<p>To elaborate on what cosmicfish has said: generally, your advisor is in your home department, but you may add another professor, in a different department, as a secondary advisor. The role of the primary/departmental advisor is to make sure you answer important questions in the program’s field, even if your research is interdisciplinary. For example, you can’t get a PhD in Physics if your work is of primary interest to computer scientists – or vice versa.</p>
<p>For this reason, you should consider the direction of your career. Chances are, you are going to branch out from your dissertation work into related areas of your field. Your colleagues and graduate students will be in that field. Do you want that basis to be physics, math, computer science, or engineering? Your broad interests will serve you well when approaching research since often scientists have to draw on other knowledge sets to arrive at innovative approaches; however, you need to decide where you want the depth.</p>