Help Understanding the PhD Application Process

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YoshiMoshi
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So if I'm understanding correctly, companies provide grants to universities to conduct certain research. The professors oversee the grants and have PhD student help contribute to the research under that grant.

Based on this, it seems that PhD students don't conduct research "any question" they want, but topics in correlation with the grant from companies?

So in a Research Interest Summary that I'm supposed to submit along with my application, am I supposed to propose conducting research to answer a specific question that may or may not correlate with a grant, or just a topic that correlates with a lab that exists within the university?

For example, a university has a plumbing lab that a professor oversees. Should I express interest in conducting researching in the:
  1. Plumbing industry
    1. Example, just express interests in doing research on the plumbing industry
  2. A specific subtopic in the plumbing industry
    1. Example, alternate materials in plumbing industry
  3. A specific question under a subtopic in the plumbing industry
    1. Example, what are the effects of using polyurethane as pipes in plumbing (I'm just making this up)

I would seem to me that I should just describe my interest in doing research in the plumbing industry? I don't know which research grants will be available to me to work under when I actually start my research and am done with my course work. The grant may dictate certain topics or questions have research done on them.

I think I might be overthinking this.
 
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It depends, ask the potential supervisor.
If it's a company-funded program it might come with specific requirements, if it's government-funded there is usually a larger range of options. A PhD program can be a mixture, too.

It's not uncommon to change the exact scope of the research over time. I don't want to clog the thread with plumbing analogies: In experimental particle physics it's common to start with some sort of service task - something necessary to run the detector like calibration, work on hardware and so on. During that time someone else might start the data analysis you planned to do, or some other analysis is now more interesting because some theorist published new predictions, or the accelerator didn't collect enough data to make this analysis promising, or whatever else might happen. In the US where the PhD doesn't start with research the time span where things can happen is even longer.