I finished my Bachelor’s in math about a year ago, and have been working since. Now I want to go back to school for a PhD, but there are a few subfields that I’m interested in, so I’d like to be at a school where the “pure” math department works closely with the applied math department so I can explore. Ideally there would even be an option for switching between programs, since I’m still not entirely sure which track I’m more interested in. A lot of schools have these as entirely separate departments, with no overlap whatsoever between programs, and I want to avoid that. I’m interested in schools across the spectrum of high-ranked to low, as long as they meet this criteria. This isn’t easy to search by, though, and a lot of programs don’t have much information online, so if anyone has any recommendations I’d appreciate it!
It may depend on what applied area(s) you are interested in.
Some math departments include the applied areas, rather than having applied math as a separate department. But some applied areas may be in other departments, such as statistics, (theoretical) computer science, and operations research / industrial engineering.
Probably best to look on the web sites of each math department to see whether its areas of research (which may require looking in each faculty member’s web page) include those which you are interested in, and check if the college has other departments which may include the areas you are interested in.
Basically, it is not hard to find. I just googled.
There are a lot of misconceptions about “Applied Mathematics”. For one thing, the field covers a continuum of research activity with varying degrees of emphasis on theoretical proofs. Many applied mathematicians (like me) prove theorems (just like pure math) for problems that initially had their origins in an another field. Others create and research sophisticated computational techniques. And so on. Also, there are folks in CS departments that base a lot of their research on what many would consider “pure math”. Furthermore, OR departments and Stats departments are usually separate, but also do what could be termed “applied math”.
Second misconception is that an applied math PhD degree is marketable in industry. This is only marginally so - it really depends on the subfield, and the kind of connections that the doctoral adviser and/or department has. And academia is flooded with math Ph.D.'s looking for tenure track positions.
Before you start looking at graduate programs, you need to ask yourself why you need a PhD. Have you done research as an undergrad? You will need some evidence of that if you want to get into a decent PhD program. Also, there is a lot of variability in PhD programs, and I would advise you to apply mostly to R1 institutions. Those will have the most percentage of faculty with an active research agenda in a variety of fields.
As the other posts have already outlined, you will need to look at the each program’s website to see their departmental research agendas. Also, look at the math departments of big flagships with well regarded math departments: UMD, UMN, Univ Wisc- Madison, Penn State, Rutgers - they all have lots of information on their websites.
What are those subfields that you’re interested in?
Differential geometry and graph theory and combinatorics were my favorite topics in undergrad, but I’m also looking to explore other areas! I also want to at least try a few more marketable topics though I’m not really sure what that would be yet 
Definitely don’t want to apply to R1 universities, actually! There’s one R1 school that I really like for other reasons so I’ll be applying to it, but I want to avoid the super academic, ambitious schools in general. Thanks for your advice on the difference between applied and traditional, though, that’s interesting… if applied doesn’t necessarily mean marketable, are there specific subfields that are more marketable? If so, what would you say they are?
At the end of the day, I’m doing this because I really want to, and if I just went and got an industry job now, I’d still want to go to grad school when I retire someday… so may as well do it now when there’s at least some small career benefit! So I’m not exactly looking for the most marketable thing ever, but if something I’m interested in happens to be marketable, that would be a nice bonus.
This is really helpful, thanks! There are a lot of schools out there, though, even if I narrow it down by location, so it’s a pretty big task to look at each professor on each website… I guess it makes sense that there’s no real shortcut, but I was hoping for one anyway.
Try searching a research database like Mathscinet or the American Mathematical Association search engine to find papers in your area of interest and note the authors and institutions.
Also, ask your undergrad instructors.
You’re going to need LOR’s from your undergrad math profs for your applications.
They’re also the best source for advice on where to apply based on your interests and academic record.
You’ll have to contact them anyway for the LOR’s, might as well ask them for recommendations.