Mathematics GRE

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?"

That email sounds silly. You should ask such a question to someone in your undergraduate department. If it's such a small field and no one at your department knows stuff on it, ask here or on stackexchange.

If you have something insightful to say about their research, say it, or if you know someone who knows that person, go ahead. Otherwise, just apply and mention you're interested in their work in your SOP. Your email will probably sound silly when you start trying to talk about algebraic geometry, say, but really know nothing about the field.

I'm certain that professors are very busy people, so it may be the case that they get bombarded with email questions just like yours during application season. I would definitely ask your letter writers about this. This is sort of a grey area, and there's no rule of thumb on how you should approach this situation from what I know. But i'd be very interested in an answer for this as well.