Should I remind my potential PhD advisor to know his feedback regarding my solutions to a problem set given by him? About a month ago, I emailed a potential PhD advisor in the field of theoretical physics expressing my interest in a PhD position he had advertised. He replied back by saying that he normally gives a problem set for PhD applicants to solve within 2 days. A week later, while sending me the questions, he added that he wanted to understand how reliable I am with calculations. The problem set included questions of increasing difficulty. I solved the problem set in the specified deadline and sent back the solutions but he hasn't replied since then.

Should I send him a reminder email to know the status of my application process?
Yes, absolutely. If it's been more than a week, it's definitely appropriate to send a polite reminder. He probably gets a lot of e-mails and may have just missed yours.

In general, I've found it's best to follow up after a week if there's any chance that the other person missed or forgot about the e-mail. If he's aware that he still needs to respond, no harm done. If he's not aware, then it's a good thing you're letting him know.