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I am a year away from completing an undergrad degree in Computer Science and I find myself increasingly interested in more advanced subjects from pure mathematics that sometimes lie outside the scope of my major, and I think I would enjoy doing graduate work in pure math. But I feel like the amount of formal education I've had the opportunity to have in my undergrad isn't quite sufficient for graduate work in math.

If I were to go and buy a book or two on some subject I'd like to learn more about that I can't take a course on (say, Abstract Algebra or Number Theory), and spend this summer getting comfortable with it, is it likely that a graduate admission committee take my efforts into account?

My situation is worded specific to a CS --> Pure Math trajectory, but it's part of a more general question. Do graduate school admissions take into account personal study, or do they only care about formal university education?

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I have limited knowledge about this and I speak from experience as a graduate student who has gone through a similar situation (from the outside) as well as being the student representative on our graduate admissions committee (on the inside). Obviously, a professor or academic with much more experience than I have can attest to this in better ways.

In general, formal education is given preference. Grades in relevant courses are given more importance than others. Note that there is a lot of competition for slots in a PhD program. Self study is a very fuzzy area and there is little scope for the graduate admissions committee to evaluate it. There exists of course, two mutually non-exclusive ways exceptions to this general norm:

  1. A respected recommender in the area attests to the fact that you did engage in significant amounts of self study and that has contributed to your overall knowledge.

  2. You do self study. Then you do research based on it. Then you publish in a non trivial journal or conference. That automatically attests to your knowledge in certain ways.

Otherwise, ask yourself this, why would graduate admissions committees believe you, especially when there is usually bound to be a few more well qualified applicants with good scores in relevant courses?

  • This says pretty much everything I wanted to say on the topic. – aeismail May 31 '13 at 21:25
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The other answers are of course reasonable, but my own viewpoint, perhaps an outlier, is that energetic/extensive self-study is a much more positive indicator about motivation and self-discipline and genuine interest than course attendance with good grades.

Yes, there is the obvious point mentioned in other answers: how to measure or certify self-study? The GRE subject test is very iffy, in a variety of aspects. Thus, the ideal circumstance is coursework and self-study, to have certifiable conventional ("passive") education as well as demonstrating initiative and interest. Further, very often the available undergrad curriculum really doesn't prepare people for grad school, so I'd strongly recommend substantial self-study in any case. Perhaps best under the aegis of math faculty who can provide some certification in letters of recommendation.

The thing that might be missing from self-study, if that's all one has as mathematics background, is the "regular drill" on basic reflexes that, in any case, routine coursework does cultivate. If one has to stop and think too much, the slowdown/cognitive-load can make routine things effectively impossible.

Also, beware, the usual first year-or-two of math grad school include "routine" coursework and exams that do presume a "standard" undergrad background, including routine drill on a fairly standard body of material. With an extremely thin or idiosyncratic background, one must play a lot of catch-up. This can have several bad effects: may give the impression of incapability, may make you tired and discouraged, may stifle natural interest. So "brace yourself" if that's the path you find yourself on.

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If possible, I would recommend first seeking out a math professor at your university, preferably someone you know and who holds you in high regard. If you are home for the summer, e-mail would suffice, but an in person meeting would work much better.

Anyway, explain your plans, and:

  • Ask the professor's advice about your self-study plan (maybe the prof has different suggestions)

  • Ask if he or she would be willing to evaluate your progress at the end of the summer (e.g., you show up to his/her office and take an informal oral exam), and to write you a rec letter if you've done a good job.

Good luck!

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    I strongly agree with this comment. My husband is a math professor, and if you approached him in this manner ("I just love math, would you please go through this number theory book with me?") he would leap at the opportunity to do independent study with you. People like you are why professors go into academia. :) – guestwife Jun 2 '13 at 0:54
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Another option for the special case you describe is to take the math subject GRE after your self studies. This would validate your knowledge of the undergraduate curriculum in pure maths. They might take into account that you did not take any of the courses (you probably won't get the same score you would have gotten as a math major, because you don't know real analysis, topology, etc.)

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