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**what fields of math progress the most "continuously"?**

i have a physics BS, and have left acamedia to work. but i have an interest in going back some day and doing research, probably in math. i do some amateur math tinkering on my own, but i try hard to keep it valid and to avoid straying into pseudomath territory. i think i have done a good job in that last respect, because none of my "great ideas" have turned out to both new or better than standard methods.

anyway, my impression of mathematical research is that it's hard to pick the right problems to do. it's difficult to find something that is reasonably tractable, not already solved, and possibly of non-negligible importance. if you pick the wrong problem, you can easily get stuck in an all-or-nothing situation where you spend a lot of time on it, and if you don't solve it, you have nothing to show for your effort. i'm wondering what field(s) of math are the

*least*like that. which field(s) admit smaller steps of progress, such that if one problem isn't working out for you, you can find a related issue to chip away at? or if it's your preference, you can

*focus*on small problems, which are of manageable difficulty, but are not so small as to be worthless.

my interests are in PDE's and numerical methods. originally it was just PDE's, but i'm becoming more and more interested in numerical methods as a way of chipping away at them.

any advice is appreciated