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Stage 2

This stage commences with successful completion of the Qualifying Review and finishes with admission to Candidacy. Most students complete this phase in 12 to 18 months.


After passing the Qualifying Review Examination, students consult with the Chair of the Doctoral Committee about the choice of a Advisor. Ideally, the Advisor will supervise the student through the completion of the student's degree, but the student may make a change of Advisor, after consulting with the Chair of the Doctoral Committee (if for example, the student's field of interest changes).

Long Range Plan of Study

Each student entering Stage 2 should prepare a long-range study plan in consultation with his or her Advisor. This means making tentative choices about the topics for the Preliminary Examination (including a detailed list of subjects) as well as for a foreign language and cognates.

Requirements for Candidacy

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

  1. Knowledge of six mathematics courses chosen from at least three of five areas.
  2. Passing the Preliminary Examination.
  3. Passing the Departmental Reading Examination in one of the following three languages---French, German, Russian.
  4. Completion of four hours of cognate courses.

The Course Requirement

Each student must earn a grade of at least B- in six advanced mathematics courses, in addition to core courses or substitutes used to fulfill the requirements for the Qualifying Review, and 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 and the more advanced 500 level courses. Transfer students may petition the Doctoral Committee to receive credit for upper level courses taken at another institution. With the approval of the Doctoral Committee, certain courses taken outside the department, for example, in physics may be allowed to count for this requirement (under area 4).

The department may allow the student to proceed to candidacy before fulfilling this course distribution requirement. If the Chair of the Doctoral Committee determines that the student has not fulfilled the requirement at the time the student applies for candidacy, he or she will notify the student, who must then meet with him to agree on a program to complete the requirement.

Preliminary Examination

This examination is normally in the area of the student's thesis. The material covered should be equivalent to at least two 600 or 700 level courses, but should include material the student has learned individually.

In order to pass the Preliminary Examination the student must pass an oral examination. The Preliminary Examination is administered by a Preliminary Examination Committee and is based on a detailed list of topics prepared by the student in consultation with the advisor. The advisor usually serves as Chair of the Committee. The advisor also advises the student on the choice of an additional faculty member of the Committee. The list of subjects for the oral examination must be approved and initialed by all members of the Preliminary Examination Committee and by the Chair of the Doctoral Committee before the examination can take place.

When a date for the Examination has been agreed on by the student and examiners, it should be reported to the Graduate Studies Office, which will arrange for a room and officially remind each examiner as the date approaches.

Students who fail the Preliminary Examination are granted at most one opportunity to take it again. This exam must be passed by the beginning of student's eighth term in the program.

Language Requirement

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. The examiners may be willing to give exams at other times if students have a special need. Students are strongly advised to satisfy this requirement as early in their program as possible.

The department recommends that students develop some sort of proficiency in all three languages.

Cognate Requirements

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

These may be either:

  • Graduate courses offered by a Department other than the Mathematics Department, or
  • 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
    1. Treat ideas, techniques, or patterns of problem solving distinctly different from those of the student's major area;
    2. involve significant intellectual content that is important in an area of science other than Mathematics;
    3. 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).

A document giving advice on the selection of cognate courses is available from the Graduate Office.