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I am a high school senior planning to major in mathematics next year at college. My question is this: Are there any fields of mathematics which do not have a wide body of literature (maybe even just a few papers or monographs), yet have potential for further research? Please give some sources for learning about these fields, if you can.

I am mainly looking for an area of mathematics where I can reach the frontier of knowledge reasonably quickly, and hopefully even contribute during my undergraduate years. I have looked into graph theory as a possibility (is it a good one?), but I was just wondering if there were any other fields out there which might work as well.

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    $\begingroup$ [This is a better forum for your post, and I'm glad to see positive reception of it.] Graph theory is good. I would also recommend combinatorics more generally. (I do discrete math, and as I said on MO, feel free to email me if you'd like any resources in that area: prd41 [then put at symbol] math.rutgers.edu.) $\endgroup$
    – Pat Devlin
    Commented Dec 19, 2016 at 5:34
  • $\begingroup$ Another thing to do might be to look at some student-friendly journals (assuming you can find access to them). $\endgroup$
    – Pat Devlin
    Commented Dec 21, 2016 at 13:03

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In line with your identification of graph theory, I suggest you might look into what is now known as "Discrete and Computational Geometry." Although there is much to learn (there is a 1937-page Handbook of Discrete and Computational Geometry, CRC link, now in its 3rd edition), one can reach frontiers in narrow areas without mastering the whole field. In particular, you might find browsing the titles in the recent conference collection below. The papers include,

  • "Mario Kart is Hard,"
  • "Continuous Folding of Regular Dodecahedra,"
  • "On Evasion Games on Graphs,"

among other accessible topics.

Akiyama, Jin, Hiro Ito, and Toshinori Sakai, eds. Discrete and Computational Geometry and Graphs: 18th Japan Conference, JCDCGG 2015, Kyoto, Japan, September 14-16, 2015, Revised Selected Papers. Vol. 9943. Springer, 2016. Google books link.

Akijamap19

Fig.9: "Reversible Nets of Polyhedra," p.19.

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