Is there an incentive to submitting a math PhD application before the deadline? I was aware that there are a few REU programs that began offering positions before their stated deadline. Is there an advantage to submitting my application early, besides the reassurance that my supporting materials like letters of recommendation will be received on time? For example, if the admissions committee starts reading applications early, they may look at my application more favorably when they aren't inundated with applications.
Is there an advantage to submitting my application early, besides the reassurance that my supporting materials like letters of recommendation will be received on time?


I'm not aware of any advantage to submitting a math Ph.D. application early, beyond knowing for sure that you have not missed the deadline.  Not even for letters of recommendation, since it's not clear that submitting early will lead your recommenders to do likewise.

There are certainly advantages to preparing your application materials early.  You'll have more time to revise them, and you can share polished versions with your recommenders before the last minute.  However, this is a different issue than whether you should submit them early.  You might as well submit everything once you're done making changes, but there's no need to rush.


  For example, if the admissions committee starts reading applications early, they may look at my application more favorably when they aren't inundated with applications.


I doubt that many people will look at applications early.  For example, it's common to assign each application to one or more faculty members, who are supposed to carry out an initial evaluation.  (This ensures that no application gets overlooked and that the readers have equitable workloads.)  In such a system, there's no advantage to starting early since the assignments will not yet have been made.

Even if someone does start reading early, I'm not convinced they will necessarily evaluate the applications more favorably.  One can imagine the opposite scenario, in which early readers are harsh, in the hopes that the upcoming applications will be better, while later readers have a more realistic assessment of the overall applicant pool.  Ultimately, I have no idea whether early readers would be more or less harsh, but I'd bet the effect would be very small.  If it's unpredictable and small, then it's not worth worrying about.