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What is an efficient strategy to find fruitful research problems. So far the best advice I have heard about choosing a problem is to "talk to as many people as possible and go to as many talks as possible and when you find yourself engaged in something that interests you, then work on that." I forget who said this but its on the internet somewhere.

But I don't have many people to talk to or many talks to go to so how can I efficiently find problems. Any advice?

Joe
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  • If you're not hanging around universities with departments doing research in the discipline you think you might want to work in, you can also start from many "popular" books on open and active problems; they aren't generally very technical, but they can give you an idea of what the "landscape" is like. (And there seems to be something of a small cottage-industry in writing them.) Of course, there's also the internet for "nosing around" in the field. There is no "efficient" system for doing this if you're just starting out: "when you don't know where you're going, any way can be good"... – colormegone Jul 30 '15 at 04:54
  • I'm not starting out though, I'm in grad school. There's just no talks and not many people around. – Joe Jul 30 '15 at 04:56
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    Can you give an example of one of these books. – Joe Jul 30 '15 at 04:57
  • There are loads of talks (at a wide range of levels) on YouTube, however. We live in a wonderful age where you don't have to "be somewhere" to get an idea of what people are working on. But you have to spend the time looking around; there's no shortcut for that. – colormegone Jul 30 '15 at 04:58
  • Entering open math problems on Google would be much easier than posting this entire question here. – barak manos Jul 30 '15 at 05:03
  • I have in my hand here Ian Stewart's Visions of Infinity (2013), which discusses at a "popular" level over a dozen of the open conjectures (and a few recently settled) under active investigation. There are several books on the Riemann Hypothesis that were written about ten years ago or so (I think when it was in the news because Clay Mathematics Institute put it on their "prize list"). There are a fair number of others you can find if you live near a reasonably large library. – colormegone Jul 30 '15 at 05:03
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    I'm looking for more manageable problems than that. – Joe Jul 30 '15 at 05:05
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    These are more CS-oriented, but many can be cast as pure (discrete) math: http://sublinear.info/index.php?title=Open_Problems:By_Number – Clement C. Jul 30 '15 at 05:10
  • Not many people set out to solve the entirety of those problems in one go. I made these suggestions because they might provide ideas for "smaller", more manageable problems to work on. – colormegone Jul 30 '15 at 05:19
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    There has to be a reason you're interested in math - so do that! You don't necessarily have to wait for somebody to hand you an open problem. There's something in the mathematical universe that interests you. So hone in on that, focus your interest, and if you can't find a problem from somebody else, you can always make up your own. – Jair Taylor Jul 30 '15 at 05:49
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    @jrodatus "MathOverflow is meant to be a hub for open mathematics research problems[...]" No, it isn't. – quid Jul 30 '15 at 12:01
  • @quid, then the Wikipedia article on MathOverflow should be updated not to contain the following statement: "It is primarily for asking questions on mathematics research – i.e. related to unsolved problems and the extension of knowledge of mathematics into areas that are not yet known" –  Jul 30 '15 at 16:06
  • @jrodatus yes this is a bit misleading. I never edit Wikipedia, though. – quid Jul 30 '15 at 16:14

2 Answers2

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One possibility is to look what gets published at arXiv.org. Skim the abstracts to get an overview and if something interests you, you can find more on that topic.

Manuel Schmidt
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I think that part of this question answers itself when you get "into" a field. When I was writing my MSc. dissertation I really had no idea what I was interested in, but my supervisor pointed me in a few directions and I ended up working in the one I found most interesting. Then, as you read more and more papers and see the most recent research you find yourself enquiring as to how something was done, or if a certain construction could be altered, or whether it can be applied to a different setting. With the apparent explosion of (higher) category theory into almost area of maths, many open problems can come of the form "does this apply to a general category", or whether certain categorical results can be applied.

I am also in grad school, and I've found that the more talks / conferences you can attend the better. From listening to other people's research areas, you can take hints or maybe find a method / area of maths that you can apply to something you find interesting, to hopefully find a new research area / problem.

Stephen Nand-Lal
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