I am currently a 4th year CS undergrad student.
If I tailor my Statement of Purpose (SOP) around my research interests (which are related to the research project I am working on with my professor), I feel like I may be limiting myself to professors who are in my field of interest, but may not have any funding, or may not accept any PhD students. Is this detrimental?
If you seem to absolutely rule out all other options, then yes, it could hurt you since it lessens the likelihood of finding an advisor.
However, it is good to narrow the field somewhat (languages, algorithms, AI,..., whatever). And it is good to say that you have some specific interest and experience with that.
So, be flexible, but focused.
In my own case (math), I'd have been happy in either Topology or Classical Analysis and had some insight in both. My dissertation was the latter.
According to me, whether or not you should focus your research interests around your current project will depend on whether: (a) you want to continue working in that field/domain, and (b) whether you are open to exploring other domains during your PhD.
When I look back to when I was an undergraduate student (I am currently doing a PhD in quantitative psychology in Europe after following a two-year research master and working as a full-time research assistant for a year), I had a broad curriculum and learned a great deal, but didn't (and still don't) know all the possibilities for research. Learning about other interesting topics for research requires constant exploring on your part. You can do this by reading, attending conferences, or speaking with your field's experts. But this is also what your PhD is for :)
In general, while I think it is important to elaborate on your current research interests and your work with your professor, discussing things you don't yet know and want to learn about in your SOP is a big plus. I would delve into detail about (a) what you have already done and how you could apply/extend this work during your PhD, and (b) things you are curious about but have no experience with yet. It's alright to be less specific when it comes to topics you're not familiar with - but expressing an active interest is what matters.
Before you apply at all, consider emailing them with some of your own questions or ideas and ask if they're accepting students. They might not always reply, but it could save you some time during the application process. This worked quite well for me!
I would also tailor the SOP to the university/professor you are applying to work with. Look up the people you want to work with and read about their current research. In the SOP, explain to them why you're interested in their topic/why you'd be a great addition to their lab. Not only will this show your motivation, but it will also already give you a foundation for what will be discussed during the interview.
Best of luck!
