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.