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Since my university doesn't have algebraic geometry courses or professors (*), I'm self studying algebraic geometry with the goal of finishing prerequisites before transferring. I've already started on commutative algebra and algebraic toplogy, next my goal is to read Royal Road to Algebraic Geometry. After that, I'll finally start Hartshorne.

How do I put my efforts on my transfer application for U.S. schools? I'll mention how I self-studied and what exactly I did in more detail on my statement of purpose, but is that enough? My primary goal in studying is to prepare, but my secondary goal is to convince universities that I'm putting in a ton of work on my own and thereby boost my application. I'm certain to succeed in my primary goal, but I'm unsure about the secondary goal since even if they do believe I studied, they have no reason to believe I carefully studied. I could type up exercise solutions, but they're not gonna spend time reading through all of them when there's so many students applying.

A similar question was already asked here but there are 2 differences: the topics are more advanced and I'm applying to transfer. The answers given there are to excel in the math subject GRE and get a professor to vouch for your knowledge through a letter of recommendation. The 1st difference makes the former answer irrelevant (besides, I already got 96%), and the 2nd difference + (*) makes the latter impossible unless I get someone from another university to vouch for me. What's the best way to do that, knowing full well that I would be approaching faculty just to get them to check my work?

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    What country is your desired university in? If it's the U.S., then knowledge of things such as commutative algebra and algebraic geometry (even if intending to pursue these fields) is less important than the overall sense of your mathematical promise/potential regarding both intellectual potential and academic perseverance. Maybe the answers to How does the admissions process work for Ph.D. programs in Country X? will help.
    – Dave L Renfro
    Apr 7 at 9:23
  • @DaveLRenfro Applying to all U.S. schools, thanks for the clarification
    – Anonamouse
    Apr 8 at 0:42
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    A professor who doesn't work in algebraic geometry can still write a letter about how you are studying it.
    – Nate Eldredge
    Apr 8 at 6:27

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If this is the United States, this might merit a line in your statement of purpose, but only to state that you have been studying on your own, showing self-sufficiency, in areas that will help fulfill your future research goals and/or fill in any gaps in your undergraduate education. However, I will say that this is fairly common for incoming graduate students to study on their own; it does not need to merit more than a few words.

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I might disagree with @parever about the universality of self-directed study... in people going to math grad school in the U.S.

Yes, some do... and, yes, it is highly desirable, both for the sheer information, and, for me, as a grad-admissions-committee person, to see that an applicant doesn't "just take classes", but has some self-motivated interest in the subject!!!!

Sure, it is possible to skim through things "without mastery", but, hm, given that we are finite beings, it is often optimal to skim through lots of things without mastery, rather than look at just one thing to "master" it... especially if/when that thing turns out to be just some gnarly little technical point, and of no great consequence.

Again, if anything, I'm more interested in applicants' self-motivated study rather than in their transcript. :)

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  • Why is graduate admissions so...secretive? I have read that when it comes down to narrowing down a list, it starts to come down to minutiae. For example, one sentence in a recommendation letter might not be as strongly worded as it could have been, which leads to rejection. Or the applicant might have said something that could be interpreted in a certain way, leading to rejection. When I applied to grad school one professor told me that there are so many factors that go into hiring that it could seem random.
    – cgb5436
    Apr 8 at 3:23
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    I do get where you're coming from, Paul. I think yours is a helpful addition, especially coming from a field more akin to the OP's. If the applicant writes in a letter that they have studied on their own, that's a sign that they're self-motivated and self-directed, which is helpful for pursuing a PhD. However, it's difficult to tell whether or not they studied well or carefully, or if they just read multiple textbooks without doing many assignments.
    – Parever
    Apr 8 at 14:13
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    @cgb5436, it's not that grad admissions (or hiring) are deliberately secretive, most of the time. Rather, the difficulty in quantifying the desirable features, combined with HR's and higher-adminstration's seeming demand to do so, and different interest groups competing for the same positions... often lead to very messy causal chains. People have tried to make it all "objective" by numerical rubrics, but I have not ever seen that really work, since the numbers are either already subjective, or are precise measures of things that are not of great interest...
    – paul garrett
    Apr 8 at 20:43
  • @paulgarrett Not relevant to your answer, but do you have advice on how I can write a paper that's good enough to be accepted in the Annals? I was attempting to understand your 1987 paper on triple product L-functions.
    – cgb5436
    Apr 8 at 22:34
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    @cgb5436, ah, well, that paper of mine was actually a semi-random spin-off of stuff I'd been doing since 1978, that I couldn't "get published" (no internet at the time) ... :) I really do believe that one should just follow one's own curiosity... despite the poor correlation with fundable-ness or (traditional-) publishable-ness. Otherwise, one drives oneself crazy trying to guess and comply with the hasty-random dictates of hierarchies... or trying to "join the gang" of some "programme". Just try to do what you like, and try to present it as well as possible to get paid for it. :)
    – paul garrett
    Apr 8 at 23:05

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