Doctor of Philosophy

General Requirements
In addition to satisfying the University regulations, the candidate for the Doctor of Philosophy (Ph.D.) must:

1. Complete 72 credit hours with minimum of six full terms in residence.
2. Complete at least 10 courses at the 2000/3000 level with an average grade of B or better, excluding the Orientation Seminars, Independent Study, and those courses designated as preparatory to the Preliminary Examination.
3. Achieve an overall course average of B or better.
4. Pass a written Preliminary Examination.
5. Pass a Comprehensive Examination.
6. Complete an oral Overview/Prospectus Examination and be admitted to candidacy.
7. Write and defend a dissertation embodying an extended original investigation of a problem of significance in an area of mathematics.
8. Participate in a teaching assignment of at least one lecture or recitation section per term for a minimum of two terms. This requirement can be waived by the Graduate Committee.

It is, of course, the satisfactory writing of the dissertation that is the most important goal of the Ph.D. candidate. All other requirements, except that of item 8, support this objective.

The 72 credit hours may be obtained through formal course work, directed study, and thesis research. Although most of this will come from the Department, a student may take courses from other University disciplines if they are in areas relevant to the proposed program of study. The advisor must submit a request to the Graduate Committee prior to course registration.

Special requirements for a Ph.D. in Mathematics:
To receive a PhD in mathematics a student must complete at least 10 mathematics courses at the 2000 and 3000 level with grade B or better. This excludes reading courses, directed study courses and the (preparatory courses for the Ph.D. Preliminary Examination - Progress in Mathematics, Matrices & Linear Operators I and II - Math 2020, 2370, 2371)
These courses must include at least three one year sequences chosen from a list developed and kept up to date by the Graduate Committee. The courses should adhere to a coherent plan of study for the students interests. The plan must be designed by the students advisor with input from the student and consulting examples of such plans of study. It is to be submitted to the graduate committee for approval.
Up to two courses outside the mathematics department are permitted in these ten courses provided all the following are satisfied:

• The programmatic necessity for the course is explained in writing to the Graduate Committee by the students advisor and the courses are approved by the Graduate Committee.
  
• The courses are at the 2000 and 3000 level and are completed with a grade of B or better.
  

The course requirement at the 2000/3000 level is to insure that the graduating Ph.D. has a reasonable breadth in mathematics. Students with advanced degrees may be allowed credit for up to four courses.

The Overview/Prospectus involves the formation of the student's doctoral committee and the presentation of a proposed research topic. The research should lead to a dissertation describing an extended original investigation and containing a definite contribution to mathematics. The thesis defense constitutes the final step in achieving the Ph.D.

A well-prepared student who is unencumbered by other obligations can complete the Ph.D. requirements in four years. This can be done by enrolling for six courses per year, passing the Preliminary Evaluation after one year, the Comprehensive after two years, and the Overview/Prospectus at the end of the third year. The dissertation is then written during the final year. For most students this ideal minimum time will not be possible. However, no more than two years should be needed to write the thesis and finish the remaining requirements. Thus a maximum of five years is anticipated to accomplish the doctorate. In exceptional cases the Graduate Committee will grant an additional year.

For part-time students, no rigid schedule can be imposed. Such candidates are encouraged to complete the requirements as rapidly as possible, but the only upper limit is the University's statute of limitations of 10 calendar years from the date of admission to graduate study.

The Ph.D. Preliminary Examination (PE)
According to the Graduate and Professional Bulletin as published in the FAS Graduate Bulletin, the Preliminary Examination ``should be designed to assess the breadth of the student's knowledge of the discipline, the student's achievement during the first year of graduate study, and the potential to apply research methods independently. Students who have full graduate status should be evaluated formally by approximately the end of the first year of full residence''. The PE ``is used to identify those students who may be expected to complete a doctoral program successfully and also to reveal areas of weakness in the student's preparation''.

Accordingly, the examination will cover advanced undergraduate and beginning graduate material; specifically, material covered in Math 1530 and 1540 (Advanced Calculus I and II) and Math 2370 and 2371 (Matrices and Linear Operators I and II). The PE is given in April (in conjunction with the final exams for Math 1540 and Math 2371) and August of each academic year; it is a written examination, and consists of questions pertaining to the areas studied in the four courses just mentioned. The examination is prepared by a committee in which recent instructors of Math 1530, 1540, 2370 and 2371 will be involved, and consists of one single set of questions related to Advanced Calculus and Linear Algebra. It is based on a syllabus previously prepared and announced by the committee. Sample exams are available.

The examination is given in two sessions of three hours each, and will be graded as a unit by the entire committee that prepares the questions. A student will receive credit for passing one of the exams if he or she receives a passing grade on this exam and scores above a certain threshold in the other exam. The student will then need to pass the other exam in the next two times the exams are offered. If not successful, the student will lose the credit for passing the first exam. Each time the exams are offered, the exam committee will determine for each of the exams both the passing score and the acceptable threshold.

Under normal circumstances Ph.D. students are allowed to attempt the Preliminary Exam for a period of three years, starting from the time of their enrollment. If a student has not passed the Preliminary Exam by the beginning of the fourth year, his/her enrollment in the Ph.D. program will be discontinued. At that time the student will have an option of switching his/her degree objective to M.A. or M.S.

The passing of the PE automatically fulfills the M.A. Comprehensive Examination requirement. Thus, a student who has passed the PE and satisfied items 1 and 2 under General Requirements for the Master's degree is entitled to a M.A. degree.

The Ph.D. Comprehensive Examination (CE)
Thet Graduate and Professional Bulletin states that ``The Comprehensive Examination should be designed to assess the student's mastery of the general field of doctoral study, the student's acquisition of both depth and breadth in the area of specialization within the general field, and the ability to use the research methods of the discipline. It should be administered at approximately the time of the completion of the formal course requirements and should be passed at least eight months before the scheduling of the final oral examination and dissertation defense. In no case may the comprehensive examination be taken in the same term in which the student is graduated.''

In contrast to the Preliminary Examination, the CE is suited to the individual needs of each student and is created and administered by a specially formed examining committee of three faculty members which includes the student's advisor. The CE should maintain a proper balance between specialization and generality.

The CE will consist of a written component and, at the discretion of the examining committee, a subsequent oral component.

The content of these components should be determined by the examining committee taking into consideration the chief interests of the student. The examining committee shall also decide whether the student has passed the CE. Prior to administering the examination, the examining committee will submit an outline of the exam's areas and the questions comprising its written component to the Graduate Committee for approval.

It is the responsibility of the student to verify that the necessary approvals have been secured before taking the CE.

Students are expected to pass the Comprehensive Examination by the end of the their third year of full-time study.

Overview/Prospectus Examination and Admission to Candidacy
After passing the Comprehensive Examination the student should have secured a prospective thesis advisor in an area in which he or she wishes to perform research. The thesis advisor will guide the student in further course work and directed study in the appropriate areas. At this time the student may begin research toward the doctoral thesis.

In contrast to both the Preliminary and Comprehensive Examinations, the Overview Examination is an oral examination whose form may vary from student to student. In order to take the overview a student must have worked long enough with her/his advisor to have begun work on a thesis topic. At a mutually agreed-upon time (but no later than one year after the CE and at least eight months prior to the thesis defense) the thesis advisor will form an overview committee consisting of at least four members of the Graduate Faculty. At least one member must be from outside the Department. If a committe member is not a member of the University of Pittsburgh faculty, prior approval must be granted by the Graduate FAS office. The Graduate Committee will be informed in writing and must approve the selection prior to the overview.

The exam begins with a presentation of approximately one hour in which the student discusses her/his thesis topic and gives some ideas on where the research may lead. Then there is a period for questions and comments by the committee. Since this question and comment period can take a variety of forms, the length and format of this exam cannot be predicted.

If the student is deemed to have passed this examination, admission to candidacy forms are submitted by the examining committee to the Graduate Committee. Normally, the Ph.D. Thesis Committee for the Final Oral or Defense of Thesis will be the same as the Overview Committee. If there is a change in committee members, then the student's advisor must request this in writing, and the Graduate Committee must approve it, as well as the Dean of FAS.

If the committee is not pleased with the student's performance, then he or she will be asked to retake the examination at a later date. After this second examination, the committee will recommend whether the student will be allowed to continue toward the Ph.D..

Thesis Defense
The Thesis Defense is conducted by the student's doctoral committee as outlined in the the Graduate and Professional Bulletin. Successful completion of this final examination marks the end of studies for the Ph.D. degree

Thesis Format
The proper format for a doctoral dissertation at the University of Pittsburgh is described in detail in the Style and Form Manual.