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. :)