> How to self study pure math ... Did a lot of that, before, during, and/or after formal courses, and the results were good. In one case, in a subject I'd car

> How to self study pure math ... Did a lot of that, before, during, and/or after formal courses, and the results were good. In one case, in a subject I'd carefully studied on my own, I was required to take a course given by the department Chairman. The course was mostly filtering, not education, and as I did really well totally torqued the Chairman, angry that he was not able to filter me. Eventually I got my degree, and he got fired -- his attacks on me were not the only reason he got fired. Here is something of a standard sequence: (1) first year algebra (2) plane geometry (3) second year algebra (4) trigonometry (5) solid geometry (6) first, second year calculus (7) abstract algebra (8) linear algebra (9) analysis (10) multi-variate advanced calculus (11) differential equations (12) measure theory, Banach space, Hilbert space, Fourier theory (13) Probability, stochastic processes (14) statistics Keys: Get the best books, 1-3 books per subject. This point is crucial since actually most of the books are not very good and the best books are MUCH better than the rest. Then read the definitions, theorems, and proofs, and work some or all of the exercises. Doing this, obtain intuitive explanations and the reasons the material is important. While it is good for each proof to see why it works, commonly people are flexible on just how hard they study the proofs. Maybe as a check on the quality of your progress, sit in on a few sessions of a few courses at a good math department. This sequence (1) -- (14) is a good start for more specialized topics and research and/or applications. One good research career direction is to have such a sequence and then leave math for research in some other field where can apply the math. E.g., I had a fellow student who did that and ended up with a good career applying his math in biology and ecology in an Ivy League university. For a research career, need to get a sense of what constitutes good research: The material in (1) -- (14) is the finely filtered and polished result of 200 years or so of math and too good for most research now. In nearly any applied field, take one of their problems, look a little more closely than usual, and will see where want some new tools and results.