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In less than a year from now I'll be applying to pure math PhD programs. I'm going to have a competitive application. I'm attending a well regarded school and will no doubt get good reccomendations; I took a whole slew of math undergrad courses receiving all A's; I took (and am taking) a handful of graduate courses receiving a mix of A's and B's; TA'd some undergrad courses; I have two solid REUs; participated in Budapest semesters in mathematics, etc

But these are all things that every top (20 or so) graduate school is going to expect from all it's applicants. I don't have any research independent of my REUs, I've never done notably well on the Putnam, and so on. My main concern is that I'm not going to stand out in any way to a graduate school admissions committee.

Looking at application threads, this seems quite worrisome. There are consistently applicants in a similar position as I who get almost universal rejection from good graduate programs. I love mathematics and so that I can pursue a fruitful research career someday, I want to attend a serious graduate program.

What can I do in under a year to improve my graduate application? How can I stand out?

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    But these are all things that every top (20 or so) graduate school is going to expect from all it's applicants. This is just false. Top 5 schools okay, top 20 no.
    – Kimball
    Mar 7, 2015 at 12:31
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    Your score on the Putnam is irrelevant. Having original research outside of any REU is not crucial either. Are you writing a senior thesis? The recommendation letters are the key. Why don't you seek advice from faculty at your school instead of strangers on the internet? They'd know better than us where students from your school have gone for graduate school and what the background of those past students was.
    – KCd
    Mar 7, 2015 at 13:41
  • @Kcd I'm away from my school at BSM now, so I figured this was a good alternative. I plan on writing a thesis that will synthesize a couple of recent papers in a very active field which is of great interest to me, but I so far have no original contributions which would be seen as nontrivial in the eyes of anyone who has any experience in the field.
    – user31409
    Mar 7, 2015 at 15:26
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    Relax. You don't need original contributions in math as an undergrad to get into top PhD programs. It's more important to learn a lot of nontrivial math and have letters from good people. Write to faculty at your home institution with your concerns.
    – KCd
    Mar 7, 2015 at 19:49
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    Showing that you understand that mathematics is not a "school subject", and that you think about it on your own, of your own volition, are willing to take risks, are curious, ... as opposed to success at simple compliance and obedience ... matter a great deal to some admissions committee people.
    – paul garrett
    Mar 8, 2015 at 0:42

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Consider the viewpoint of the admissions committee. How would they know who to pick? One way is to trust others opinions -- this is why they seek recommendation letters from those who know your abilities.

If you still want to "stand out", and aren't satisfied with your performance in the standard ways (grades, exams, REU programs), consider what makes you different. You should seek advice from those who know you better than us for ideas here.

In addition, if you're afraid of not getting in anywhere -- apply to a spectrum of schools, and simply pick the top one(s) you are accepted to. Having backups is a good way to be safe.

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