When applying to grad school, is it a good idea to mention professors you want to work with? For example, "I am applying to this program because University X has leading experts in the field Y, such as Professor Z. I studied some of Professor Z's papers and would like to learn more about this particular topic in field Y."
EDIT: In addition, what if I were to email Professor Z to ask if he is taking students (in order to avoid an awkward situation where the Prof is leaving/not taking students)? For example, "Dear Prof. Z, I am applying to University X for grad school and I am interested in field Y. I am wondering if you are taking any students at this time, or if there are other researchers in field Y that are taking students?" Also, is it normal/not awkward for a math phd applicant to email a professor?