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I am a mathematics student and have been extensively reading textbooks cover-to-cover in my own time. Thus, my transcript does not say that I have (for example) taken differential geometry, even though I know diffgeo as well, if not better, than if I took the course.

How would you suggest that I display such efforts so that grad schools can be aware that I am more knowledgeable than my transcript might suggest?

Thanks!

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I faced this exact problem when I was applying to grad schools. Here are three ways that worked for me (in order of importance):

  1. Include this information in your statement of purpose. I do not recommend going through a laundry list of all the books you have read, however, I would mention the new topics that you are familiar with, how you have been learning them (independent study), and how you are looking to continue furthering your knowledge (which is presumably by study at the university you’re applying to).

  2. Have your letter writers vouch for your independent study. Most universities require letters of recommendation as part of your application. It is a good idea to choose letter writers who can speak to your different strengths, and I think it would be beneficial to have at least one of them who is familiar with all of your self-study. I met with my professors who agreed to write letters for me so that we could talk about my application and preparation for grad school. One professor asked me specifically to prepare a list of topics I’d studied independently so that he could write about it in his letter to the university.

  3. Contact professors at the schools where you want to apply. Getting professors familiar with your background can be helpful so that they can point you to even more resources for self-study, and they may also be helpful when going through the application review process. There is no guarantee this will influence your application, however, it possibly has some bearing on it, and it more importantly will help you become better at self-study regardless.

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You could discuss your independent learning in your personal statements.

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Another option is to take Credit By Exam or Competency-Based Education (CBE) credits. Credit by exam is pretty simple - you sit the final exam (or an exam equivalent to a final exam) and if you pass, you get the credits as if you had attended the full course. Competency-based education credit is a little more complex. You don't necessarily get to sit the final exam right away, but you can bypass the weekly homework and lectures and knock out unit tests and projects at your own pace.

The effect of this would be to translate your informal knowledge into actual credits. You would then be able to say that yes, you do have university credits in Differential Geometry and whatever else you have been learning on your own.

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As an undergrad, I used to make lists of errata for texts I have read and post them on my website (I still keep updating the list when I read new stuff with a nontrivial amount of errors). Back then I wasn't thinking of it strategically (it just felt like the obvious thing to do, like reporting bugs in software), but now that I've had some bits of experience with assessing students' applications, I have come to the conclusion that such lists of errata are fairly good evidence of actually having read and understood the books/papers you claim to have read. (I have seen my share of high-schoolers claiming to have read graduate textbooks, but somehow failing to remember much of use from them afterwards. It appears to be an inflationary currency, where everyone keeps claiming more and more achievements, just as with most other extracurriculars in student applications.)

Another thing that can help is posting collections of solutions to textbook exercises. But this is somewhat more hit-and-miss; the copyright status of such collections is unclear (they are derivative works most of the time), and some professors (older ones, in particular) might find it unwelcome (as they want solutions to not be easily available to students). A variant of this that lacks these drawbacks is posting little notes about the text, such as "a different proof of Proposition 2.4.1" or "why the second assumption in Theorem 2.5.2 is needed".

These days, you can also ask questions about the textbooks on forums like math.stackexchange, although of course these better be good questions.

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Formal Qualifications mean formal learning with various formal assessment methods.
For Example: BE electrical at University of Canterbury NZ.

Informal learning comes under a different sub-heading on your CV. This informal learning comes under two subcategories: Subject Related and Other.

An example of Subject related would be "CBFC101 a computer course with no Exam". An example of other that is clearly not subject related is "lifting paraplegics in and out of wheelchairs". You can break down the qualifications into Academic and Non-Academic if you want because you may have a useful non-Academic qualifications that you want to mention.

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