Independent undergraduate research -- what to do?

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.