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Ph D Program Requirements

The Ph.D. Program in Mathematical Sciences Requirements

A student entering the Ph.D. program is assigned an academic advisor who will assist the student in choosing courses for meeting the admission to candidacy requirements. The role of the academic advisor is assumed by the thesis advisor, once the student finds one.

A full-time student must be enrolled in at least three courses each semester. To remain in the program a student must show sufficient progress, which includes achieving an average grade of B- in courses taken each year. The progress of every student is reviewed on a yearly basis by the Graduate Student Retention Committee. Students who are making satisfactory progress towards completion of their doctoral degree and whose teaching/research performance is satisfactory can expect their support to continue for a total of five years. Support past the fifth year is decided on a case-by-case basis as described here

The requirements for obtaining the Ph.D. in Mathematical Sciences are

A. Admission to Doctoral Candidacy [details below]

There are three requirements for admission to doctoral candidacy:

  • Basic Examinations

  • Course Requirements
  • Qualifying Oral Examination

B. Thesis Requirements [details below]

  • Write and publicly defend an original doctoral thesis.


A. Admission to Doctoral Candidacy

Basic Examinations

A student must take and pass two out of the following six Basic Examinations: Each exam is based on the graduate courses whose numbers are given in parentheses. 

  1. General Topology (21-651) and Functional Analysis (21-640)

  2. Measure and Integration (21-720) and Probability (21-721)
  3. Discrete Mathematics (21-701) and Probabilistic Combinatorics (21-737)
  4. Set Theory (21-602) and General Topology (21-651)
  5. Algebra (21-610) and Model Theory (21-603)
  6. Set Theory (21-602) and Algebra (21-610)

Follow these links for Exam Syllabi and Sample Exams.

For instance, a student whose primary interest is Mathematical Analysis would most likely take the exams 1 and 2 while a student with primary interest in Discrete Mathematics would probably take the exams 2 and 3. Important: The subjects on the exams chosen cannot overlap. That is, one cannot obtain credit for both 1 and 4, or for both 4 and 6, or for 5 and 6. Academic advisors will help students make appropriate choices from the various options.

Examinations are offered every January and August. Each basic exam consists of two subjects. To pass a basic exam a student can pass both subject at the same time, or pass each of the subjects separately. Each subject exam is 2 hours long. The next Basic Examination will be offered in January 2014. The exams take place in Wean Hall 7201, from 4:30pm to 6:30pm.  The schedule of the exams is:

Mon, 1/13: Measure and Integration
Wed, 1/15: Probability
Thu, 1/16: General Topology
Fri, 1/17 : Discrete Mathematics

Tue, 1/21 : Functional Analysis
Wed, 1/22: Set Theory
Thu, 1/23: Model Theory
Fri, 1/24 : Probabilistic Combinatorics

Mon, 1/27: Algebra

To register for the examination contact Stella Andreoletti <sd2e@andrew.cmu.edu> by January 2nd. Please specify which subjects you would like to take.

A student must pass at least one of the exams by the end of January of the second year of studies. A student who does not pass a single examination by that time will be supported for the spring semester of the second year, but will not normally be retained in the Ph.D. program beyond that semester.

A student must pass both examinations by the end of January of the third year of studies. A student who has not passed both examinations by that time will be supported for the spring semester of the third year, but will not be retained in the Ph.D. program beyond that semester.

If a student fails a Basic Examination without having taken the appropriate graduate course that prepares for the examination, then the student is required to take the course before being allowed to take the examination again.

Course Requirements

Students are required to complete 6 additional courses, beyond those covering their Basic Examinations maintaining a grade of B- or above in each. Typically these are graduate courses in the Department, at level 700 or above. Other choices of courses may be made, inside or outside the Department, subject in all cases to approval by the student's academic advisor.

The full list of departmental graduate courses is available here. Further information about the courses offered in each research area can be obtained on the pages that describe the Areas of Research represented in the department.

Qualifying Oral Examination

The primary purpose of this examination is to establish the breadth and depth of the student’s knowledge in general areas related to the research area.

The format and content of the Qualifying Oral Examination is decided jointly by the student and the student’s Doctoral Advisory Committee which also administers the examination. At least one month before the scheduled date of the examination a document describing its format and content will be submitted to the Department. 

Every Doctoral Advisory Committee has at least three faculty members and is chaired by a faculty member chosen by the student. The chair must be chosen by the end of the student’s second year in residence.

The format of the Qualifying Oral Examination varies according to subject area. In addition to a part related to the area of the proposed thesis it may include a minor topic deemed to be of interest or relevance. In exceptional cases the committee may choose to make part of the examination written rather than oral.

The examiners may choose to require the student to repeat all or part of the examination. An unsatisfactory performance on the second examination normally results in the student not being retained in the program.

A student in the Department of Mathematical Sciences is expected to have passed his/her qualifying oral exam by the end of his/her third year in residence at CMU.

B. Thesis Requirements

Selecting a Doctoral Thesis Advisor

After admission to doctoral candidacy, a student must select a doctoral thesis advisor. Acting as an academic advisor is not a commitment to act as a doctoral thesis advisor. The advisor/advisee relationship is long-term and not to be entered into casually by either party. It's important to establish a clear understanding of commitment from the start.  

Usually, the thesis advisor is a member of the Department of Mathematical Sciences at Carnegie Mellon. On occasion students are permitted to choose an advisor from outside the Department or even outside the university. However, when the thesis advisor is not a regular or research faculty member at Carnegie Mellon, the head of the Mathematical Sciences Department or the faculty member who supervises the Department's doctoral program shall appoint, after consulting both the candidate and his/her thesis advisor, a faculty member in the Department to serve as the candidate's Departmental sponsor. It is the Departmental sponsor's responsibility to monitor the candidate's work and to assist him/her, the thesis advisor, and the Department in assuring that all work conforms to the candidate's doctoral program. 

After the Qualifying Oral Examination has been passed and a thesis advisor has been selected, a new Doctoral Advisory Committee (which may or may not differ from the previous one) shall be formed. The members will be nominated by the thesis advisor with the agreement of the candidate (and of the Departmental sponsor, if there is one), and their appointment approved by the Department head. The purpose of the committee at this stage is to serve as a resource for the student and to monitor the student's progress. The responsibility for advising the student lies with the thesis advisor. 

The committee may include members from outside the Department. The committee must have at least three members, including the thesis advisor and the Departmental sponsor, if there is one (and thus will have at least one member from the Mathematical Sciences Department). The student is responsible for maintaining contact with the members of the committee. 

The Doctoral Thesis Committee

Often the Doctoral Advisory Committee will serve as the Doctoral Thesis Committee, although this is not required. The duties of the Doctoral Thesis Committee are described below. 

The thesis committee should be appointed no less than two months before the estimated date of the final examination. In order to permit an orderly performance of the committee's functions, it shall be the responsibility of the candidate to keep the committee informed about the progress of his or her work, from the time the committee is appointed to the time the thesis is submitted. The committee may specify whether this should be done individually or collectively by formal or informal presentations. 

The purpose of a Doctoral Thesis Committee is to judge the validity, originality, significance, and proper presentation of the candidate's doctoral thesis. To that end, the committee shall examine the thesis submitted by the candidate, conduct the public oral final examination on the thesis, prescribe corrections or revisions to the thesis before or at the time of the examination, and certify to the dean its finding on the acceptability of the thesis in its final form. 

The Doctoral Thesis Committee shall consist of no fewer than four members, and shall include the thesis advisor, as well as the Departmental sponsor, if there is one. 

At least half of the members of the committee shall be regular or research faculty members in the Department of Mathematical Sciences; one of these, who must be a regular faulty member with the rank of assistant professor or higher, shall chair the committee. If qualified under the preceding provision, the thesis advisor will ordinarily chair the committee; the same is applicable to the Departmental sponsor if there is one.

At least one member of the committee shall be a "visitor," i.e., a person not affiliated with the Department nor with any Department participating in the candidate's thesis research; the thesis advisor may not serve as "visitor." To be eligible to be a "visitor," a person should be familiar with academic standards and procedures and be especially qualified to judge some aspect of the thesis. A "visitor" may come from another Department at Carnegie Mellon, from some other university, or from outside academic institutions altogether. 

A vacancy on the doctoral thesis committee need only be filled if the remaining members would not constitute a valid committee. When a vacancy is filled, care shall be taken that the new committee member has the time and opportunity to participate effectively in the performance of the committee's functions. 

The final examination may proceed only if the committee members present would, by themselves, constitute a valid thesis committee according to the preceding provisions. A committee member is counted as present if he or she participates via a video-conference connection. A member of the committee who is unable to be present at the final examination may, if he or she wishes, submit a written recommendation. 

When the thesis advisor (and the Departmental sponsor if there is one) is satisfied that the thesis is ready, it shall be submitted to the committee. The final examination shall be scheduled so as to provide the committee with two weeks to study the thesis between its submission and the date of the examination.