"The Ph.D. program of the Harvard Department of Mathematics is designed to help motivated students develop their understanding and enjoyment of mathematics. Enjoyment and understanding of the subject, as well as enthusiasm in teaching it, are greater when one is actively thinking about mathematics in one’s own way. For this reason, a Ph.D. dissertation involving some original research is a fundamental part of the program. The stages in this program may be described as follows:

- Acquiring a broad basic knowledge of mathematics on which to build a future mathematical culture and more detailed knowledge of a field of specialization.

- Choosing a field of specialization within mathematics and obtaining enough knowledge of this specialized field to arrive at the point of current thinking.

- Making a first original contribution to mathematics within this chosen special area.

Students are expected to take the initiative in pacing themselves through the Ph.D. program. In theory, a future research mathematician should be able to go through all three stages with the help of only a good library."

https://www.math.harvard.edu/graduate/guide-to-graduate-study/