![]() ![]() In the course of mathematical research, students will inevitably encounter areas in which they have gaps in knowledge. The minor thesis is complementary to the qualifying exam. After passing the qualifying exam students are expected to find a Ph.D. Students are expected to pass the qualifying exam before the end of their second year. More details about the qualifying exams can be found here. Students are required to take the exam at the beginning of the first term. The qualifying examination covers algebra, algebraic geometry, algebraic topology, complex analysis, differential geometry, and real analysis. The department gives the qualifying examination at the beginning of the fall and spring terms. There is no prescribed set of course requirements, but students are required to register and enroll in four courses each term to maintain full-time status with the Graduate School of Arts and Sciences. Please review the program requirements timeline. On the other hand, five years in residence is the maximum usually allowed by the department. The University requires a minimum of two years of academic residence (16 half-courses) for the Ph.D. The presence of other graduate students of comparable ability and level of enthusiasm is also very helpful. In addition, it is not at all trivial to find one’s way through the ever-burgeoning literature of mathematics, and one can go through the stages outlined above with much less lost motion if one has some access to a group of older and more experienced mathematicians who can guide one’s reading, supplement it with seminars and courses, and evaluate one’s first attempts at research. In practice, many of the more subtle aspects of mathematics, such as a sense of taste or relative importance and feeling for a particular subject, are primarily communicated by personal contact. In theory, a future research mathematician should be able to go through all three stages with the help of only a good library. Students are expected to take the initiative in pacing themselves through the Ph.D. Making a first original contribution to mathematics within this chosen special area.Choosing a field of specialization within mathematics and obtaining enough knowledge of this specialized field to arrive at the point of current thinking.Acquiring a broad basic knowledge of mathematics on which to build a future mathematical culture and more detailed knowledge of a field of specialization.The stages in this program may be described as follows: dissertation involving some original research is a fundamental part of the program. 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. ![]() program of the Harvard Department of Mathematics is designed to help motivated students develop their understanding and enjoyment of mathematics. ![]()
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