I am a high schooler who really likes math, and I am interested in pursuing it in my undergraduate years. I have a basic facility with proofs, and I am currently exploring several different areas of math (analysis, graph theory, and topology). I am interested in seeing if I can contribute to mathematics even if it has no applications, mathematical or otherwise, however most areas of math don't seem to have any "low hanging fruit" so to speak, so I'm looking for a different, more obscure area where I can learn all that is known, and maybe contribute myself.

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    $\begingroup$ That "fruit" that you mention is indeed hard to come by nowadays, but one "piece of land" where you can grow some new fruits is combinatorial set theory. Check out the book by Van Vu & Tao called "Additive Combinatorics". $\endgroup$ – DeepSea Apr 16 '16 at 21:53
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    $\begingroup$ There's really no imperative to immediately/quickly "contribute" in this sense. Especially, I'd think it'd be ill-advised to specifically avoid (learning about, looking at, ...) well-developed (=useful, effective, whether "pure" or "applied", ...) mathematics just for the sake of novelty. Looking for untouched ecological niches might lead you to end up like the emperor penguins standing still for months in Antarctica through blizzards... cuz no one else wants to live there. $\endgroup$ – paul garrett Apr 16 '16 at 21:59
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    $\begingroup$ I think you're best bet is to start reading more mathematics in whatever area you are curious about, and trying your hand at some problems. Doing exercises is always a great idea, but also see what happens if you try to attack some open problem. It will help you understand what the problem is about and build your intuition, even if you don't end up solving it. Contributing original content to mathematics is usually not something that happens until the dissertation, at the end of the doctorate. But there's no harm in setting your goals high as long as you're aware of this. $\endgroup$ – j0equ1nn Apr 16 '16 at 22:04
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    $\begingroup$ dev.geogebra.org/trac $\endgroup$ – mvw Apr 16 '16 at 22:10
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    $\begingroup$ I'm also a high schooler with a deep passion for math - though more in applied math. If your interested in talking (would be nice to meet another similarly interested peer) id love to chat. I recently discovered the generalized tower of Hanoi problem optimal solution is still an open problem if that interests you $\endgroup$ – theideasmith Apr 17 '16 at 2:19

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