b@r!um

The top programs (which it sounds are the ones you are interested in) will be super-competitive both for pure and applied math. MIT had joint admission events for the pure and applied math programs. The admitted students seemed equally accomplished (in different ways, of course).

All of them! It's not the only important element of your education (your letters of recommendation are crucially important as well), but you can bet that all graduate math programs will pay a lot of attention to your upper-division math coursework.

mathboy98

"Young" is relative - certainly, mathematicians are generally less likely to prove their most ground-breaking theorems at age 60. However, it often takes (at least in pure mathematics) years and years of getting acquainted with a research specialty before a researcher's best work may actually come forth. So, it's really a question of what aspect of applied mathematics you want to contribute to. It sounds like if you really don't want to teach, you might not like the university professorship lifestyle. Then again, a lot of these decisions needn't be made until you're actually done with the PhD, assuming you have other options (that is, besides work at a university).

Mathematical sophistication in the applied sense can win you very nice careers - I think if you strictly want to go where the money is, finance might be your best bet though. There's a lot of other cool stuff you can do with applied math, though, which can lead to nice careers.



If your target is making an original contribution to mathematics, a master's degree in mathematics is not the ideal option, at least as far as I can tell. Generally PhD programs in the US are looking for applicants with strong foundations so that they can dust off cobwebs and really iron their foundations out at the start of the program, plus begin doing things relevant to what they'd specialize in. So the master's program is in a sense already built in, and further, having a master's degree will not really speed up the initial requirements of the PhD (however, having strong foundations, however it is you acquired those, will).

If by "make an original contribution to mathematics" you mean publishing papers for a living, it seems hard to avoid the academic career or some such approximation (there are definitely places you can primarily research for a living, at least in mathematical fields, though I'm unfortunately unfamiliar with mathematical research that happens outside of universities), and further, you probably won't be able to just do this initially "while you're young" and leave - it'll take a lot of time and effort just to get the foundations down, and then a bunch to navigate the research literature enough to make an interesting contribution (except in basically a negligible number of cases).

A master's degree in applied mathematics could be useful for you personally if you want to explore further, and actually develop a specialty within mathematics ... which may then lead you either to careers within or outside of academia. To be clear, all I meant by the above is that if your target is ultimately original contributions, apply to the PhD programs promptly once you feel you're prepared.

And yes, all graduate programs in mathematics are primarily interested in your upper level coursework in mathematics (and much less in work done in non-mathematical disciplines). A pure mathematics program, I'd imagine, would take the foundational mathematics courses more seriously than, say, a class on electrodynamics, even if that course is in a sense mathematical. Even in applied mathematics, I'm pretty sure a solid grounding in at least some of the foundational pure mathematics courses is essential - this is probably tailored depending on what kind of applied mathematics you're going to do.

MD Mom

OP, you might also join SIAM. It is free or very inexpensive if your university has a chapter. I think it will give you lots of ideas about types of positions applied math majors get.

juillet

juillet, your post demonstrates a very narrow view of academia. Sure, if you are passionate about something very specific, move across the world to attend the graduate program best for that particular specialty. But that's really not the only way of doing things.

It demonstrates a very realistic, current way of doing things. Of course it's not the only way, but it is the best way, and the way that has the largest chance of getting you an academic job in the future. I know a few people who were tied down because of house and family, too, when they applied for a PhD. Everyone has to make choices that are right for them. Some choose to be less competitive for future jobs because of those reasons - and when you select programs based on geographic location instead of opening yourself up to the best programs across the nation, that's essentially what you are doing, unless you are in a high-demand faculty field. But how are they going to compete in a national search? Some choose to move their families, and some will decide they never want to move. That's ok, but most of those people realize that they will not be as competitive as someone willing to go anywhere. It makes even less sense to limit yourself if you don't have children and a house somewhere.

I used science-based language in my post (which was wrong and doesn't broadly apply to math) but the point is, you don't go into getting a PhD because you want to be a lawyer, or because you don't know what to do next, or because you want to use math in applied jobs or even simply because you like math. You get a PhD because you are passionate about math and want to do mathematics research as a career (even if you aren't leading a team). If you aren't sure you want to do math research, then a PhD is not really worth it until you ARE sure.

Last time I checked, the national median income of a full-time worker was $40,000/year. My professors seem to make significantly more than that.

That is because they are tenured and probably more advanced in their careers. First of all, median individual income is actually closer to $32,000. Median *household* income is $45,000. Secondly, median income in this case is misleading because it's the median income of everyone, from high school dropouts to college grads to professional degree holders. The median individual income for doctoral degree holders is $70,853. But most importantly, when I said "not much money" I meant in comparison to other professional jobs, especially ones that require training as long as PhD work, and in response to the term "financial opportunities" used by the OP. If your other choice is patent lawyer, I think it is worth mentioning that professors rarely get rich and are sometimes struggling to stay in the middle class.

Let's also not forget that a substantial portion of professors are actually adjuncts, who are often stringing together a full load of classes for $2-5K per class. Even if you are on the high end and teach 6 classes a year, that's $30K with no benefits and no office.

mathboy98

I think perhaps relevant to the discussion are the following three lines -

These are somewhat at odds. In my long posts, it may get lost that a primary confusion on how to advise here is that the typical route to making original contributions in mathematics is getting a PhD, even if your contributions will be made outside of university.

While you may be able to get accepted to NYU, Northwestern, UCLA, UCB, or some such school on that list of geographic locations, it's much less predictable (if you continue in academia) where you will end up after that. Getting accepted to those schools is also no joke and no certainty - it's a matter of a lot of things falling into place, after you develop a great profile. I would not link geographic stability with the career associated with publishing mathematics research.

That said, an applied mathematics PhD, depending on his or her skillset, can be extremely valuable outside academia, and one may shoot for careers with a little more certainty. If finance appeals, there's a good chance you'll end up in NY, which was on your list. Other than that, I can't think of good ways to make sure you end up in one of those locations. Main thing is to go for a career that doesn't seem highly likely to push you elsewhere.

You can talk to me, b@r!um or others about what the mathematics track is like career-wise, but I strongly suggest you also talk to someone who would actually know about what actually going into the legal profession (NOT just going to law school) entails. A lot of people go to law school not knowing what they really want. It's a lot better if you don't have regrets halfway through. You sound like someone who has interest in different areas, but might be much more rigid when it comes to actually selecting a career, particularly because that involves constraints beyond what your work actually involves. I think you'll actually check off a lot of these options once you get a good idea what the careers are like.

b@r!um

And that's there we disagree. I am in one of the most selective math PhD programs in the country, and most of my classmates aren't sure that they want to do math research for a career. There is no way to be sure until you're at least half-way through your PhD. We spend 2 or 3 years in graduate school taking courses and reading papers before we can even begin to think about research problems.

MOST math grad students started their PhD program simply because they liked math. They can decide if they want a research career while they're in graduate school. If not, they are well trained for a range of quantitative and analytical careers outside of academia.

juillet

I mean every person has to make their own choices and people will have differences in opinions. IMO, it's a waste of time to go to get a PhD if you aren't relatively sure that you want to do research. Of course, you don't know what research is really like until you really get deep into the program, but I feel like you can be reasonably sure ahead of time by doing some research with professors and perhaps an independent study or senior thesis. And it's a lot harder, psychologically speaking, to leave a PhD than most people think.

Of course, an MA/MS in mathematics is a pretty useful degree and if you can extricate yourself after 2-3 years with that in hand if you decide you hate it, then no harm done.

Again, difference of opinion here. Personally, I don't think you should go to a PhD program thinking that there are a wide variety of careers in which a PhD would be useful, because the truth is a lot of employers (willing to bet most) would prefer an MA/MS with experience to a PhD, especially in fields like finance and operations research. You should go...if you want to do research...because that's what a PhD is preparation for.

b@r!um

This may work in the sciences. It won't work in mathematics. 99% of all senior theses are reading courses and students cannot easily "do research with a professor." What would that even mean? There are two main steps in mathematical research:
- knowing the literature well enough to form a hypothesis extending current knowledge
- knowing the techniques to prove this hypothesis

Most undergraduates are not in a position to do either. (Keep in mind that the undergraduate math curriculum ends in the 19th century. The first half of a graduate program is dedicated to learning some subset of 20th century mathematics...) Unlike in the sciences, there's no lab work or easy tasks that an ambitious-but-untrained undergraduate could do. If you are not doing the theoretical work, you are not doing research.

mathboy98

Ialso think the situation is somewhat more similar in engineering or CS or whatever than one may be inclined to believe initially, where you might enjoy doing a few serious projects as an undergraduate and be good at it, get admission to some solid PhD programs, but realize after 6 years that your steam to do it is not the same. The relationship with an adviser and doing several years' worth of making mistakes and slowly learning to make a contribution is generally not possible to replace in terms of realizing if the path is for you. People leave academia even after some not-at-all-unsuccessful postdocs. People, actually, rework their careers a lot in various settings (doesn't have to be academia).

Now "relatively sure" can mean a lot of different things. I think basically, the best definition I can think of is knowing the pitfalls of leaving a PhD, knowing that one could have spent that time a different way, knowing that you don't have to do a PhD to learn and appreciate mathematics all your life, etc etc .... and still having a strong feeling it's worth it and that one might very well end up happy doing mathematics research at the end.

I'd say that my stance would be to try to avoid a PhD pretty hard, inform oneself, break down biases both toward and against the path, etc etc, and then if it still seems the best way to go, just go for it and give it a very good shot.

Now this varies. EVEN in mathematics, sometimes students do relatively a fair amount of research as undergraduates. They are certainly in the minority, and I myself was too all over the place to be one of them. But even so, I claim that given the length of an undergraduate degree as compared with the amount of time it takes to really get to one's best work, it's often hard to tell if a career in research is something the individual wants. Doing what it takes to be successful in a research career is, in my view, still quite distinct from knowing what research is (and even liking it a fair bit). I have come to believe this after seeing people with plenty of talent, at the very strongest programs, doing very nice work, still deciding that doing what it would take to not flop (well, by their definition, this means do pretty well as opposed to barely scrape by with something not quite acceptable to them) at the career is not really in line with their goals.