Stage 3 begins with promotion to Candidacy and finishes with the successful completion of the dissertation.

The main goal of Stage 3 is to conduct mathematical research, and then write up the results into a doctoral dissertation. This is the most difficult part of the program. No student should expect to complete it in less than twenty-four months.

### Thesis Research

Students normally will be well-prepared to begin thesis research after the intense study they conducted in preparation for their prelim exam. Still, research requires patience and perseverance, and the time required to complete this stage of the Program should not be underestimated.

Most students find that doing research is very different from attending courses and participating in seminars. Students may feel mathematically alone for the first time, trying to develop ideas from unfamiliar material without the security of knowing that the answers have been worked previously by others. Students should take the lead in maintaining regular contact with the dissertation advisor, even when---or perhaps especially when--- they feel like they are not making progress. Discussing mathematics widely with senior mathematicians and with peers is essential to mastering techniques and new tricks in the ever-evolving literature, and it will also make research more sociable and fun.

### Post Candidacy Course Requirement

Upon entering Candidacy, students are required to take one graduate mathematics course for credit during the first two semesters in residence. This course may NOT be a Reading Course (MATH 700).

### Dissertation

It is important that the student communicate regularly with the Thesis Advisor about their research progress and post-Ph.D. career plans. Eventually, when the Thesis Advisor is satisfied that a student has done sufficient research for the dissertation, the student prepares a draft. This usually takes at least a full semester to write.

Students should formulate a Dissertation Committee by the time they begin writing the thesis. Thesis committee members can often make valuable contributions to the mathematics and its presentation, so it is important that students seek feedback from their committee on the drafts.

As the dissertation nears completion, students should schedule a Thesis Defense. Before the defense can proceed, two committee members (one of whom must be the Thesis Advisor) submit independent evaluations (“reader’s reports”) of the thesis to the Doctoral Committee, on which basis the Committee votes to allow the defense to proceed or not. Students select these readers in consultation with their Thesis Advisor. In addition, all committee members must submit formal approval to Rackham. It is important that students provide a complete draft to the committee with plenty of time for them to make a critical evaluation.

There are various steps surrounding the production of the thesis that the student must carry out with Rackham in a timely way throughout their final semester. It is important that the student inform the math Graduate Office at the beginning of the semester they plan to defend, and pay careful attention to all communication from the graduate office and from Rackham regarding the format check and other details. Steps are detailed in the Dissertation Defense Timeline document.

### Dissertation Committee

Students should assemble their Thesis Committee at least six months before their planned defense. This is done in consultation with their Thesis Advisor, who will serve as Chair of the Dissertation Committee, and according to the rules prescribed by Rackham. Students should complete the Dissertation Committee Form and submit it to the math Graduate Office. The math Graduate Office will then submit the formal petition for approval to Rackham.

The Dissertation Committee consists of at least four but typically five members. One of these, the “cognate member,” must be a graduate faculty member from a University of Michigan department other than mathematics, and at least two should be from the math department, though typically all remaining members are. Mathematicians from outside the University of Michigan can also serve with special permission from Rackham. (See the math Graduate Office for details on Special Members.) The Dissertation Committee is usually chaired by the Thesis Advisor and includes both readers of the thesis.

Students are well advised to choose the Committee early so they may be consulted as the research progresses.

### Thesis Defense

When the thesis is in final form, the student is examined orally on its contents and related topics by the Dissertation Committee. The thesis should be given to the Dissertation Committee at least one month before the Thesis Defense.

The math graduate office helps the student plan the time and place of the defense and submit the necessary paperwork. It is advisable to make these arrangements as far in advance as possible as a matter of courtesy and to avoid dates when members of the Committee may be out of town. There are several other important deadlines which must be met before a Dissertation Defense can be scheduled---see the "Dissertation Handbook''. Students should always pay attention to communication from the math graduate office and from Rackham, so they do not miss any important deadlines.