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Stage 2

This stage commences with successful completion of the Qualifying Review and finishes with admission to Candidacy. Rackham rules require that students achieve candidacy by the end of the third year.


Math Ph.D. students will receive course counseling from a member of the Doctoral Committee until they match with their advisor. The Graduate Program Coordinator will process departmental and Rackham paperwork.  Information about minimum credits, cross-listed courses, etc. can be found in this Course Enrollment section of the Graduate Student Handbook.

Finding an Advisor

The most important task of the Stage II student is to get settled working with a dissertation advisor. Students will make progress toward the goal of finding an advisor by attending seminars and actively talking about mathematics with a wide range of faculty and students. It is never too early to begin thinking about what field of study and what potential advisors might be of interest, even for students who have not yet entered Stage II.   With permission of a potential advisor, students may take a 3-credit Math 700 Reading/Research course during the Fall and/or Winter semesters.  Math 700 does not fulfill distribution requirements.

The Chair of the Doctoral Committee consults with each student upon completing the Qualifying Review to discuss the various other requirements for candidacy and offer guidance about finding an advisor. The Chair of the Doctoral Committee is a good person to consult if students experience difficulty finding a dissertation advisor, or if they discover at some point later that they need to change advisors for some reason. Changing advisors after becoming a candidate is unusual and requires the approval of the Chair of the Doctoral Committee. Students should also alert the graduate office about any changes and difficulties.

The Dissertation Advisor, once the relationship is established, serves as the student’s primary academic advisor, although the Chair of the Doctoral Committee as well as the office staff are always available to explain some of the details of the requirements for candidacy.

It is always a good strategy to talk with many faculty and build mathematical and personal relationships with faculty beyond the advisor. These faculty may serve as prelim or dissertation committee members in the future, as informal mentors, or even as collaborators.

Requirements for Candidacy

Admission to Candidacy is contingent upon completing the following three requirements:

  1. Satisfying a Distribution Requirement of 6 600+ level courses in 3 out of 5 areas to ensure breadth of knowledge within mathematics. 
  2. Completion of the Cognate Requirement (four credits in non-math courses). 
  3. Passing the Preliminary Examination.

I.  Distribution Requirement

Each student must earn a grade of at least B- in six advanced mathematics courses. These must be chosen from three of the following five areas:

  1. algebra, algebraic geometry, algebraic number theory
  2. analysis, analytic number theory, probability
  3. topology, differential geometry
  4. applied analysis, numerical analysis
  5. applied discrete mathematics, combinatorics, logic

Eligible courses include those at the 600 level or above. Math 700 reading courses may not be used toward the distribution requirement. Certain advanced 500 level courses may also be eligible, subject to prior approval by the Chair of the Doctoral Committee. Core "alpha" courses (or any substitute) are typically not eligible, but a student who has already taken four alpha courses may count one or two additional alpha courses towards the distribution requirement. With the approval of the Chair of the Doctoral Committee, certain courses taken outside the department, for example in physics, may be allowed to count for the distribution requirement (under area 4).

II. Cognate Requirement

The Graduate School requires that every student successfully complete four hours of cognate courses before achieving candidacy.

These may be either:

  • Rackham Graduate Level Approved Courses (typically 400 or above) offered by a Department other than the Mathematics Department, including cross-listed courses such as Math 625/Stats 625. Please register for these courses under the Cognate Department i.e. Stats 625 rather than Math 625.
  • Courses at the 500 level or above offered by the Mathematics Department for which a grade of B or better is earned, provided these courses:
    • Treat ideas, techniques, or patterns of problem solving distinctly different from those of the student's major area.
    • Involve significant intellectual content that is important in an area of science other than Mathematics.
    • Are approved in advance by the Doctoral Committee of the Mathematics Department.

For courses taken within the Mathematics Department, a written statement must be provided by the student at the time approval is sought, cosigned by the student's advisor, explaining how the specific course meets conditions (1) and (2). In certain cases, the Doctoral Committee may be willing to approve a mathematics course as a cognate after it has been taken, provided the course clearly satisfies the conditions (1) and (2). This Cognate Approval Form must be completed by the student, approved by the course instructor, thesis advisor, and Doctoral Chair and turned into the Graduate Office.

It is worth noting that the LSA required professional ethics course, UC 415, provides 1 cognate credit. Typically, Ph.D. students need only three credits of cognate credit beyond this required course.

III.  Preliminary Exam

Soon after finding an advisor, students should prepare a long-range plan of study in consultation with their dissertation advisor.  This means making tentative choices about the courses, papers, and topics the student will study in preparation for the Preliminary Examination.  The Preliminary Examination is the most important requirement for achieving Candidacy.  This is an oral exam on a list of advanced topics, as well as topics deemed preparatory for research in the chosen specialty, chosen by the student in consultation with their advisor.  The amount of material should be equal to two advanced courses and may include books or papers beyond coursework as well as some of the student’s own research.  The exam syllabus must be approved by the Preliminary Examination Committee, which consists of the advisor, one further examiner, and the Chair of the Doctoral Committee.  Students are expected to complete their Preliminary Examination and advance to Candidacy by the start of the fourth year

Term Wish to Achieve Candidacy Prelim Window
Fall 2023 Saturday, January 7, 2023 through Wednesday, May 31, 2023 OR Monday, August 28, 2023 through Wednesday, August 30, 2023
Winter 2024 Thursday, August 31, 2023 through Friday, January 12, 2024
Fall 2024 Saturday, January 13, 2024 through Friday, May 31, 2024 OR during the first three days of Fall classes.

Once the Preliminary Examination is completed, a graduate student will be eligible to advance to candidacy. The Rackham Graduate School mandates that a student in the candidacy stage of the Ph.D. degree must enroll each fall and winter term for 8 credits of Math 995 under the supervision of their advisor. Students may elect to register for additional courses not totaling more than 4 credits per term. Any coursework beyond that may result in tuition charges for which the student would be responsible to pay.