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.