No reason you couldn't learn a bit about copyright law yourself. You can actually get a decent understanding of the basics by reading the law itself, title 17 of the United States Code. I'm not a lawyer, but I've read a good chunk of this stuff. I'd point you to these sections to learn the basics:
- Sections 102 and 103 describe what copyright applies to: any original work "fixed in a tangible medium" (e.g. written down, recorded, stored on a hard drive), as well as compilations and derivative works (i.e. modified versions of someone else's work), but only to the added value you created by compiling or modifying the original. Copyright does not apply to the ideas contained within a work. Also, section 105 says that official works of the US government are not covered under copyright (though there are subtleties in what is considered an official work of the government).
Section 106 describes what activities are controlled by copyright law. There are six:
- creation of derivative works
- distribution to the public
- public performances
- public display
- public broadcasting, in the case of sound recordings
When non-lawyers talk about "copyright", they are typically talking about the right to undertake these activities. To say that someone "owns (or holds) copyright in a work" means that they may reproduce, modify, distribute, perform, display, or broadcast that work, that they may authorize others to do so, and furthermore that other people are not allowed to do these things without authorization from the copyright holder - although there are some exceptions.
Probably the most relevant exception for academics is fair use, described by section 107. Fair use means that, under certain conditions, people can do the things that would normally be prohibited under section 106 even without authorization from the copyright owner. The catch is that just what those conditions are has to be decided by a court on a case-to-case basis. The law instructs judges to consider four factors when deciding:
- "purpose and character" of the use: commercial? educational? for research, criticism, parody? etc.
- nature of the work, e.g. how complex it is
- how much of the work is used
- the effect of the reuse on the market for the original
All four factors matter, so you couldn't just say "I'm doing this for educational purposes, clearly it's fair use".
- Section 201 says that the authors of a work own the copyright when the work is created, or if it's a "work for hire", whoever did the "hiring" owns the copyright. They can transfer it to others, or transfer only some of the six protected rights, or do so only under certain conditions, or so on.
- Sections 302, 303, and 304 describe how long a work is covered under copyright law, i.e. how long people are prohibited from doing the six things without permission. It's an absurdly long time: for works published after 1978, the life of the author plus 70 years, or for works published before 1978, 28 years from publication with the option for renewal for another 67 years. (In your case, I think it's likely the copyright was renewed, so the book is probably still protected.)
There's a lot more to the law but I think most of it is less relevant in your case.
Obviously if I do this for myself and don't share it with anyone, there's no problem.
There's "no problem" in the sense that nobody would ever find out. (Or if they did, let's be honest, 99% of people wouldn't care.) But technically, I think even that might not be legal. After all, I'm pretty sure what you're doing is preparing a derivative work, which is one of the protected rights, even if you never share it with anyone. It might fall under fair use but that would have to go to a court to decide - again, if anyone ever cared enough to go after you for it, which they (almost certainly) wouldn't.
But if I were to do this, it would really be for the benefit of other grad students or mathematicians.
This could easily be irrelevant. It would probably be considered in a decision on whether your project constitutes fair use of the original book, but other than that, copyright law doesn't care who the work benefits.
If I were to post it on a blog or something, it might violate some kind of copyright law
I suppose technically it's the preparation of the derivative work that violates the law, and I'm not sure if distributing the work counts as a whole separate violation on top of that, but the important factor is that putting this on a blog makes it public, which means your copyright violation is in plain view of anyone who would want to hassle you about it. This hassling would probably take the form of a DMCA notice - something like a cease-and-desist letter for suspected copyright infringement - sent to your hosting provider, who would either pass it on to you along with a demand to remove the content (if they're "cool") or would just remove it themselves without warning (if they're not). Now, I'm not going to go in detail about the DMCA here, as this post is getting long enough already, but you should know that these notices are fairly cheap and easy to send, they do carry legal weight but there is a process for challenging them, and if the issue isn't resolved by that point, then you might well wind up in court.