Contacting potential PhD advisors while not knowing research topic?

Question

In December I will finish a Computational Sciences M.S. degree and plan to pursue a PhD in Applied and Computational mathematics afterwards. I have a B.Sc. in Applied and Computational Mathematics so have some exposure to the field but my Masters degree has largely been focused on Data Science and Machine Learning.

I would like to reach out to potential advisors over the summer whose research interests align with mine but I'm having trouble doing so in a meaningful way. Based on my coursework I would like to study math modeling/stochastic processes, numerical methods and efficient implementation of numerical methods, and/or PDEs, as these are the topics I found most interesting during my B.S. and M.S. degrees.

Most answers to this question indicate I should have a research question in mind before reaching out but beyond stating my interests as I've done in the previous paragraph I don't really have anything to go off. I've brought this up to a mathematics professor (who served as an advisor for my undergraduate math research) in the department of my current university and they agreed that mathematics papers are very difficult and time consuming to read unless you're up-to-date in that particular field and probably in that niche area of the field due to their density.

My question is how can I come even close to a research question prior to applying to PhD programs when it would take me an incredible amount of time to muscle through even a small fraction of the work by a single professor, let alone many professors from many universities? Do I need to fully understand the mathematics in their papers before I can completely determine the advisor is right for me? Is it OK to reach out with less information than an exact research question?

This is in the US, if that information is pertinent.

Answer 1

You definitely don't need super specific research topics to reach out to potential advisors to gauge their interest in working with you. When I first applied to PhD programs, I also got in touch with potential advisors beforehand to introduce myself and tell them about my research interests just to see if they would be responsive/be interested in working with me should I be admitted. At the time, I had a couple of ideas about what I wanted to do for my dissertation, but they weren't fully formed at all. Still, I did get some positive responses from several professors which encouraged me to apply to their programs.

At the end of the day, they know that most students don't join PhD programs with well-developed research agendas, that research interests change, and even that the person you wanted to work with originally might not be the person you end up working with in the end (the last scenario happened to me in some very ugly turn of events). Point being, I wouldn't worry too much about presenting developed research questions in the initial emails to faculty. If anything it's just a good way to get an idea of professors' availability and responsiveness, and to narrow down your list of schools to which you're applying.

And actually, given the above reference to my own negative experiences post-admission, I would add that it doesn't hurt to look for programs that have at least a couple of potential advisors to choose from.

Answer 2

In the UK, where you go straight into focusing on research, it is very normal for a potential mathematics PhD student to not have much idea what they want to research (arts subjects can be very different). When I was applying I was only required to say which research group I wanted to be a part of. Indeed, in maths the specific question generally comes from your supervisor, and having a very definite idea would potentially make it harder to find someone who wanted to work with you. Within the supervisor's interests, the topic should be matched to the interests and strengths of the students (unless other factors, eg funding, restrict the choice).

In the US you usually don't start the research portion of the programme immediately, and you don't always (usually?) need to have a specific supervisor agreed at the point of starting the programme, so there is even more space for flexibility.