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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.

Finding an Advisor

The most important task of the Stage II student is to get settled working with a dissertation advisor. Finding an advisor can be stressful, but by attending seminars and by actively talking about mathematics with a wide range of faculty and students, students will make progress towards this goal. Of course, 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.

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 when 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. 

Once you have obtained a dissertation advisor, please notify the Graduate Office of Student Services.

Requirements for Candidacy

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

  1. Passing the Preliminary Examination.
  2. Satisfying a Distribution Requirement to ensure breadth of knowledge within mathematics. 
  3. Passing the departmental reading Language Examination in one of the following three languages: French, German, or Russian.
  4. Completion of the Cognate Requirement (four credits in non-math courses). 

I.  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 Preliminary Examination by the end of the third year.  

II.  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. Certain advanced 500 level courses may also be eligible, subject to prior approval by the Chair of the Doctoral Committee. However, core "alpha" courses are not eligible, nor is any course used to satisfy the QR 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).

III. Language Examination

Students in the Doctoral Program are required to pass Departmental Reading Examinations in one of the following languages:

  • French
  • German
  • Russian

These examinations are given twice a year, in mid-October and mid-February, by members of the faculty. Students are strongly advised to satisfy this requirement as early in their program as possible.

IV.  Cognate Requirement

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

These may be either:

  • Graduate 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 required training course for GSIs, Math 993, typically taken by incoming students upon arrival in August, provides 1 credit. Likewise, the required professional ethics course, UC 403, also provides 1 credit. Typically, Ph.D. students need only two credits of cognate credit beyond these required courses.