I'm an undergraduate student who's interested in possibly pursuing graduate school for mathematical optimization or something adjacent to that.

I'm going to participate in an REU this coming summer, and I'm wrapping up the year-long undergrad analysis and undergrad optimization sequences this quarter. Since I've more or less completed my graduation requirements, I was thinking of petitioning into some graduate courses next year, ideally in order to get to know some faculty whose areas of research are close to what I'm aiming for. The instructors that I'm closest to are unfortunately more focused on the teaching side of things rather than research, and I've been advised that for graduate admission purposes, it's ideal to have a letter written by a tenured professor.

Some of the grad classes I'm considering are:

1. convex optimization with algorithms emphasis (CS department)

2. convex optimization with modeling emphasis (EE department)

3. convex analysis and nonsmooth optimization (Math department)

4. mathematics of data science (year-long sequence under Math department)

The 2nd one follows the textbook by Stephan Boyd. I think the 4th one will most likely follow the high-dimensional probablity textbook by Roman Vershynin and the high-dimensional statistics textbook by Martin J. Wainwright.

My main questions are:

1. Is trying to foster a relationship with a grad course instructor to get a letter of recommendation or for the distant possibility of having them as an advisor doable or ill-advised?

2. would it be reasonable to ask any of them if I could get involved in their research in some form, or do I lack the experience to be of any real use to them at this point?

3. Which of these courses among those listed should I prioritize? (I don't think it'd be a good idea to take any more than 2 classes a quarter, and that's already probably pushing my luck)

4. Should I focus on classes directly related to optimization, or would it be better to take more foundational classes? (e.g. measure theory or functional analysis)

5. For the Mathematics of Data Science class, do I have the appropriate background to take it? I've taken analysis at the level of Baby Rudin and the first 6 chapters of Royden. I took half of the probability sequence, so while I've seen stuff like Chernoff bounds, transformations, and MGFs, I haven't formally seen stuff in the multivariate setting or things like conditional distributions. I also haven't taken an actual machine learning or data science course yet.