How to best utilize a gap year to enter mathematics graduate school

Question

I have the following two options available on how to utilize the next year. First some background:

I'm an Economics and Mathematics major at a college in Pakistan. I'm, however, interested in going to graduate school in mathematics and working with mathematical physicists. Starting from the end of my second year, I started taking physics course. I'll have a dozen or so physics courses by the time I graduate; I have the basic, and a few upper division math courses. I continue, however, to study mathematics outside of class.

First Option:

I can choose to take an additional year before going to graduate school. I doubt I'll be able to fulfill the requirements of a major in mathematics (and my school's policies are a bit ambiguous as well); however, at the very least, I could take as much mathematics courses as I can. If I were to do so, I'll probably be able to complete the department's requirements of math courses, so one could assume that I have completed all the requirements necessary to fulfill the department's choice. I may be offered a RA position in the physics department; so I could fund my additional year with that money, along with all my savings.

Second Option

Just take the year off. Continue to study mathematics informally with the instructors. That is, cover material outside of class/college, follow up with them regualrly so they can write good letters. Depending on whether I get to work on some other problem, I could still take the RA position and save money to apply for graduate schools etc.

Here are some more details that'll help you advise.

Mathematics courses I have taken:

I have taken single variable calculus, multivariable calculus, linear algebra, real analysis (two semesters), probability and point-set topology.

I have also been studying material outside of class. I finished linear algebra II on my own, and I am working on multivariable analysis right now. Regardless of which option I choose, I will continue studying more mathematics, revising old mathematics etc.

Here's my assessment of the options thus far.

RA Position:

1. Frankly, I don't think I'll want to work in this position for an extended period of time. I wrote a physics paper with a professor on (open) quantum dynamics; since he wishes to work on similar problems, he offered me a position so I can continue working on similar problems.

2. I don't think it'll be (extremely) helpful in my eventual application because the path I wish to take is somewhat different.

3. I, however, can get paid and use this money for either option.

First Option

1. I get to cover my cores -- ODEs, Complex Variables etc.

2. I'm not too sure how much it'll help, but I'll have graded coursework.

3. I'll still, however, be covering some of the same material outside of class on my own.

4. It'll be hectic; I'll have to complete 2 semesters worth 35-40 credit hours. I will also have the responsibilities of the RA position; on top of that, add applying to graduate schools in the mix.

Second Option

1. Flexible schedule designed at my own will and pace.

2. If both the above options would have the same effect, then I can just study material on my own. For instance, assuming I take the RA position, I'll be on campus the next year. I can audit courses which my advisers will be teaching next semester -- for instance, on of my advisers will be teaching a graduate level course in algebra. I can sit through classes, work on problem sets, and hopefully then the instructor can compare my work when she writes the letter; of course, I won't be graded relative to my peers, though.

3. Another local university provides short, intensive courses in mathematics. It's a graduate school, so most courses are advances courses -- for example, in advanced algebra, differential geometry. Since I'll have more time on me, I can perhaps take 1 or 2 of these courses, and cover the more advanced material. You can see the this link for a description of such short courses that have been offered in the past.

4. Also, with the first option, I'd have to go through the boring, plug and chug courses -- ODEs etc. Very important to train yourself, though.

It'd be great if someone could weigh out both these options, especially considering the graduate admissions process at math departments.

Answer 1

What's the best way to make sure admissions committees can verify what I have done?

Your transcript and your GRE score.

No one ever became a good mathematician without doing plenty of self-study. But it is hard to get admitted in the U.S. as an international student without a strong transcript. So, if the question is, take courses or study on my own, the answer is both.

It sounds like you find the idea of sitting in a cave learning math on your own quite appealing (which is understandable). Well, this is your decision and I doubt I'll have much influence, but I will ask you to imagine yourself in sitting on the admissions committee looking at Junaid's application. From the committee's point of view, Junaid who has spent the last four to six months doing self-study might not be as good a bet as Junaid who has been continuing to work his way through the available math courses in his local university.

Are there some periods, perhaps a summer break, perhaps a couple of weeks here and there, when you can do some cave visits, without setting yourself up there for the entire duration?

Why are you so down on ODEs? In a classic ODEs course, you lay groundwork (e.g. Laplace transforms) for some great stuff to come (e.g. Fourier transforms -- so useful in physics). Are you basing your negative remarks on things you've heard from students who already took ODEs in your university, or what?

Apparently you haven't run out of courses to take yet in your local university. This is good news.

I can't possibly weigh in on the research assistantship and the financial question, because I have no idea what the alternatives are, or what your general financial situation is.

Having followed your questions here for a while, I am glad to see from your comment that you have taken a close look at at least one math department's admission application guidance. Good work! Are you planning to take a first look at a math GRE practice exam some time soon? Are you looking for a mentor who can help you structure your GRE preparation? Have you started to look at some faculty bios in a couple of departments, to identify some departments that have several people doing the kind of work that appeals to you?

You give the impression that you have done good work so far, you are a hard worker, you are motivated, you enjoy studying math, you have found good study techniques, and you have a vision (a dream). Now add a soupçon of patience with yourself, and you'll be all set.

Answer 2

I like option 1 (do an extra year of school) better. It gives you some organized support for your study (most people lack the discipline to really do that for themselves effectively). Also it proves the work objectively to an applications committee.

Yes, the RA will cause a conflict but at least it keeps you in touch with physics. You also need to talk to the person providing the funding and get some clarity on how many hours he expects or conversely how many hours you will need to study math. Just man up and talk to him about that. This actually applies in both options, but the self study option has the disadvantage of the professor tacitly assuming you are dedicated to his RA and not honoring your self study time the way he would course work.

It would have been helpful if you also listed what coursework you did in physics. I'm baffled how you can do several physics courses without ODE for instance.

I'm also confused with how/why you took all the physics courses and didn't get your math (or econ!) degrees done. Nobody is perfect and the situation is the situation but I just wonder if there is some learning here. At least you appear to be more strategic now.

It would also be helpful to know status of your econ major.

And then is some sort of BS going to happen at the end of this year or is it just a case of you played with 3 subjects and never got enough done to get a degree in anything? If so, this is another strong rationale for doing the extra year of coursework (however funded) and just getting the darned sheepskin.

I agree that your math background is insufficient for math grad school. Probably also for physics grad school. A semester of ODE is expected and usually an extra 1-2 semesters of "engineering mathematics" (a brief coverage of linear algebra, PDEs, and complex analysis, with emphasis on key calculation methods and problem types, not proofs).

If anything maybe your background would be a good fit for econ grad school in the states. Econ tends to be more mathematical in grad school than undergrad (which frustrates many undergrads who love micro but hate derivatives). You would have better than average math background in econ but much worse than normal in math grad school and somewhat worse than norm in theoretical physics.

I also suspect that your math ability (intrinsic ability to learn math) somewhat correlates to your education level. If you head to econ, you will have a comparative advantage. If you head to math or physics, the opposite. Also, econ grad school has traditionally been somewhat receptive to people moving cross field (e.g. from engineering undergrad to econ grad school).