The whole world of mathematics is open to you! Choose your favorite flavor (or at least one you are intrigued by).
To get started, here is some general guidance:
Start with a narrow focus and broaden out if time permits. It can be helpful (but is not necessary!) to begin with a specific problem in mind. For instance, one group in the past studied RSA encryption. That was the focus and the number theory details were filled in as needed.
Remember that this is not a class! The mentee is expected to devote only 3-5 hours per week on the topic. Given that constraint, it is not possible to, e.g., discuss the whole field of topology. See the bullet point above -- best to have a focused aim.
Don't feel pressure to learn everything or even useful things. If you find something interesting, dig in and enjoy!
Here are some more specific ideas:
Get ahead on a class you are going to take in the future (and stop to smell the roses on the way).
Dive deeper into a topic that you skimmed in a course that you liked.
Learn the history of how a branch of mathematics was developed.
Learn to write code and conduct computational experiments.
Study a non-math topic of your interest using mathematics (ex: games, art, politics, sports, food, etc.)
At the end of each semester, mentees are asked to give short (~10 minute) presentations that showcase what they learned during the semester. These are intended to be fun and casual, and offer the mentee an opportunity to share the great work they did! Presentations can either be "chalk-talks" or slide-based.
General recommendations: in our experience, the most effective presentations are focused. You don't need to review everything that was discussed over the course of the semester. Instead, perhaps choose one or two of the most interesting problems you worked on. In the course of explaining that problem, you can give the necessary background for other students to understand what you did. In other words, rather than the presentation being a list of everything the happened over the semester, we recommend choosing a particular problem, and in the course of explaining that problem, fill in the background material as needed.
Examples of past presentations are below.
Muhideen Ogunlowo and Vladislav Taranchuck: Application of Linear Algebra to Statistics through the Mean Vector and Covariance Matrix
Schuyler Brennan and Mary Taranchuck: A Model for Tear Film Thinning with Osmolarity and Fluorescein
Andrew Kallai and John Byrne: Investment Science
Silo Murphy and Jerome Roehm: Sequences in the Calkin-Wilf Tree
Yiming Wang and Tyler O'Grady: Graph theory and Spanning Trees
Daniel Bowers and Niko Schonscheck: Number Theory and RSA Encryption
Below are some articles on mentoring and teaching in an inclusive way. One of the goals of the DRP is to increase participation in mathematics by historically underrepresented groups. As mentors, the way we interact with our students can have a huge impact on their future development, and we believe it is important to teach from a place of compassion and empathy. Students should feel empowered by their participation in the DRP. If there are other articles you know of that should be included on this list, please contact Yuan-Chieh (Jimmy) Chen (ycchen@udel.edu)!