>>11424111Basic answer is boring and labor-intensive: walk through the theorems, reproduce them, and do the problems.
Boring but slightly less so: it's also good to formulate your own questions while going through well-trodden material. E.g. if a theorem's one-directional, ask yourself why the converse is/isn't true. Try to weaken assumptions and see why a proof may or may not be possible.
I'm not sure it matters to think about technique. Just diving into the material headfirst might produce better results than trying to find the optimal way to learn.
>>11424130Depends on the school, but I think there's generally two types: (1) the program where you don't start with an advisor and have nasty qualifier exams; and (2) the program where you start with an advisor and are expected to pass the qualifiers, if there are any to begin with.
For (1) all you can do is pump up your stats. Get publications if you can, take advanced classes, and do well on the subject GRE. The usual, boring stuff. I will particularly emphasize though the importance of having a good reference letter---the name of the referer is important, but the content of the letter should attest to your research/advanced math abilities.
For (2) you should contact a POI at a target school and discuss research. Don't be like "I want to do research with you" but instead try for "I read XYZ paper and I think one can extend the results this way".
>>11424128I settled for less than the best because I'm not the best. ¯\_(?)_/¯
>>11424204Switched in. I started undecided with a heavy humanities leaning. I don't dislike humanities but math was more fun to study.
>>11424389If I'm successful in my research? Academic. If not? Quant probably. I'm told I'll know by the time I graduate.
