The doctorate in mathematics is a degree which certifies both a high level of scholarship and the ability to make original contributions in one's own field. Students must take several advanced courses, pass certain qualifying exams, and write a dissertation containing original research. The course of study falls naturally into 3 stages, although in some cases, these stages may be passed over or merged together. Most students spend their first two years taking basic courses and passing various written qualifying exams. They also use this time to increase their breadth of knowledge in mathematics.
Students take courses, read, and talk with fellow students and faculty.
By the end of this first stage, students have a fairly clear idea of the general area of mathematics for their dissertation topic. Students with superior undergraduate educations or prior graduate experience can often shorten considerably this first stage of graduate study. If students know their principle research area, they should take advanced courses as soon as possible. This process involves taking advanced courses, sitting in seminars, enrolling in reading courses with individual faculty members, talking with other students, and general reading and thinking. Most successful students have a good idea of their research area and problem within a year of passing their qualifying exams.