# Ph.D. Regulations

The regulations discussed below include the requirements established by the Department of Mathematics as well as the most commonly applicable of the Graduate School’s requirements. The latter appear in the publications entitled Graduate Catalog and Theses and Dissertation Guidelines. Students should read both of these publications thoroughly and be aware of the requirements stated therein.

## The standard timeline of a typical doctoral program can be found here.

### I. ADVISOR AND DOCTORAL COMMITTEE

Students in the mathematics Ph.D. program take courses and prepare for their preliminary and qualifying examinations during the first two to three years of graduate work. After passing the qualifying examination they take additional courses and write and defend their Ph.D. dissertation. Throughout this process, each student is guided by an adviser.

A new student is assigned an initial adviser, usually the Director of Graduate Studies. Eventually the student will decide upon an area of mathematics in which to specialize and will choose an adviser to supervise research in that area; this decision should be made no later than the end of the second year.

The student’s doctoral committee (also known as the qualifying committee) is to be formed shortly after the student has chosen an adviser, and no later than two weeks after the beginning of the third year. The doctoral committee is to contain four members of the mathematics graduate faculty and one member of the graduate faculty of a department other than mathematics. (For a list of graduate faculty members, consult the aforementioned Bulletin.) Usually the committee will be chaired by the student’s adviser. It is possible, under unusual circumstances, to have a sixth member placed on the committee. The final step in the committee formation process is to have the DGS certify compliance with graduate school’s policies on committees.

The functions of the doctoral committee are: (a) to define the student’s area of competence and administer the qualifying examination; (b) to approve the dissertation subject: (c) to aid the student and monitor the progress of the dissertation; and (d) to read and approve the dissertation and administer the final oral examination.

### II. PRE-QUALIFYING REQUIREMENTS

The pre-qualifying requirements should be fulfilled within the first two to three years. They include some course requirements and passing the preliminary and qualifying examinations.

**A. Pre-qualifying courses**

Seventy-two hours of graduate courses are required for the Ph.D. Thirty-six of these hours must be completed before the qualifying examination and include the required core courses listed below.

**1. Core courses**

The core courses are Math 6200 and 6201 (topology), 6300 and 6301 (algebra), 6100 and 6101 (real analysis), and 7100 and 7101 (complex analysis). Each Ph.D. student is required to take seven courses chosen from these. These seven courses must each be passed with a grade of B or better. Moreover, upon entering the Ph.D. program, the Director of Graduate Studies may require a student to take a diagnostic test in analysis. Students who do not perform adequately on the diagnostic test will be required to take Math 5100 (introductory analysis) before taking Math 6100.

In some cases the Director of Graduate Studies will require that lower level courses be taken for no credit in order to make up for deficiencies in a student’s mathematical background. Students who have taken the equivalent of a core course elsewhere may be exempted, by the Director of Graduate Studies, from taking that course provided they either pass the final examination in that course with a grade of B or better or pass a more advanced course in the same area with a grade of B or better. In some cases the Director of Graduate Studies may waive these requirements.

An entering student will typically take Math 6200 (topology), 6300 (algebra) and 6100 (analysis) in the first semester. Two or three courses chosen from among Math 6201, 6301, and 6101 will then be taken in the second semester; if only two are chosen, then a course lying outside the core will be taken.

Several variations are possible. As noted above, a student may need to take Math 5100 as a result of the diagnostic test in analysis. A student with a particularly strong background in algebra might start with Math 9300 and 9301. A student with a strong interest in an area such as graph theory or applied mathematics might take courses in that area in the first year, saving for the second year those core courses that would thereby be displaced. It should be noted, however, that the only mathematics courses carrying graduate credit for students in mathematics are Math 5640 (probability), 5641 (statistics), and those courses numbered 6000 or higher.

**2. Additional courses**

A total of thirty-six hours of graduate courses in mathematics, exclusive of Math 9999, must be taken before a student will be permitted to take the qualifying examinations. For one or two of these courses, the student may, with the approval of the Director of Graduate Studies, substitute mathematically relevant graduate courses outside the Mathematics Department.

**B. Preliminary examinations**

Each student must pass a written preliminary examination in two out of the three areas of topology, algebra, and analysis. The preliminary examinations in topology, algebra, and analysis will cover the material of Math 6200 and 6201, 6300 and 6301, and 6100 and 6101, respectively. The preliminary examination in the area will be prepared by a committee of three graduate faculty specializing in that area and will be administered by the Director of Graduate Studies.

Topics Covered in the Preliminary Exams and Sample Exams:

1. Algebra Topics—Algebra Sample Test 1—Algebra Sample Test 2

2. Analysis Topics—Analysis Sample Test 1—Analysis Sample Test 2

3. Topology Topics— Topology Sample Test 1— Topology Sample Test 2

**1. Scheduling**

The preliminary examination in two out of the three areas of topology, algebra, and analysis must be taken no later than the fall of the second year. The preliminary examinations in topology, algebra, and analysis will be offered just prior to the beginning classes each semester with at least three days between each area examination. The examinations will be of three hours duration and are written. No reference material may be used during the examination.

**2. Evaluation**

The preliminary examinations will be graded by the same committees that prepared the examinations. The committees in the three areas will decide by a majority vote who passes in their respective areas.

**3. Re-examination**

A student will be given two opportunities to pass an area preliminary examination. If a student passes one area exam and fails another area exam, then the student does not have to retake a preliminary examination in the area that was passed. If a student fails to pass preliminary examinations in two out of the three areas of topology, algebra, and analysis by the Spring semester of the second year, then the student will leave the Ph.D. program.

**C. Qualifying examination**

The qualifying examination is administered by the student’s doctoral committee and is based on the student’s area of competence. The student will choose between two options for the form of the examination. Option 1 is an oral examination of two hours duration. Option 2 is an hour presentation of a qualifying paper and an hour questioning about the paper.

The purpose of the qualifying examination is to demonstrate to the doctoral committee that the student is prepared to begin research in the student’s area of competence leading to a dissertation. When the student has passed the qualifying examination, the committee will recommend to the Graduate School that the student be admitted to doctoral candidacy.

**1. Scheduling**

The qualifying examination must be taken no later than the end of the Spring semester of the third year of residency. The qualifying examination must be scheduled by the Graduate School at least two weeks in advance. If a student elects Option 2 of the qualifying examination, each member of the doctoral committee must receive a copy of the qualifying paper at least two weeks before the examination.

**2. Area of competence**

The area of competence will be defined by the student’s doctoral committee and will include the topic of the qualifying paper. (For example, if the area of competence is on some aspect of Banach algebras, then the area of competence might include the material of Math. 6100 and 6101, 7100 and 7101, and 9100 and 9101.) Shortly after the doctoral committee has been appointed, it will prepare a written statement of the area of competence. Copies of that statement will be given to the student and to the Director of Graduate Studies.

**3. Qualifying paper**

The qualifying paper is an expository or research paper in the student’s area of competence. The paper must be typed (preferably in TeX form) and conform in format to the guidelines for a master’s thesis as set forth in the aforementioned Instructions for the Preparation of Theses and Dissertations. The doctoral committee may permit a student who has written a long paper, honors thesis, master’s thesis, or research paper to offer it as part or all of the qualifying paper.

**4. Attendance**

Attendance at Option 1 of the qualifying examination will be open to all faculty of the university, but student attendance will not be permitted. Attendance at Option 2 of the qualifying examination will be open to all faculty and students of the university, but attendance of students during the subsequent questioning will be at the discretion of the committee.

**5. Evaluation**

The doctoral committee will decide by a majority vote whether the student has passed or failed the qualifying examination.

**6. Re-examination**

A student who fails in the first attempt to pass the qualifying examination will, upon request, be granted a repeat examination. In no case will a student have more than two opportunities to pass the qualifying examination.

A repeat examination may be taken under either Option 1 or Option 2 independently of the option chosen for the first attempt at passing. The qualifying examination must be passed by the end of the Fall semester of the fourth year of residency.

### III. DISSERTATION AND FINAL EXAMINATION

An original dissertation is required. The student must give an oral public presentation of the dissertation and undergo public questioning by the doctoral committee about it. The presentation and questioning occur at the final examination, otherwise known as the defense of the dissertation.

**A. Dissertation**

Each member of the student’s doctoral committee is to receive a copy of the dissertation at least one month before the final examination. Two copies of the dissertation bearing original signatures of at least a majority of the committee must be registered in the Graduate School office no later than two weeks before the last day of finals in the term in which the student plans to receive the degree. The dissertation must be typed (preferably in TeX form) and conform in format to the guidelines of the Instructions for the Preparation of Theses and Dissertations. The dissertation must be accompanied by an abstract, signed by the student’s adviser, of not more than 350 words.

**B. Final Examination**

The final examination must take place at least fourteen days before the last day of finals in the term in which the student plans to receive the degree. This event must be scheduled at least two weeks before it is to occur.

A student who fails the final examination may take it again. In no case will a student have more than two opportunities to pass the final examination.

### IV. ADDITIONAL REQUIREMENTS

**A. Courses**

Seventy-two hours of graduate courses are required. Twenty-four of these hours must be earned in formal course work at Vanderbilt; the remainder may include transfer credit and dissertation research hours (Math. 9999). A maximum of twelve of the seventy-two required hours may be earned in courses in departments other than mathematics; such courses must be approved by the Director of Graduate Studies.

In addition to the pre-qualifying courses discussed in II.A, the student must take eight three-hour mathematics courses numbered higher than 7000, exclusive of Math. 9999. These courses must be in at least two of the following areas: algebra, analysis, applied mathematics, geometry-topology, and graph theory-combinatorics.

**B. Maintaining a B Average**

An average of at least B is required for graduation. Grades received in no-credit courses and in Math. 9999 are not included in computing the grade-point average.

**C. Teaching**

Most graduate students in the Department of Mathematics at Vanderbilt University receive financial support in the form of a teaching assistantship. The duties and training of teaching assistants (TAs) are outlined below.

FIRST-YEAR TAs

All first-year graduate students participate in a teacher training program which consists of the following components:

1. TAs proctor a two-hour evening tutoring session for calculus students each week for both semesters.

2. TAs provide limited one-on-one tutoring through Tutoring Services.

3. In the spring, the Teaching Seminar meets weekly to introduce the teaching program in our department, the resources available for teachers at Vanderbilt, and good teaching techniques.

SECOND- AND THIRD-YEAR TAs

In their second and third years at Vanderbilt, graduate students serve as teaching assistants for instructors in our first-year calculus courses. Students conduct two weekly problem sessions for the class, present an occasional lecture, assist the instructor in grading, and hold office hours. Before the beginning of the fall semester, the Center for Teaching conducts an orientation session for all new TAs in the College of Arts and Science.

FOURTH- AND FIFTH-YEAR TAs

In the fourth and fifth years, students with good evaluations may teach a first-year calculus course. Usually, teaching duties require about 12 hours a week. During their first year of independent teaching, TAs will be mentored by members of the Teaching Committee who will conduct class observations and offer support as needed.

EVALUATION OF TEACHING

Teaching performance is evaluated via student questionnaires in mid-semester and at the end of the semester. These evaluations are reviewed by the Associate Director of Graduate Studies, who makes recommendations on renewal of teaching assistantships. TAs interested in improving their teaching can ask to have faculty observation of their classes, videotaping, or Small Group Analyses.

TEACHING SKILLS FOR INTERNATIONAL TAs

The University’s Center for Teaching has a staff member devoted to the teaching needs of international teaching assistants, particularly those whose first language is not English. Teaching assistants who are experiencing language problems can participate in programs to improve their speaking and listening skills. In the past this program has proved to be a very effective means of improving language abilities.

However, the Department may accept other activities as a substitute, and may waive the teaching requirement for graduate students with previous teaching experience.

**D. Continuous registration**

Registration must be continuous except for summer sessions. Any interruption in registration must be authorized by the Dean of the Graduate School as a leave of absence. Thus, except when granted a leave of absence, the student must register each Fall and Spring semester, even if all course and hour requirements have been met. Failure to maintain continuous registration will result in loss of student status.

**E. Residence requirement**

The student must be in residence a minimum of three years.

### V. MISCELLANEOUS

**A. Minor**

It is not required that a student have a minor, but it may be desirable in some cases. A minor is a body of related courses either in a mathematically relevant area outside of mathematics or in an area of mathematics outside the area of the student’s dissertation. Twelve hours of such courses constitute the minor. If the minor is in an area of mathematics, courses taken toward the minor may include courses used to satisfy the requirements of II.A and IV.A.

**B. Transfer credit**

Transfer credit can sometimes be granted for graduate work completed at another institution. As many as twenty-four hours of transfer credit can be awarded to a student who has completed a master’s degree in mathematics at a comparable university.

**C. Math 9999**

A student ought not to register for Math 9999 (Dissertation Research) until the qualifying examination has been passed. Grades in Math 9999 are not counted in calculating a student’s quality point ratio.

**D. No-credit courses**

After seventy-two hours of credit have been earned, additional registration for research or course work will be permitted, but will earn zero credit hours, unless exception is made by the Dean of the Graduate School.

Document last updated: 8/11/06 by M.D. Plummer, DGS