I'm beginning to run into work where I have to do a significant amount of learning of math by myself, with a book rather than with a teacher. Now, I do know that doing problems tends to be the best way to learn these things, but my question is a bit different.


How do you pace yourselves when you're learning new mathematics? Or reading a paper, let's say. Any pointers for reading large swathes of mathematics? (I'm running into this problem with my category theory references).


One answer per post.



Pacing for learning new material
Here are my advise that are mostly based on experience:


If I start a totally new math, especially in the graduate level. I'd give my self at least 1.5-2 years (especially if it's an area in which a lot of lot of reading is involved.. say algebraic geometry). One of the things I find important, is not getting frustrated that you haven't learned to the level you need to learn. It is indeed frustrating, but when I look back in my PhD years.. I indeed took 2 years of just reading before even being able to start any new ideas of my own. You just don't have enough knowledge to make a ground-breaking mathematics and until you do be patient and learn it and try new ideas and create new examples (out of the book). I personally hardly answer excercises in the book, but created my own questions and tried to answer them first. If you were able to make new ideas and even publish a paper or two during these 1.5-2years then thats a bonus but you shouldn't feel incapable during that time.


The other advise I'd give is the references. Never stick to one or even just two single reference. Especially if they are the references that are difficult to digest. You should collect as many of the references in that area of mathematics as possible. If this is math that people have done already, chances are there are many many references about it that you don't probably know yet. And never read linearly through the references (esp. textbooks). I don't know of any professional mathematicians that has actually finished reading an entire book that he has not written himself. Switch from the different references as much as possible and try to get as much goodies from each as possible. There is no ONE book in homological algebra and different mathematician find different book suitable, you should find one that writes in a style your prefer and every now and then look at the other books as well. There is NO one book in commutative algebra, you can read certain characterization of testing for flatness of modules/algebras in commutative algebra books but you can hardly find ALL of them in ONE single book and some of them don't even have all of the proof.


Third advise, is to collaborate or speak as often as possible with people very knowledgeable in the topic. Attend seminar and conferences in that area of mathematics, even if you don't understand a pea. Chances are you learn something new or you learn about a question you think you find interesting in that area that is unanswered. There are some people who are knowledgeable in some area and make me feel like sh\*t when I speak to them, I tend to avoid them.. but sometimes I mingle nevertheless. For me, true authentic mathematicians must good educators as well, so that if they find someone not knowledgeable in one thing they actually help him become knowledgeable instead of making him feel bad about it.


