The Ph.D. Program in Mathematical Sciences Requirements
For students enrolling in 2025 and later
The requirements are admission to doctoral candidacy, submission of a thesis, and fulfillment of teaching and language requirements.
A student entering the doctoral program will be assigned an academic advisor, who will assist the student in meeting the requirements for admission to candidacy. Once the student has found a thesis advisor, they will assume the role of the academic advisor.
A full-time student must be enrolled for at least 36 units each semester. To remain in the program a student must show sufficient progress. The progress of every student is reviewed twice a year by the Graduate Student Retention Committee. Students who are making satisfactory progress towards completion of their doctoral degree, and whose performance in teaching is satisfactory, can expect their support to continue for a total of five years. Support for a sixth year is decided on a case-by-case basis.
A. Admission to Candidacy
There are four requirements for admission to candidacy (note that the ACO program has its own requirements for admission to candidacy, which are slightly different from those given here. Details can be found at the ACO program page)
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- Fulfillment of the Breadth Requirement.
- Passing a set of Basic Examinations.
- Fulfillment of the Course Requirements.
- Passing the Oral Qualifying Examination.
The six areas of the Breadth Requirement are: algebra, analysis, discrete math, logic, probability, and topology. Students are required to demonstrate competence in at least four of these areas, either by successfully completing an associated course or by passing the corresponding Basic Exam. The associated courses for each area are:
- Algebra: 21-610 (Algebra)
- Analysis: 21-720 (Measure and Integration)
- Discrete Math: 21-701 (Discrete Mathematics)
- Logic: 21-602 (Introduction to Set Theory I) or 21-603 (Model Theory I)
- Probability: 21-721 (Probability)
- Topology: 21-651 (General Topology)
Successful completion of a course requires a grade of B- or above. It is not permitted to satisfy the Breadth Requirement by taking a course with a Pass/Fail grading scheme.
Students are expected to complete the Breadth Requirement by the end of the fifth semester. Students will be evaluated at the end of the fourth semester and may be placed on academic probation if deemed unlikely to complete the Breadth Requirement in the expected timeframe. Students may be removed from the program if the Breadth Requirement is not completed after one semester of probation.
There will be six Basic Exams, offered in the same areas as the Breadth Requirement: algebra, analysis, discrete math, logic, probability, and topology. Students are required to pass at least two Basic Exams by the beginning of the third semester. If two Basic Exams are not passed by this time, students will be placed on academic probation. Students may be removed from the program if the Basic Exams are not passed after one semester of probation.
Basic Exams will satisfy at least two of the four Breadth Requirements. Students may pass additional Basic Exams to fulfill remaining Breadth Requirements.
Each examination is three hours long. Examinations are offered at the start of each semester. Their schedule and instructions for registration will be sent by email. Usually one needs to register by writing to Christine Gilchrist at least 10 days before the start of the semester.
Students are required to successfully complete 60 units of advanced graduate courses in mathematics, or graduate courses with sufficiently strong mathematical content. A course may not be used to satisfy the Course Requirement if that course, or the corresponding Basic Exam, was used to satisfy the Breadth Requirement. Moreover, Reading and Research, Doctoral Thesis Research, Practicum in Mathematical Sciences, and Internship in Mathematical Science cannot be used to fulfill the Course Requirement.
A typical semester-long course used to satisfy this requirement will be 12 units, while a corresponding mini course will be 6 units. Successful completion of a course requires a grade of B- or above. It is not permitted to satisfy the Course Requirement by taking a course with a Pass/Fail grading scheme.
After passing the Basic Exams, students are expected to choose a Doctoral Advisory Committee and select a chair by their fourth semester. In consultation with this Committee, students will create a syllabus for the Qualifying Oral Exam, consisting of a major area and at least one minor area. 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 fourth semester of graduate studies.
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.
Students are expected to pass the Qualifying Oral Exam by the end of the fifth semester. If the student fails the exam, they are placed on academic probation. Students may be removed from the program if the Qualifying Oral Exam is not passed in the subsequent semester. Only two attempts are permitted, and the committee cannot change between attempts (unless approved by the Department Head or the Director of Graduate Studies).
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 shall appoint, after consulting both the candidate and their 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 the candidate, 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 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 thesis 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 faculty 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", that is 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.
C. Teaching and English Language Requirements
All graduate students are required to perform the duties of a teaching assistant (TA) for at least one semester before receiving a doctoral degree. Students will benefit from the experience gained by explaining mathematical concepts in an efficient and understandable way and by responding to questions. The classroom performance of TAs is monitored by the departmental TA Supervisor. Students will receive feedback on their teaching performance based on comments received by students and supervising faculty, and on classroom observation by the TA Supervisor.
Students are required to complete 21-605 (Teaching Mathematics). In addition, students are required to conduct recitation or teach a course for at least one semester. Non-native English speakers must pass the International TA (ITA) Language Test by the end of semester 4.
Proficiency in spoken English is extremely important for communicating mathematical ideas. After enrolling at 一本道无码, students whose native language is not English are required to take the International Teaching Assistant (ITA) Test administered by the Intercultural Communications Center (ICC) at 一本道无码. Students are required to take this test in order to be certified as Teaching Assistants. Students whose native language is not English must receive either a score of PASS or RESTRICTED I by the end of their second year in order to receive financial support from the department.
Students who have not yet received a score of PASS or RESTRICTED I are required to attend workshops and training sessions at the ICC.