We are delighted that you are considering Duke Math for your graduate work. To ensure that your application is submitted and reviewed successfully, read the How to Apply page carefully.
Broadly speaking we are looking for graduate students who will engage with and contribute to the department, its research community and its educational mission. When reviewing applications we seek to identify students whose research interests are a good fit for the areas of expertise within the department, who are well prepared for graduate work, and with the potential for independent thesis work.
As you prepare your dossier, keep the following items in mind:
 The letters of recommendation are one of the most influential components of your application: they provide reviewers valuable perspective on your mathematical work to date, and your potential for PhD thesis work. Form relationships with faculty, so that they get to know you and your work; this will help them write strong letters on your behalf.
 Take advanced/reading courses, and if possible take part in an REU, or similar research experience. A strong letter of recommendation from a research supervisor can be very influential.
 Writing a good letter takes time, and you might be surprised by how busy your letter writers are: give them at least a month. A week before the letter is due, send a followup email reminding your writers of the due date(s).

The personal statement is also important. Think of it as an opportunity to introduce yourself as a mathematician: it should summarize your mathematical work/background and interests, and explain why Duke Math is a good fit for you. It is generally beneficial to keep it under two pages.

Indicate on your application any Duke Graduate Math Faculty whose work is close to your current area(s) of interest(s); this will increase the likelyhood that a faculty member in the area will read your application. (It does not obligate you to choose that faculty member as a thesis advisor.)