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Fall 2018 application cycle has started and I have a couple of (hopefully not too specific) questions regarding recommendation letters. I am applying for a PhD position in pure mathematics in the USA, with lots of top schools included.

(1) Does the status of the letter writer matter greatly to admission committees? For example, I have done research with a full professor, an assistant professor and a lecturer in my department. The latter two got their PhDs in 2011. If the three of them write me a letter, does that automatically put me at a disadvantage because two of them are not full professors or one of them is a lecturer? I have heard that, in principle, letters from people that are known in their field are looked upon more favorably by committees, particularly because they might be familiar with the letter-writer and can 'read between the lines' and get more information out of the letter.

(2) Should I bother asking for a letter from professors that only know me through classes? I have a potential fourth person in mind for a letter - he is a professor I took two classes with and did well in both. In my understanding, "so-and-so was a good student in my X and Y class" will probably impress no-one, even if it comes from an established professor. All programs require 3 recommendation letters, so would you think it is worth the effort to have an extra letter if it comes from such a professor?

Thanks in advance for any responses.

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Many, if not most, successful PhD applicants in mathematics have no research experience, and even when an applicant does have "undergraduate research experience, " it's usually not worth much as an indication of talent. If the first three letters are from the same project, it's probably better to skip one of them and go with the course professor instead. Rank, or at least reputation, of the recommender does matter, but what's most important is what they have to say about you.

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A letter from someone who has only taught you in classes and cannot testify to your ability to do research carries little weight with an admissions committee, so it is probably not worth your time to ask for the fourth letter.

As for the level and rank of the recommenders, I believe that also does not make that much of a difference in determining admissions, either. It would be unfair to penalize people based on the fact that they're working for an assistant professor instead of a full professor. (Not to mention that many admissions committee members are junior faculty themselves!)

Instead, what is most important is the quality of the referee letter. How well does the referee know you, and how strongly can they comment on your skills is what is most important.

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    I very much disagree for math in the US that a letter from a professor who has only taught you in classes will carry little weight. Especially at a top program where professors tend to favor highly abstract problems, an undergraduate student who has not taken several graduate classes will not even be able to understand any research problem the faculty would consider serious and worthy of (being part of) a PhD dissertation. (I mean this seriously, as in the statement of a problem will contain vocabulary that requires years of graduate courses to understand.)
    – Alexander Woo
    Commented Sep 10, 2017 at 5:11
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    Hence, a pre-requisite to doing research is several more years of study, so your ability to learn more mathematics will be a serious consideration.
    – Alexander Woo
    Commented Sep 10, 2017 at 5:13

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