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 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. The presence of other students of comparable ability and level of enthusiasm is also very helpful. The University requires a minimum of two years academic residence for the Ph.D. degree. On the other hand, five years in residence is the maximum usually allowed by the Department of Mathematics.
Many students finish in four years.
This is strongly encouraged. A fifth year is allowed if during the fourth year there is some positive progress towards a thesis, meaning written results comprising a significant fraction of a thesis and sufficient to give the advisor confidence that the student will complete the thesis within a year, assuming steady progress thesis has been made. If a student has not completed her or his dissertation while here, she or he can finish it elsewhere and submit it at a later date. The qualifying examination is given in three installments on three consecutive days, at least once a year, early in the term. One's first goal should be to bring one's basic knowledge up to such a point that one can pass it.