I was helping a math PhD who worked in computable ramsey theory prepare his teaching and research statements for his job application. One of the questions various schools wanted to know was
How can you use your research as the basis for undergraduate projects?
His answer was kind of a cheat- Ramsey Theory was combinatorics which ugrads can work on without too much prior knowledge (true), computability theory could be picked up (not true), and the chance to combine the two should thrill ugrads (it doesn't even thrill most Ramsey Theorists!).
I don't fault the PhD student, I fault the school. What they really want to know is:
Can you come up with ugrad projects?
They should realize that some math requires too much bacground and is not appropriate for ugrads AND that a math PhD will have picked up OTHER knowledge for projects. They may relate some to the research (e.g., in the case above just Ramsey Theory) or they may not even be close.
In Comp Sci it is more common for an ugrad to work on research related to a profs research since Comp Sci is a younger field and needs less background knowledge. And students can often code something up of interest related to the profs research.
SO- do you have ugrads work on your research program OR on something else?