I hope this question is within the scope of this website. I am currently a rising senior, and need to decide on a topic for my independent undergraduate research/thesis. I was hoping to get some suggestions from people knowledgeable in the field. I do not have a very deep mathematical background; I have had differential equations, analysis, algebra, an advanced treatment of linear algebra, probability, statistics, and some general treatment of mathematical coding theory. If it comes to that, I would be fine studying something extra for the project, as long as it's not too much (for example, I don't think I could pull off a whole course in topology).

Ideally, I am looking for something very theoretical that'd be accessible with my background. If it's very theoretical, but still has some kind of application, even better! So far I have been considering octonions and their application to quantum physics (became fascinated with the topic after reading a paper by Baez/Huerta on the issue), and perhaps exploring the Riemann hypothesis and maybe even making a tiny bit of a progress (naivete, I understand) in some direction (though I would need to study quite a bit of complex analysis for that). Something very theoretical and probability-related would be awesome as well! I would be very thankful for any suggestions, perhaps even an open problem that would be accessible to me as an undergrad? Maybe not too hard to solve, but no one got around to it yet. :)

Thanks again for any suggestions -- this is very open-ended, so please don't hesitate to throw out ideas.

If I end up using your idea I would be happy to credit you in my paper, just let me know of the best way to do so. :)

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    $\begingroup$ You should probably be asking this question to professors in your department who can give you much more tailored advice than you will get on a website. Presumably you will have an advisor, right? $\endgroup$ – user21725 May 4 '12 at 0:38
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    $\begingroup$ @EricGregor: I am, but I figured that getting as much feedback on this from various sources as possible would be a good idea, since this is a pretty serious project for me, and not something I'd be doing ever again. :) $\endgroup$ – LookingForLeads May 4 '12 at 0:40

Sundry advice on looking for ideas:

  • If you have not done so already, cultivate relationships with your faculty. If you think there is one that you would like to study with, they would probably be your best bet for a fast orientation to a topic you might pick. This is especially important if they can guide you on the type of work they're looking for in your thesis.

  • Go to all the math colloquia and talks that you can possibly attend. Chat with other students and professors about math topics whenever possible. Consider giving talks on what you have learned or what you are thinking about, also.

  • Read more papers on the things you like (not necessarily thoroughly, but just to get a feel for what you could learn about.) Just expose yourself to more stuff!

  • Don't let learning something new/hard put you off topics you like. In fact, there is even a possibility you might find it enjoyable to learn an entirely new thing to motivate your research. (Don't embark on that without deep reflection, though, if you are tight on time.)

  • Search the references of papers you like for other good things to check out. Make a binder. Highlight stuff.

  • If you find out about a mathematician who is working in the thing you like, ask around to see if they will bite your head off. If the consensus seems to be "no", work up the courage to strike up a conversation with them.

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    $\begingroup$ This is pretty good advice. I'd also add that you should consider speaking in a seminar as a possible goal for your efforts. $\endgroup$ – Antonio Vargas May 4 '12 at 2:21
  • $\begingroup$ That's a good one that I omitted, @AntonioVargas, I hope you don't mind if I add it up there. $\endgroup$ – rschwieb May 4 '12 at 13:26

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