When one does not have well-defined research interests for PhD Programs 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?
In mathematics it is common to have little specific idea of what you want to do in a PhD at the point of applying. Unlike in many arts subject, for example, your supervisor will largely choose the research question for you (hopefully with your input). At a graduate open day, we were told that SOPs range from 'I want to do a PhD in pure maths' to 'I want to prove the Riemann Hypothesis'. Indeed, my own loosely said 'this is what I've done, so something a bit like that might be good'. Exactly how much detail you need to give will depend on the structure of the PhD programme, and local rules/funding arrangements.

I think the thing you will need to set out clearly is why you've chosen to switch to mathematics, and how you will achieve the same standard as those who have done a maths undergrad/major.