Let's say you are a prospective graduate student and you have some ideas about what you want to study. Is it a good idea to email potential professors you want to work with?
I know that for Math PhD programs, you get admitted into the program first, and then you find an adviser. Still, should you email people you want to work with, just so they know who you are (and potentially vouch for you in the admissions process)? What should you write?
Typical answers I've seen on other sites (like stackexchange) is to make a remark about the professor's research. But in math, to really understand someone's research, you need to have a lot of working knowledge. It's not like chemistry or biology where you can at least read experimental procedures, etc.
What about asking for a reference request? (in a more nuanced way.) For example:
"Dear Professor ___,
I am a senior at ____ university and I am applying to math phd programs this fall. I am interested in your field ____, and I want to learn more about this topic ____. What would be some good references to learn about topic ____? I've found reference A, which focuses more on X, and reference B, which focuses more on Y. Which one should I start with?"