Importance of domain selection for undergrad applying to a PhD program (USA CS)
Actually, in the US, any research experience at entrance with a bachelors is a plus, possibly a big plus. Having an idea of where you want to go in research is a big plus, even if not entirely settled. So, yes, it is good to mention the specific areas that most interest you and any experience you have with those.
But acceptance depends on many things, from GPA to letters of recommendation.
I'd suggest that, other things being equal, having an idea of what you want to do in grad school will be an advantage and will help you choose more precisely when it comes time to do so.
But doctoral studies in the US normally start with a lot of advanced coursework leading up to the comprehensive exams so that candidates have a fairly broad exposure to mathematical concepts even before they specialize (and specialize...) on a dissertation topic.
It might be a mistake, however, to refuse to consider other options. Keep an open mind. You will have plenty of time to develop a formal plan.

My own experience was that, at entry, I would have been pretty happy with anything in analysis or topology, but nothing specific beyond that. And pretty unhappy with algebra. My only &quot;research&quot; experience was in the philosophy of math and it was pretty embarrassing. Long time ago, however.