engineer" into monster.com and compare the results. You can try searching for "mathematics" or variants thereof too. Actually, I would guess that a degree in statistics is more marketable than one in mathematics. –  Stefan Smith Nov 21 '13 at 0:00
Well there are not many jobs "mathematician" in the title and would be something like "analyst" I guess but it all deals with data etc which I am not really interested in. But I see your point. Is it better to take some stats courses or some PDE courses then ? –  Hypem Nov 21 '13 at 8:39
@Hypem : in addition to thinking about what kind of math courses you want to take, I suggest you make friends with students and/or profs in physics, engineering, or related fields and ask them what kinds of math problems they are interested in. These are not necessarily the problems that mathematicians spend their time on, to the detriment of many mathematicians. I think that much interesting mathematics these days is driven by applications as opposed to mathematicians' curiousity. Even if you are not interested in the applications, the math problems may be interesting to you. –  Stefan Smith Nov 22 '13 at 16:58

4 Answers 4

up vote 3 down vote accepted

While "applied math" is not pure math, it can still be intellectually rewarding to study - especially if you like pure math, you might enjoy some graduate-level courses in applied math (statistics, computer science...). I would suggest taking a good mix of both - remember that the background in more thorough pure math can make the applications much easier.

Keep in mind that you are still young, and you may find your interests changing over time. At your stage, I would advise trying to keep up being at least a little mathematically well-rounded.

Ultimately, I can't tell you what to take or when to take it - neither can anyone else, this is a choice you must make on your own - but I can advise you confidently to take as many classes as possible.

EDIT: As an anecdote, some of the more advanced undergraduates I know took very challenging courses in pure mathematics during their first two or three years of college, which gave them a foundation solid enough to just burn through graduate-level classes in applied math during their third and fourth years. Pure math is pretty good. If in doubt, take entry-level pure math instead of entry-level applied math.

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I think it's important to have a balanced diet. My personal recommendation would be to supplement your pure mathematics courses with some applied math that interests you no matter which path. Personally, I find that studying applied math, physics, and computer science is a great means of providing motivation and inspiration in my work in pure mathematics.

I love math for its own sake, but I admit that sometimes it can be a little discouraging to study tons of definitions and theorems about things you can never see or touch. It's very stimulating to see functional analysis is realized in quantum mechanics, or algebra realized in cryptography and coding theory, or category theory in programming. It scratches a certain itch. Furthermore, applied math can give you inspiration for problems in pure mathematics. Clifford algebras, for example- and more broadly, lie theory- both originally developed out of physics.

Perhaps most importantly, I wouldn't think of choosing to take applied math courses alongside pure as a "safe" option, anyhow. If your backup plan is to go into industry, you need to do a lot more than just take applied math courses. Among other things, you'll need to develop a different attitude about math centered around efficacy and efficiency, rather than just elegance. You should read this article if that's something you're considering. I'm not saying you can't do both- but if you do, treat applied and pure as a double major, and fully commit to each. I walked this line myself as an undergraduate.

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Note, though, that some of us don’t feel such an itch and genuinely don’t care one way or the other whether some branch of mathematics finds application somewhere outside of mathematics. –  Brian M. Scott Nov 21 '13 at 16:45
@BrianM.Scott I don't mean to imply that application is required to be "good math," by any means. What I mean is more that being exposed to a pure math construction in a an applied context can enhance one's understanding of it, kind of like seeing a very detailed example. I'd argue this would be helpful to any mathematics student, regardless of whether or not the application is useful. –  Alexander Gruber Nov 21 '13 at 22:00
I understood what you were saying; I’m just pointing out that it doesn’t apply to all of us, at least in the terms that you actually used. –  Brian M. Scott Nov 21 '13 at 22:01
@BrianM.Scott Sure :) Just thought I'd clarify. –  Alexander Gruber Nov 21 '13 at 22:02

I've heard some professors say before that "You can switch from pure math to applied math, but not the other way". If I remember correctly, it was Munkres that said this.

There's nothing wrong with studying pure mathematics. Even with a pure math focus, you will have the ability to look at any other piece of mathematics, learn the definitions and gain a bit of intuition for it, and from there you should be able to grasp the material without much effort; mathematics isn't simply the study of a bunch of material to memorize theorems or solve problems. It teaches a way of looking at problems, about how to abstract things and how to find the relationship between two seemingly disparate topics.

You can certainly find careers that are not in academia even if you have a pure math focus. As an example, at Microsoft there is Microsoft Research New England and the Theory Group (though admittedly this borders on theoretical CS). I've known people that were pure math majors and ended doing Law School (their application was based on the idea that "Every law school needs a mathematician!"). The NSA and like organizations are also friendly towards pure mathematicians.

In the same regard, you shouldn't immediately blast applied math as being a sell-out (if I'm understanding what you mean well). Take an applied math course and see how you like it.

However, perhaps most importantly is the fact that you find pure math more interesting and invigorating, then you should almost definitely go with that.

As a current student myself, I can't say much about how taking certain classes has guided my career, but I can at least say that I'm happier doing pure math than not.

A class that has been incredibly helpful to me in the study of mathematics has been topology, which provides a new and important way at looking at different fields of mathematics, and makes precise a lot of intuitions gained over the years in conventional math courses. Other classes that would be helpful include proof based classes such as a discrete math course or (at least this is what it is at MIT) an analysis course. Most importantly, however, take what you think you'll enjoy. There is a lot of mathematics out there, and you'll almost definitely be able to find a niche in what you enjoy!

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You write "You can certainly find careers that are not in academia even if you have a pure math focus." I think the OP could benefit from some examples/suggestions, since such careers are hard to find. For example, if you look at the job ads at the back of SIAM's newletter, the large majority of them are for adacemic jobs. And that is SIAM. –  Stefan Smith Nov 21 '13 at 0:08
@StefanSmith I added some of the things I could think of. Agreed that it certainly isn't the range of options available to an applied mathematician, though. –  Hayden Nov 21 '13 at 0:19
Thanks. From what I have heard, groups like Microsoft Research New England are tiny groups of people who are both extremely brilliant and lucky (not to mention might be actually working in TCS), and such groups are shrinking (I think Bell Labs (?) had a research group that disappeared almost or entirely). Unless you are Richard Feynmann, I don't think you can count on finding a job at such a place. Government funding of pure science research of all types is shrinking. I also know a math Ph.D. from a top-tier school (also a chess International Master who could play blindfolded!).. –  Stefan Smith Nov 21 '13 at 0:39
...who ended up being a lawyer. But this may require law school, and may not be possible if one is married and/or with children. The NSA indeed hires lots of mathematicians and I don't think they are interested just in number theorists as one might suppose. What do you mean by "like organizations"? The NSA is very unusual in many ways and I'm having trouble thinking of a similar organization. Again, thanks for adding to your answer, and I don't mean to pick on you. –  Stefan Smith Nov 21 '13 at 0:43
I strongly recommend, however, that anyone thinking of working for NSA as a mathematician get a PhD first; judging by the experience of a friend and former student of mine, if you go in with less, you may find yourself a bit of a second-class citizen. –  Brian M. Scott Nov 21 '13 at 16:54

Actually there are quite remunerative jobs in the area of finance for highly skilled math students. Especially in risk management at financial institutions.

Also actuaries make an excellent income and have high job security. Although eventually with more responsibility, you would also need to become familiar with insurance regulations and accounting. But that is usually accomplished on the job.

A degree in math with excellent grades will be recognized as an indication of your capacity.

In that you say "follow your heart" it seems pretty clear what you would like to do.

Have you taken a real analysis course? That's usually the entry point to pure math and could be a "decider" for you.

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As a discrete mathematician, I'm gonna have to disagree about real analysis being "the" entry point to pure math. (If that were my "decider," I absolutely would not have continued in math!) It is for some people, but abstract algebra is for others, or combinatorics, or logic & set theory. Different people are inspired by different courses. I think it takes several courses from different areas at that level to decide whether you really love math or not. –  Alexander Gruber Nov 21 '13 at 0:10
The "decider" for me would be linear algebra/abstract algebra more than analysis indeed. As you said, there are lots of opportunities in finance etc, but having studied a majority of pure math does not disqualify you to hired for those jobs ? (because you are not in "reality") –  Hypem Nov 21 '13 at 8:42

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