I'll be applying to tier II PhD programs in Mathematics. I've made a switch from another field to Mathematics, and, given that my primary major was not mathematics during my undergraduate, I don't have well-defined research interests. I have some idea that I'd like to work in analysis, but I'm not sure of which field -- differential equations, functional analysis, geometric measure theory etc. I still, at the very least, need to study all these subjects in more depth to be able to make an informed decision. Moreover, I come from a background where I studied multiple subjects in college -- Economics, Physics and Mathematics -- and I consistently realized I'm more interested in the mathematics of all the subjects I have studied thus far. So, right now, I am more interested exploring more mathematics out there, so I can make informed choices about my research interests at a later time.
So, my question is: how do students like myself sell their research interests in the personal statement addressed to graduate admissions commmittee?