Advice from former UMN REU mentors and student participants

1. Set aside time each day when you are alert and free of distractions. If you are working with others, you may need to adjust your schedule some days to align with theirs, just like they may need to adjust to fit yours.
2. Be courteous to others in your group; listen to their suggestions and comments as you would have them listen to yours.
3. Write up your work as you go along. Include all details. Many times there may be a small mistake in more involved calculations. Have someone else look over your work -- it is very valuable to have a new set of eyes on it.
4. It is great to read the literature, but at some point you want to start pushing it forward. For summer research you are not expected to be an expert in everything; trying to become one will result in doing nothing but reading. Play with simple cases to build intuition. Ask questions and try to generalize.
5. Look at multiple problems or parts at the same time. Sometimes if you're stuck on one, it helps to put it aside and look at something different (which can sometimes provide insights to the original problem).
6. Different people prefer different technologies; explore a few and choose the one that works well for you. For example, some people prefer to type during a meeting, others to use a stylus / pen, and others a camera phone pointed at a piece of paper.
7. If you see conflict arising in your group, talk to someone about it; don't let small points of contention fester and become large ones. Is there a third person who can give an impartial view? Can you talk to the person directly? The mentors and teaching assistants are also here to assist.
8. Keep good notes. Sometimes an approach that didn't look promising before turns out to be useful later (this happened in Wiles' proof of Fermat's Last Theorem). Date your notes. If you write by hand put page numbers. I often scan or take pictures of my notes and save on my computer. Have good file and directory names. Document your progress carefully for others using the Slack channels.
9. Persevere. Math is hard. You may be fortunate and prove something quickly, or it may take a long time, or perhaps the problem was too hard or you did not have the right tools. Do not be discouraged. If the original problem is resisting analysis, consider pivoting. Is there a related problem you could try first? Is there an easier case? Talk to other people and see if they have ideas, and listen to what they are doing and perhaps you can help them.
10. Explore! It's okay if when you start you don't know if the idea will pan out. Try it! Suggest it to others! Even if it doesn't work, it might resonate and suggest something to a colleague.
11. Have fun! You are going to play with some interesting problems with kindred spirits. Enjoy!

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