How to make sure graduate admissions committees are able to recognize/verify the material a student has self studied

Question

I posted a similar question at the following link yesterday; since, however, the focus of the question was different, I'm posting this question in a separate post.

It seems as if I'll be taking the next year off and studying a lot of mathematics on my own and apply to graduate schools in the next cycle. There's a lot of advice both on this website and Academia StackExchange for such students. However, I'd like to know how can students who have studied material outside of class better convince the admissions committee of the work they have done thus far. I know it's generally difficult to do this as an international student; the situation must be different for U.S citizens, I suppose.

Assuming I don't get to take courses which I'll be self studying over the next few months, how should I go about making sure the admissions committee take due notice of the work I have done, and more importantly, they can recognize it, if not verify it completely. Of course, I will keep in touch with my math adviser. In addition, what else can I do? Say, I am studying abstract algebra on my own in the summer, working out problems in, say, Dummit and Foote's book. Should I, perhaps, make a website (a Google site), periodically type the solutions to the work I have done and post it online, so when I apply to graduate schools, I can refer admissions committees to this portfolio of sorts.

In a nutshell, I'd like to know, especially for students who have gone through such a process/situation, of the list of best possible set of actions one can do to make sure the work one has done outside of class is duly considered, if someone who is interested in mathematics schools doesn't have a large number of math courses.

Edit:

I have an additional query: I'll be taking next year off, but I'll be able to visit my college. Even though I won't be enrolled in classes, preferably with the instructor(s) who already know me, sit in their classes, work on problem, and perhaps even take exams. even though I won't get a grade, I could ask the instructor to mention in the letter that I completed this task. She could also comment on how well I did relative to my peers. Thoughts?

Answer 1

Letters of recommendation which attest to this, and also perhaps the math GRE. You can't believe how much garbage I see in personal statements about how much students have "mastered" a particular mathematical topic studying on their own, only to find out later they had "forgotten" what they had learned months ago after I meet them in the autumn and they're banging on my door asking me to take them on as their potential advisor. So this is why I trust letters of recommendation over the word of the student by and large these days. Please get one of your letter writers to attest to your self-study. Generally, the more senior and influential the letter writer who says this, the more likely I will believe you've self-studied what you say you've self-studied.

Answer 2

I got into a computer science PhD program based almost entirely on work done outside of class. My academic CS education was limited to a master's degree awarded over 25 years earlier. My bachelor's degree was in mathematics, not CS. In 2002, I don't think anyone really cared that I had known how compilers were built in the early 1970's.

At the time, the GRE had a CS subject test. The combination of a general effort to keep up with what was happening in computer science and some independent study before taking it got me into the 94th percentile. I got letters of recommendation from colleagues who knew I was doing research-type work on the job, and could point to patent applications.

Translating my experience to your different situation, I agree with the comment recommending taking the GRE subject test in mathematics, even if it is not required. Apply at least some of your study time to the topics that will be tested in it, and to taking practice tests.

Ideally, find some courses you can take with controlled, graded tests. Think of the grading as a service you are paying for with your tuition, even if you don't need lectures.

Keep in touch with the professors you are hoping to use as references. See if one or more of them will let you help with their research, or discuss your independent study. For this purpose, the research does not have to be what you intend to pursue in the future, just something that will let you demonstrate your skills.

Your objective is to have the following:

• GRE subject test
• grades on courses you have taken
• letters of recommendation

accurately reflect your level of suitability and preparation for graduate studies. Those are the things you can be sure an admissions committee will consider.

Answer 3

Wait, what? I feel like I stepped out of the theater for a moment and missed an important plot development. There was a sudden leap from wondering whether To Take More Courses Next Year Or To Take Next Year Off, "that is the question," to "I'll be taking next year off," and I don't know how you got from there to here.

But if I assume you will not be enrolled in any courses next year, and that you have some rational reason for arriving at that plan, then:

The answer to the question you asked here is:

If your university and the instructors in question allow you to audit (either formally or informally), then you could augment your transcript with letters that say what grade you would have gotten in each audited course if you had been enrolled.

But let's back up a step.

"I chose to take more physics courses than mathematics."

Sorry I didn't catch that in your earlier questions (my fault, probably). Now that I've understood that, and remembering that in your location you will only be able to take one subject test, and that you were unsure whether to target your applications to study mathematical physics towards a math department or a physics department, and, finally, keeping in mind that as an international applicant, it would be wise to maximize your chances of admission and offer of support (typically, a teaching assistantship), I suggest the following decision making procedure:

First, take a practice test for the physics GRE and one for the math GRE. If your result is significantly better in physics, then your best bet would be to take the physics GRE, and apply primarily to physics departments. However, you could hedge your bets by choosing universities that have faculty members who do mathematical physics in both their math and their physics departments. Note that it is not impossible to change departments later on. But it would be a shame not to get the admission and the support to be able to get started in a U.S. university, as is apparently your aspiration.

I understand that right now you are feeling pulled to study math, partly because that's what fascinates you at present, and partly in order to remedy some perceived deficiencies. However, some involvement in a research project with rigorous mentorship would strengthen your application.

Comment: It sounds like you might be underestimating the importance of the GRE subject test in your application.