I'm applying to PhD programs this year. How common is it to email potential supervisors before submitting my application, asking if they have a spot available? Could this improve my chances of admission? I've heard that this is the norm in European countries. What about in the US?
I have a masters degree (Mathematics) and know roughly the area of research I'm interested in.
As @Buffy comments, the other answers do not seem so accurate about R1 and R2 grad math departments in the U.S. That is, funding is mostly as Teaching Assistants, administered through the department, not by individuals, and funding-and-admission decisions are also made at a department level, by a committee.
Yes, having a faculty person ardently promoting your case to the admissions committee is definitely a good thing, but is not necessarily decisive, depending on that faculty person's record of following through (or not), mentoring, etc.
Yes, asking faculty ... not whether they "have a spot", but... whether they are currently taking PhD students can be a relevant question. Both people with lots of students already, or who aim to retire very soon, might say that they are not currently taking more PhD/research students. It would obviously be useful to know that, rather than show up and be negatively surprised.
In any case, as other people have commented, do NOT write generic emails with a huge bcc list. Take the trouble to address people by name (!!!), at the very least! And if you describe your own interest as X, which is not much connected to what the faculty person does, don't email them at all.
True, people seem to often "reckon" that sending inaccurate emails to people is harmless, because they can just delete them. It's not that I myself keep a grudge list of people who've sent me spammish emails... but it is fairly antisocial to spam people, in my opinion.
Yes, as @Chris has described, this can be a good tactic -- but only if done properly. That answer describes some examples of proper form.
However, please don't just blindly send email to every professor you can find. You need to show that your interests have applicability to the professor, to show you've "done your homework" and chosen your targets well, and that there is a real reason for the professor to keep reading (and, hopefully, respond).
I'm a professor at a well-known US university, and I get dozens of emails every month that are obviously sent without any indication that the sender has even looked at my website to see if my research interests align with their stated capabilities and goals.
Yes, this is very common in the U.S. If you play your cards right, you will already have a pretty good idea which institution(s) will accept your application when you apply. In some cases you may get an informal offer before applying if you go this route. This has been my experience.
Contacting potential advisors / program chairs before applying for a doctoral program in the US tends to be more popular for PhDs and other research-based degrees. This is far less common for practice-based or professional doctoral degrees (especially for those degree programs in which the relationship with one's advisor isn't important). It helps to explain your research interests and to find someone whose interests align with yours...but flexibility will also serve you.
If funding comes into play (it should for a PhD program), getting to know a department and potential advisors becomes even more important. Departments want to know who they're investing in. Will you likely "stick it out" and finish the program? Do your interests align closely enough with theirs? Is funding available for your tuition AND living stipend AND research costs? Can the department trust you to teach a few undergraduate courses (or labs). Will you be pleasant enough to work with for several years?
I've gone multiple routes. In round #1 (2008) I never received interest from the PhD programs to which I blindly applied. Several programs that I contacted in 2010-2011 ("interesting" economic times) were quick to tell me that no funding was available for the following year (or probably the next). That, of course, saved time and lots of application fees. As I researched my options, I spoke with several potential advisors by email, phone, and in person (via campus visits that I set up). This ultimately led to two "informal" offers with funding. I formally applied after that point to those two universities and only those two. (Even with an informal offer...you still need the formal offer in writing before making a decision.) I'm glad I did so many phone calls and in-person visits. I eliminated several possibilities just based on these conversations. Faculty websites aren't always accurate and up-to-date. :-)
On the other hand, a few years ago when I applied for my EdD program (dissertation-based program but no funding offered) I sent a short email to the program director expressing my interest. She set up a Zoom interview with me and we discussed my research interests and professional goals. I was accepted and then assigned to a dissertation advisor. This process was much more streamlined, but the department was also much larger than my previous PhD program. And, with no funding offered, the department wasn't taking on the level of risk that a traditional PhD program would.
