What fields of math progress the most continuously ?

[carpe deum]

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

 

[carpe deum]

my first post, and already I've stumped you guys? i guess people around here aren't as bright as I've been led to believe.

*folds arms*

 

[Dembadon]

my first post, and already I've stumped you guys? i guess people around here aren't as bright as I've been led to believe.

*folds arms*

Neither of those conclusions follow.

Regarding your OP: In my opinion, decisions to pursue research in a given field should be motivated by a substantial interest in it, not by whether you'll progress "continuously."

 

[diligence]

I'm not an expert by any means since I'm just a senior undergrad taking grad courses and doing research, nor do I have much experience outside of PDE. But it certainly seems that PDE fits the mold for the type of problems that you describe. There are seemingly endless projects to work on, that while they may not be groundbreaking or make significant advances in the field, they are still on some level "important". For instance, various energy estimates in many situations have yet to be obtained. These may not seem like very important problems, but they're certainly worth the time for someone to complete, if only for the sake of thoroughness and rigor.

Then there's the big daddy of them all: the Navier-Stokes existence and uniqueness problem. It seems to me like this is either going to take a massive amount of brute force (not very groundbreaking) or it may require some new techniques in dynamical systems and chaos that have yet to be developed.

In my opinion, fields that are most likely to have the continuous progression you describe are probably closely related to physical problems.

 

[Nano-Passion]

my first post, and already I've stumped you guys? i guess people around here aren't as bright as I've been led to believe.

*folds arms*

I'm not sure if this will get you more replies.

 

[homeomorphic]

That's a difficult question because not that many people actually do research in more than one field these days. Personally, as a PhD student just finishing up this year, I've only done research on one very narrow topic (2-d TQFTs (topological quantum field theory) and related things).

Whatever your problem is, you can usually turn it into a smaller problem and solve that. Strengthen the hypothesis or switch to a slightly different context. My thesis problem was too hard, so I switched to the corresponding problem in a different setting, and I now have a pretty clear plan to solve it.

 

[carpe deum]

I'm not an expert by any means since I'm just a senior undergrad taking grad courses and doing research, nor do I have much experience outside of PDE. But it certainly seems that PDE fits the mold for the type of problems that you describe. There are seemingly endless projects to work on, that while they may not be groundbreaking or make significant advances in the field, they are still on some level "important". For instance, various energy estimates in many situations have yet to be obtained. These may not seem like very important problems, but they're certainly worth the time for someone to complete, if only for the sake of thoroughness and rigor.

Then there's the big daddy of them all: the Navier-Stokes existence and uniqueness problem. It seems to me like this is either going to take a massive amount of brute force (not very groundbreaking) or it may require some new techniques in dynamical systems and chaos that have yet to be developed.

In my opinion, fields that are most likely to have the continuous progression you describe are probably closely related to physical problems.

That's a difficult question because not that many people actually do research in more than one field these days. Personally, as a PhD student just finishing up this year, I've only done research on one very narrow topic (2-d TQFTs (topological quantum field theory) and related things).

Whatever your problem is, you can usually turn it into a smaller problem and solve that. Strengthen the hypothesis or switch to a slightly different context. My thesis problem was too hard, so I switched to the corresponding problem in a different setting, and I now have a pretty clear plan to solve it.

thanks for the replies.

i think i need to reformulate my question. OK, so there's usually a way to find a smaller or related problem that is far more tractable. what worries me is that if you switch to a problem that is too small, you can easily stray into territory where no one will ever read your paper, if it even gets published, and rightly so. it may not be worth their time to go searching for every paper where someone's solved a cauchy problem, for some peculiar IC's. in that sense, your work may not even end up contributing to the field, not even as part of incremental progress on a big problem.

so i guess what I'm looking for is which fields have a large "sweet spot" between problems that are too hard, and problems that are too small to be worth writing about.

and with the issue of switching to related problems, i worry about spending too much time shopping around for the right problem, and not enough actually solving anything.

 

[carpe deum]

I'm not sure if this will get you more replies.

, he said, replying to me

and if you argue with this, you'll be replying to me again. you can't win.

muhuhahaha

 

[homeomorphic]

i think i need to reformulate my question. OK, so there's usually a way to find a smaller or related problem that is far more tractable. what worries me is that if you switch to a problem that is too small, you can easily stray into territory where no one will ever read your paper, if it even gets published, and rightly so. it may not be worth their time to go searching for every paper where someone's solved a cauchy problem, for some peculiar IC's. in that sense, your work may not even end up contributing to the field, not even as part of incremental progress on a big problem.

I have a quirk in my thesis (a non-standard definition), which might make it irrelevant to the mathematical community. I actually see that as a strength because that means I am confident that no one will scoop me, though I could be wrong. Once you realize the extreme danger of being scooped, you might realize that, if you are just a grad student, that is a very considerable advantage, although it does come at some cost. Mere mortals shouldn't expect their PhD theses to be something spectacular. In my judgment, since the thesis is so painful--like this black cloud over your head that you are FORCED to do, it's better to just get it out of the way, so that you can move on to better things and not worry so much about making it the best thesis ever. Most of us lose a lot of our ambition once we realize the difficulty and the risks. Of course, this isn't good advice for someone who is doing really well with research, but it's pretty much okay to not do that well at it when you are just a lowly grad student.

Furthermore, a lot of what the mathematical community deems important is just driven by fashion, rather than its intrinsic importance. That can have important consequences, career-wise, but morally speaking, these fashions should be irrelevant. And this isn't me, the classic disillusioned grad student speaking. A math professor I talk to here said it's "driven to a surprising extent by fashion".

But who knows what's important, anyway? It's really hard to tell until after the fact.

so i guess what I'm looking for is which fields have a large "sweet spot" between problems that are too hard, and problems that are too small to be worth writing about.

My guess is that all kinds of fields have lots of problems of all different types. The variation between fields has more to do with how much you have to know to understand the problems. Once you learn all the stuff you need to know, I'm sure you can find all the problems you need, usually. There might be some where a lot of the problems have been solved, so there isn't that much selection left, but, again, this is a difficult question because most people only work on one fairly narrow area, and you have to be an expert before you have a good idea of what all the problems in the field are like.

and with the issue of switching to related problems, i worry about spending too much time shopping around for the right problem, and not enough actually solving anything.

That wasn't too much of an issue for me, in the sense that once I switched problems once, I was okay, although it did cost me like 5 pages of work that got tossed, and all the time it took to figure out it was too hard, which was not insignificant, I suppose. Actually, I did get somewhat screwed over by it, since I thought I was ready to graduate, but now I am spending another year here and still not even ready to finish in time to apply for postdocs this fall.

Actually, I am happy that it didn't work out because I don't want to be an academic mathematician, anyway. If I wouldn't have had so much trouble, maybe I wouldn't have questioned the value of that in the first place. I'm going to do something a little more useful to society now. I am sure I will not abandon all that stuff altogether, though, since I learned so much of it. I plan to use my topology and general math knowledge for various purposes in my spare time, if I can manage it.