Should I list my research interests that are not popular on a statement of purpose? I am currently applying to Ph.D programs in the US in Mathematics and am trying to decide what to list for my research interests. I am very interested in field Y, but I also have a background (and slightly less interested) in field X. I have enough knowledge to discuss both, but I definitely know more about field Y. That said, field Y is very small, with most departments I am applying to having only one or two people working in it, while X is definitely more common, popular topic. My question is, should I discuss more the topic which is less popular or more? Does it matter, or is it just my interests in something that is being assessed?

Edit: In this case Y=topological modular forms, and X=gauge theory, in particular floer homology.
I am impressed that an undergraduate student could speak knowledgeably about either Floer homology or topological modular forms, let alone both.  I would include both interests in your applications: by describing both, it seems implicit that you are not wedded to either one, so I don't see your interest closing any doors.