Letters of recommendation -- from postdoc or professor? Both?

Question

I am an undergraduate studying pure mathematics in an American university with a top-10 graduate program.

I am applying to PhD programs and graduate fellowships this year, and I have a dilemma about who to ask for letters. Maybe this question is too specific for academia stackexchange, but actually I think it is abstractly a somewhat common issue.

I have four potential letter writers. Two are very solid (very famous professors who know me well from reading courses / research) as far as I know, and I will ask them. Call the other two potential letter-writers "Prof." and "Postdoc"

"Postdoc" is a postdoc whose graduate-level course I TAed; I also took a graduate topics course with her and produced something semi-original (a new proof of an already-known result) for the final project. She is originally from Europe and did her PhD in the United States at a top-10 graduate program.

"Professor" is a very famous professor, though not in the same field as the other three (who are in the field that I will apply to graduate school in). I took an undergraduate course and a graduate course with her; I did very well in both courses and produced a very thorough (and long if that counts for anything) expository final project for her graduate course. I also know she is very serious and enthusiastic in general about writing letters for undergraduates, though I do not know if she actually read what I wrote in my expository final project.

For graduate programs, I have the option of including 4 letters. Should I include all 4, or just choose one of {Postdoc, Prof.}? Should I expect one of those letters to dilute my application and make it less strong? In my view, there are reasons why either letter could do that: for the postdoc, either because she is a postdoc (so some people might completely disregard her letter), or because she is very young and European (so might not be very good at writing letters yet, especially in the American style). For the professor, there is the potential downside that she is not in my field (though could it be a strength to have some diversity?), and that I didn't do anything even semi-original in her graduate course (though my understanding is that for pure mathematics this can be OK?).

For some graduate fellowships (e.g. the NSF GRFP), it seems like I can only submit three letters, so I must choose one of the postdoc or the professor to exclude (regardless of whether I ask both for PhD programs). Given this information, is there a clear choice of which one to exclude?

Apologies again if the question is too specific. However, I think there are some general themes in letter-asking embedded in my question: (1) asking postdocs vs. professors, (2) asking professors who I haven't done anything original for, (3) "dilution" of letters and whether to ask for 3 or 4 of them, (4) asking a professor whose research area in pure mathematics is different from the one I am applying for

Answer 1

To answer the general question about how to pick the last recommender:

• Ask the first two.

They know you well, they know your field, and they presumably know at least the famous professor, if not the postdoc. They can guide you to what would help your application the most. Professors beat postdocs, and a little diversity in field can help. I suspect the answer will be the famous professor, but ask them, not me!

The GRFP doesn't care at all about TA potential, so I would probably rank the research letter higher than the other. For admission, only people who know you well can help break the tie.

Answer 2

Realistically, your acceptance into a good doctoral program will depend on an assessment of your probability of success in research. Funding might depend a bit on other things as well, but not the acceptance itself. Normally that would suggest professors are to be strongly preferred.

In your case, however, the post doc has some knowledge of your research potential and that is very important to get on the record.

If you can send four letters, then send them all. If you can only send three then you need to balance what people are going to write so that you can get a complete and positive picture of your research potential. Whether "diversity" helps or not comes back to what they will say about your likely success with research in your own field. But there is little downside when people praise you and your work.

The fact that you have some experience as a TA will only give them some confidence that you will work out as a TA provided you are accepted. It will have little effect, or none, on the acceptance potential.