I am currently making my CV for undergraduate research internships in Pure Mathematics. I have read many graduate level books on the field out of my own interest. Whenever I was stuck on a difficult concept, I have sought the help of my professors and they can vouch for my understanding of these topics. Adding a section titled "Books Mastered" seems a bit narcissistic but I believe mentioning these books would add weight to my application.How can I leverage this experience and incorporate it into my CV?
How can I leverage this experience and incorporate it into my CV?
I'm not convinced there's any good way to incorporate this experience into your CV. The basic problem is that it's unclear what it means to have studied a book: there's a huge range of levels of mastery that could all legitimately fall into this category. Furthermore, many undergraduates have trouble accurately assessing their own knowledge and accomplishments, so a self-assessment of mastery is not convincing even if it's completely honest. And of course honesty and bias are important issues.
The net effect is that these claims are vague, subjective, and unverifiable, which is exactly the opposite of what belongs in your CV.
You could always try to say something more precise. For example, "I wrote out solutions to every problem in Hartshorne's algebraic geometry textbook" is much more compelling than "I studied Hartshorne" or "I mastered Hartshorne". It's still not ideal; for example, it leaves open the question of whether your solutions were actually correct. However, it at least gives a much clearer idea of how much you might have learned, and it allows the possibility of verification by requesting to see the solutions. (Don't say this if you lost them!)
I wouldn't recommend trying to make precise claims like this for all your independent reading, since it would still come across as eccentric. However, that would at least be a better option than just listing some books.
This question brought to my mind two anecdotes:
The back cover of Donald Knuth's epic book The Art of Computer Programming famously carries an endorsement quote from Bill Gates saying "If you think you're a really good programmer... read (Knuth's) Art of Computer Programming... You should definitely send me a résumé if you can read the whole thing."
True story: some years ago I was invited to join some colleagues at a conference at lunch, and found myself sitting next to a famous scientist whose classic textbook is a discipline-shaping, notoriously difficult to read work, which has greatly informed and inspired my own research. After some introductions and random smalltalk I turned to him and said something like "Professor, I thought you should know I really love your book. It has really influenced my research and my thinking." He smiled, clearly amused, and after a short pause replied "Hmm. So what's in chapter 9?" Of course I was rather taken by surprise, but I did know the book very well and after a few seconds managed to get my wits together and said "oh, chapter 9 is about [X]." He was quite pleased and impressed. (I think he did not expect me to know the answer to his question; the joke was that chapter 9 is the most difficult chapter in the book, understood at a high level perhaps by 5-10 people on earth, and certainly not by me. But I did at least know what it was about...)
Going back to the OP's question, I would say that listing books you have mastered in a CV is certainly very unconventional (I have never seen it done), but as my anecdotes illustrate, this information can certainly be valuable to a prospective employer/mentor, so I see no reason why it should be taboo to mention it, subject to the following caveats:
a. Only include the list if you truly believe your private studying of hard topics in mathematics is so extensive compared to that of your peers that it would make you a more attractive internship candidate. Everybody reads books, so your list needs to be able to convincingly demonstrate that you have a read a lot more and understood a lot more than most other internship candidates.
b. Be precise about the level of mastery you have achieved of the books in question. For example, you can include your list under the heading "Books mastered" and add a footnote saying "I have invested at least 40 hours of study in each of the books on the list, and in each one I have solved a large proportion of exercises. My solutions are recorded in a notebook that I keep for future reference."
c. Most importantly, be completely honest. Do not inflate or exaggerate your claims (and if your list doesn't seem impressive enough without such exaggerations, obviously that means you should not add it to your CV). If I were calling you up for an interview after being impressed by your list -- which I can quite possibly see myself doing -- I would make very sure to ask you (as the famous scientist did to me) some questions about tricky topics in a random sample of the books to test your understanding. Needless to say, if I found out you were exaggerating the interview would be over immediately.
If I read your question right, the answer is: there's no reason to put this in your CV. If your professors truly have a good assessment of what you've done, then this will be in their letters of recommendation (which you will almost certainly need), where they can give it the appropriate weight and you don't need to worry about doing something non-standard in your CV. (If you actually took reading courses, even informally, then this is natural to put in your CV.) As mentioned in a comment, it would be appropriate to bring up some of your readings in a cover letter and/or personal statement.
While the other answers, and the comments, make good points, I think it is quite possible to make reasonable (esp. modest) remarks about books (or PDFS of lecture nots) one "has looked at", especially if one's curiosity is strong! Sure, do not say "mastered" (even if you believe it, or even if it's true), because you'll simply discredit yourself, so readers won't believe much else in your file.
Since, apparently, self-motivated, undocumentable reading-of-books is not so common in mathematics, there seems to be no accepted tradition of how to report it or even mention it. While it is certainly true that good advice from experienced mathematics faculty can be helpful both in choice of sources and in how to benefit from them, there's simply not enough time in the day for a very-inquisitive person to ask about every thing they see, probably.
There is also a confusing pervasive mythology about "how" one should read a source: supposedly ultra-carefully, line-by-line, doing all the exercises, and so on. I do not buy into this, for various reasons. In any case, this at least indirectly would seem to completely invalidate any other notion of "looking at things", which I think is a very bad thing. That is, I think "casual" (or whatever word we can agree on) reading/skimming/perusal of mathematics books can be very helpful, productive, informative, etc. E.g., getting an idea where all the little details are going, and maybe why we'd care.
E.g., the notion that one would ONLY look at the textbooks for the few math courses one enrolls for each year strikes me as bizarre, comparable to the notion that one would play sports only when signed up for "gym class", and then only for "assignments". Or only listen to music for "music appreciation" class.
In particular, as anyone reading this answer can surmise, when I read an application file that includes self-driven reading outside of the official coursework game, it is a big positive. True, self-appraisal is a difficult thing... but/and that difficulty should not be a reason to disregard the thing that is difficult to assess! After all, the convenience of "grades" is substantially self-referential, and often reflects as much obedience as understanding.
So, yes, you might list some books you've "looked at", in a form somewhat parallel to listing advanced coursework (which should include textbook authors' names, also!). Be modest. And in mathematics, the authors' names are more explanatory than titles, so it's important to include them.