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How to study advanced math books and papers? Advanced math books don't have exercises to practice. What study methods do you use? Could you explain me in detail what you do while trying to learn new advanced math(graduate level and beyond).

PS: I know in this site there are similar questions to this one but all of them are focused on undergraduate subjects and didn't satisfy me(about how to proceed in more advanced subjects).

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    $\begingroup$ For me, trying to reproduce the various proofs on my own after absorbing it is a good technique. $\endgroup$ May 14 '20 at 22:58
  • $\begingroup$ So, isn't it necessary to write "new" proofs ? Is enough to know how to reproduce the ones in the book ? $\endgroup$
    – danilocn94
    May 14 '20 at 23:06
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    $\begingroup$ Study with a purpose. WHY are you studying? Do you plan to use a result in some other research? Is it for enjoyment? Is it to be able to teach to others? Is it to find your own new research problem? Is it.... $\endgroup$ May 14 '20 at 23:07
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While it varies on a person-by-person basis, I very much resonate with David G. Stork's comment: "Study with a purpose". I had to self-teach some graduate convex optimization for a personal problem that I was solving in an infinite dimensional setting. I had already taken courses in convex optimization, analysis, linear algebra, but this new material was out of my comfort zone (which is good!). In my case, I found that while going through the material I did not need to get caught up in the nitty gritty details of every proof and every theorem leading up to the section/theorem of interest. Admittedly, there were times where I did get caught up in the details, but it was to make sure I was applying the particular result correctly, and was that my interpretation of a particular step in a derivation was in accordance with my previous knowledge from past courses. Again, this is not always the case: I'm sure there are plenty of reasons where you would want to understand every theorem, lemma, and corollary in a section but that wasn't the case for me.

Another big takeaway from my personal self-study, which is something my math professors have told me but I shrugged off until recently, is that when you read the material, it will most likely be "non linear". That is to say, depending on your purpose, it might not serve you to read a book going from chapter to chapter sequentially. In my case I found a theorem in a publication that looked applicable to my problem at hand, but I wasn't too familiar with some of the terms (to name a few: subdifferentials, Legendre-Fenchel transforms etc). As a result, I had to go back to preceding chapters, as well as other references, to gain a more holistic understanding of what these objects were, and how they were applicable to my problem. I even made my account on here so that I could talk about where I was stuck with people who understand these subjects much better than I do. Every author explained some of the concepts in slightly different ways, with a variety of motivating examples that I had not found in my primary book. For example, I saw how these concepts, and more, manifested in finite dimensions, providing justification for intuition I had in previous courses because these were the underlying objects at work, which was very rewarding. This aided in my understanding of their applications in my setting.

Again, it varies depending on the individual and their style of learning; some may find visual plotters and videos illustrating the concepts in a more visual manner more useful, whereas others might not find that necessary. Regardless, I hope this helps! :)

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