Submitting more than three letters of recommendation to a math PhD program

The first criterion, of course, is that if a particular institution's application intsructions communicate that only three letters should be sent, then abide by that. For the ones that say "at least three" or the like:

To be honest, in practice it really depends on the internal details of each university's application storage system and how it lets reviewers view applicants' files—and this can vary wildly from place to place. At my institution, if you have four letters of recommendation, they will all be visible when I look at your file. In other places, their online portal might have specific spots for three letters and no more (I would hope such institutions would make it clear in their instructions that only three letters should be sent).

As someone who has reviewed graduate applications for nearly twenty years, I can tell you that if my file system allows me to see the letters, I will read all the letters if I am interested in an applicant. (Well, I probably wouldn't read ten letters … but certainly four.) If you have reasons to believe that the letters will offer complementary perspectives on your excellence, I would use them all.

You should email the institution to which you are applying and ask their graduate admissions team if they would accept more than three letters. Whether it’s acceptable, irrelevant, or above a threshold to send more than 3 letters is subject to the preferences of the institution to which you are applying.

A problem you might encounter if they will only accept three and you send more, is that the might just ignore or throw out some of them. They might throw out the ones(s) you most want them to keep in such a case.

Unless you hear otherwise from an institution, stick to their requirements.

Some of the rules are to limit the work they have to do in evaluation, and some of it is just establishing fair rules for all applicants.