The doctoral program in Mathematics is designed to educate students in a broad range of mathematical disciplines, with emphasis on the student gradually acquiring expertise in a particular area of inquiry. While the program is particularly strong in research training, all doctoral students gain practical skills in teaching and are prepared upon graduation to assume positions in academia as teachers as well as researchers.

During their first year in the program, students typically engage in coursework and seminars which prepare them for Qualifying Examinations. This series of three exams test students breadth of mathematical knowledge in Algebra and Real Analysis. Students are given two opportunities to take and pass all sections of the Qualifying Examinations, which are given in September and March each year. Incoming doctoral students may elect to take the September exams with no penalty- that is, if the incoming student takes and passes any portion of the Examinations, s/he is given credit for the pass and still has two more opportunities to pass other sections of the exam s/he did not pass. Conversely, an incoming student who takes and does not pass any portion of the September exam is not penalized and may still take the Qualifying Examinations twice more.

All students continue through each year of the program serving some form of Assistantship (Course, Teaching or Research Assistant.) Stanford's mathematics graduate students are very active as both leaders and participants in seminars and colloquia in their chosen areas of interest.

During their third year, students will sit for their Departmental Area Examination. This examination will assess the student's breadth of knowledge in his/her particular area of research. The Area Examination is also used as an opportunity for the student to present his/her committee with a summary of research conducted to date as well as a detailed plan for the remaining research.

Typically during the latter part of the fourth or early part of the fifth year of study, students are expected to finish their dissertation research. At this time, students will defend their dissertation as they sit for their University Oral Examination. Following the dissertation defense, students take a short time to make final revisions to their actual papers and submit the dissertation to their reading committee for final approval.