Applying to graduate program - propose research project? I am a rising fourth year chemical engineering student applying to graduate programs in statistical mechanics (condensed matter and biological physics) this fall. I have been working on research projects in two different research groups for the past two years, and will be submitting a manuscript (for my biophysics project) in a couple of months.

Some of the programs I will be applying to have advised students to contact the principal investigators whose research interests them before applying, while other programs do not mention anything of the sort.

I am wondering if it is standard practice for prospective students to propose a specific project (aligned with the PI's current research), or if I should simply express my interest in their work and leave it at that. Of course, if it is the former, I expect the PI to modify it as they see fit.
It is not standard in the US, since you are not expected to know enough about the subject to make a good proposal.  But if you can make a good proposal, it is helpful to do so.  Just be clear in your message that you are flexible, because the PI probably already promised a funding agency that they would work on something particular, and you most likely do not know what was promised.