Mentioning Professors in math PhD applications 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?
This practice is likely, on average, to help more than it hurts.  It does mark you as somewhat more knowledgeable than the typical applicant, and somewhat more focused.  It also ties your name to Professor Z's.  Should Professor Z want you as a student, and should he or she be able to exert influence, that will give your application a huge likelihood of admission.  

However, if Professor Z is politically unpopular or hated in the department (or by the admissions committee), this may actually hurt your application.  Further, if Professor Z is unimpressed with your file, then you might be rejected even if otherwise you would be accepted.



Mentioning a research area can only help, provided that the department is actually strong in that area.