The degree of Doctor of Philosophy in Mathematics is an advanced degree designed for those who wish to pursue a career involving mathematical research. It is conferred in recognition of marked ability and high attainment in advanced mathematics, including the successful completion of an original research project of merit. The Ph.D. program is designed to guide students, year-by-year, toward becoming research mathematicians. Typically the program takes five years to complete, including the dissertation (although it can be completed in less time, depending on the student). There are several stages to the Ph.D. program. The first, which is centered on the course requirements and the graduate preliminary exam, is designed to help the student acquire a broad background in mathematics. The second stage includes the oral Ph.D. preliminary exam on two topics of the student's choosing, and on participation in seminars. Breadth is one of the goals of this second stage, but the main objective at that point is to assist the student in choosing a field of specialization and in obtaining sufficient knowledge of this specialized field, including recent research developments. Ph.D. students also have an opportunity to earn a masters degree at this stage. The third and key stage of the Ph.D. program is the dissertation (or Ph.D. thesis), in which the student will make an original contribution to mathematics. The entire Ph.D. program is designed to help students move toward taking this significant step in creating new mathematics. Along the way the student is required to acquire some teaching experience.