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I intend to go to graduate school for applied/computational mathematics, specifically a program like this https://icme.stanford.edu/.

At this point, I'm trying to decide whether to take graduate level theoretical math courses in areas like Algebraic Topology, Differential Geometry, etc. (which I don't currently have any experience with, but I could still take), or just take courses in application areas (the ones I'm interested in are statistics, biology, chemistry, computer science).

My math education in the latter case will consist of basic linear algebra+calculus, a course on PDEs, a course on abstract algebra, a complex analysis course with an applied focus, a couple mathematical modeling courses, basic number theory, basic probability theory, a couple numerical analysis courses, and upper-level real analysis. So not a ton, but not negligible either.

There's also a lot of math in the non-math courses I will take (algorithms, theory of computing, quantum mechanics, optimization, stochastic processes, machine learning, etc., some of which are grad-level) However, this schedule is perhaps lighter on mathematical theory, and contains zero grad math classes.

Should I drop some of the courses in application areas (although courses with significant mathematical content that yet focus on applications are perhaps my favorite type of courses) and take graduate-level theory courses to increase my readiness for and chances of getting accepted to graduate school? Or will I have ample time for that in graduate school and should I take courses I enjoy more (and have greater aptitude for) while informally studying theory on my own?

Or do I need to take more theory even as an undergrad, and even though I'm not gunning for pure maths? (I do not plan to go into academia after graduate school, in case that matters.)

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I think that at this stage in your career, you should set yourself the goal of reaching some sort of mathematical maturity

... fearlessness in the face of symbols: the ability to read and understand notation, to introduce clear and useful notation when appropriate (and not otherwise!), and a general facility of expression in the terse—but crisp and exact—language that mathematicians use to communicate ideas.

Courses like abstract algebra, algebraic topology, differential geometry and e.g. combinatorics will give you a combination of breadth and depth that will make it much easier to quickly master any applied field that you will choose in grad school.

In my experience (PhD in string theory, currently applied economist), mastering the really advanced math courses should be done as soon as possible, when you have all the time and energy to immerse yourself. It's also my experience that it is rather straightforward to apply abstract patterns when you already know them, but the reverse (the emergence of abstractions from concrete applications) is much, much harder.

Of course, by all means mix and match your fundamental math courses with a small selection of interesting applied courses. There's no sense in not enjoying yourself.

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I would take the applied maths courses (the second option). If that is what you enjoy and would like to pursue in grad school then it will benefit you more than theoretical courses.

I take more theoretical courses to mean a lot of epsilons and deltas and theorems etc. If you are more interested in statistics, biology, chemistry and computer science then these courses will not be as useful as more applied mathematics courses. In applied mathematics you're more interested in constructing interesting algorithms or models that work without worrying about the theoretical details too much. That's not to suggest that applied mathematics is any easier, it's just that the focus is different.

Do you have any specific ideas what area you'd like to work/study in?

  • I'm not specifically sure, but most likely something in the set [chemistry, molecular biology/genetics, bioengineering, chemical engineering, materials science] with a theoretical/computational focus. After grad school, I probably want to go into quantitative finance unless I can get a good job in industry that is similar to what I do in graduate school. – user2258552 Mar 5 '14 at 17:49
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This is really quite a subjective question. However, even if you are planning to work in applied areas, it probably makes sense to get a good grounding in basic theory first. For example, you say you are interested in statistics. If so, you should definitely take at least one advanced probability course which uses measure theory, and also a measure theory course. These should definitely help later on, even if you don't wind up using advanced probability theory. Some material at the level of the Billingsley book "Probability and Measure" is kind of what I am thinking of.

More strictly pure courses like Algebraic Topology, Differential Geometry are a bit more debatable. They might be useful in applied areas, depending on what you are doing, but will probably not be. They might be worth it from a mental broadening perspective, but that is really subjective. I've worked in applied areas some, and have never needed to reference theory of this kind.

Also, I think if a course is well taught (which may not be the case, of course) and forces one to work on the material, then it is better than self study.

Disclaimer: I have a PhD in Statistics, which may cause some biases.

  • OK, I'll probably just save topology and geometry for enrichment in grad school, if I have time (my advisor did recommend geometry, but he said it's not a huge priority.) As for probability theory, it's not going to be my main area of focus, but it is an area I think is interesting. Do you think it would be a good idea to skip the undergraduate probability course and go straight to a measure theoretic probability course? Or should I try to take undergrad probability and measure theory separately? Or even undergrad probability and a grad stochastics class? – user2258552 Mar 5 '14 at 21:35
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    @user2258552 I think an undergrad probability course + a grad probability course using measure theory is good. If you are trying to learn basic probability + absorb measure theoretic probability notions at the same time, that would be too much for one course. You can and should (imo) take a separate measure theory course. As long as you take it before the measure-theoretic probability course, that would be fine. That would just be a math course. I wouldn't try learning measure theory in the context of a measure-theoretic probability course, because again it would be too much. – Faheem Mitha Mar 5 '14 at 21:41
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    Just to be clear, taking these courses if you are working with probability + statistics isn't some kind of intellectual luxury or cultural broadening exercise; they will inform the way you think about these topics. Far too many people work in statistics without even a basic mathematical grounding of the kind I am advocating. Also, no offense, but you really should be discussing this kind of stuff with faculty mentosr who know you (not limited to your advisor, necessarily), not with strangers on the net. Doesn't your university have these kinds of support services? – Faheem Mitha Mar 5 '14 at 21:42
  • I know it's not a intellectual luxury, but I'm not sure that I will actually be working with probability and statistics heavily, and if I do I can take the class in grad school (as they are graduate-level anyway). I'm just trying to figure out which ones are most crucial. As for your other comment, yes I do discuss with my advisor, he is very helpful, and I am going to meet with him again, but I don't want to keep emailing him, and the more people's input the better. It's not like nobody on this site is qualified, much the opposite, and I can weight different people's opinions as I see fit. – user2258552 Mar 5 '14 at 21:48
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    @user2258552 Ok, I agree that your perspective sounds reasonable. I'm not suggesting that asking here is a bad idea, obviously not, just not something to rely heavily on. The point is not that the information is sensitive, just that advice from strangers who don't know you is inherently limited in its usefulness. – Faheem Mitha Mar 5 '14 at 21:51

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