Strategies for Effective Self-Study

manpreet Best Answer 2 years ago

I have a long-term goal of acquiring graduate-level knowledge in Analysis, Algebra and Geometry/Topology. Once that is achieved, I am interested in applying this knowledge to both pure and applied mathematics. In particular, I am interested in various aspects of smooth manifolds, co/homology and mathematical physics. I have acquired a smattering of knowledge in all of these areas but feel that I need to become more focused to make make coherent progress. I have a very bad habit of picking up a book, reading a bit, working out a few details, and then moving on to other random topics in other random books. In doing this, I don't really feel like I accomplish much.

To rectify this admittedly undisciplined approach, I have decided to select core source material from each of the three major areas listed above and focus on it until I have assimilated all the information in that material. For analysis, I have selected Amann and Eschers' Analysis, volumes I, II, and III. I made this choice because out of the analysis texts I have surveyed, theirs seems to be the most comprehensive and treats elementary and advanced analysis as a unified discipline.

My basic strategy is to treat each theorem, example, etc. as a problem and give a fair amount of effort to proving before consulting the text. I think this is probably the best way to approach the material for maximum understanding but it requires a considerable amount of time. There are probably thousands of these sorts of "problems" among the three volumes. Ulitimately, I would like to end up with a notebook (which would probably number in the thousands of pages) that contains all of the details to all of the theorems completely worked out, as much as possible, with my own thoughts. Again, this seems like it will take forever and my time on this earth is unfortunately finite. I'm reasonably confident though that the production of such a set of notes would lead to at least a fair level of mastery of the material in question.

Can anyone suggest an alternate strategy that might be more effective in terms of time but that would lead to a comparable level of mastery?

It is also a problem that I might actually prove a fact completely on my own but then, a month later, might not be able to recall it in a time of need. What strategies are helpful for best ingraining this material (other than the obvious "Work lots of problems" approach)?

Would appreciate any tips or pointers.