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Graduate Program Requirements

**1. DEPARTMENTAL REQUIREMENTS FOR THE PH.D.**

1.1. A student must be admitted to candidacy, for which eight qualification credits subject to the distribution requirements are necessary (see sections 3 and 8). A student must earn three more qualification credits - thus the total of eleven - to obtain a Ph.D.

1.2. A student must demonstrate ability to read mathematics in one of the three foreign languages: French, German, Russian (for details and some exceptions, see section 9).

1.3. Each student must prepare for and successfully pass a Topics Examination on an advanced subject.

1.4. A student must complete a thesis which contains substantial new mathematics and which is approved by a committee of three faculty members (including possibly some outside mathematicians) and defended in an officially scheduled final examination.

1.5. Because Brown's doctoral programs train graduate students to become educators as well as researchers, teaching is an integral part of the graduate education. All doctoral students in the Mathematics program are required to train as teaching assistants for at least two semesters. In consultation with the DGS, this requirement may be fulfilled during any of the years in the program.

1.6. Concerning tuition payment requirements by the Graduate School, see the University Catalog and consult the department.

1.7. If all requirements for the Ph.D. are not completed within five years after admission to candidacy, readmission to candidacy must be approved by the Department and the Graduate Council.

**2. THE PROGRAM OF STUDY**

2.1. The program of a beginning graduate student depends on his or her background, and is worked out in consultation with the Graduate Advisor. A full-time student without previous work equivalent to graduate work at Brown, and without serious weakness in background, will usually start with three of the first-semester courses in five main areas: Differentiable Manifolds, Real Analysis, Complex Analysis, Algebra, and Topology, and continue to the second-semester courses. In some cases, special interests may make pursuit of other qualification courses (see section 7) more desirable for the individual program. In the second year, the student will take courses in the fourth area, together with other courses or reading courses designed to help in finding a field of concentration, and to prepare for work in that field.

2.2. An entering student with previous graduate-level work in one or more of the main areas is encouraged to take a Diagnostic Examination in that subject during the first week of classes. On the basis of the results of this examination, a student may receive one or two qualification credits and be advised to accelerate or even skip the qualification work in that area. A poor performance on this examination will not be held as a negative mark on the student's record.

2.3. A student who has completed some graduate work elsewhere may and should apply to transfer credits under the guidance of the Graduate Advisor. This does not lead to any qualification credit, but helps toward the tuition requirement.

2.4. Students are urged to devote considerable thought and energy to choosing an area of specialization; this should be under way as soon as possible, at least by the second year. Taking special topics or reading courses, participating in seminars, organizing seminars in areas of interest, attending colloquia, and so on help students to find what areas are of most interest, which are active at Brown, and to gain an awareness of the scope of present-day mathematics. Such activities also give an opportunity for the individual students and faculty members to have closer contact and to assess the prospects of continuing on to work on a thesis. The Topics Examination, mentioned in 1.3, is an excellent way for a student to try out a possible field of research.

2.5. Advanced students, in consultation with their thesis advisors, usually continue to take and audit courses outside their thesis areas.

2.6. Knowledge of mathematical applications can be valuable in giving another perspective on mathematics, as background for better teaching of students with applied interests, and as a preparation for certain types of employment and careers. Graduate students are urged to consider this in planning their progress. For example, courses in dynamical systems, statistics, operations research, fluid dynamics, and numerical analysis are available in the Division of Applied Mathematics; courses in relativity and quantum mechanics are given by the Physics Department; and courses in complexity theory and algorithms are offered by the Computer Science Department. A semester or two of these courses might be taken in the second or subsequent years. Computing facilities within the department are strong, and all students have access to the University's mainframe computer. Summer study or a summer job, perhaps after the first or second year, could give experience in the work based on applications of mathematics. Many of our Ph.D.'s in the past have had successful careers outside universities or colleges, namely, in government, industry, consulting, etc.

**3. ADMISSION TO CANDIDACY**

3.1. Admission to candidacy is decided by the whole faculty at a meeting (Section 5) in which all aspects of a student's performance are taken into account. The Department admits a student to candidacy when it believes he or she has a strong background in mathematics and is ready to concentrate on a special field and write a thesis.

3.2. To be considered for admission to candidacy, a student must have completed eight qualification credits (as detailed in section 8), and should show promise for mathematical research. Occasionally a student may be told that he or she has been admitted to candidacy subject to completion of some of these criteria.

3.3. As with other decisions (see section 5 on evaluation), the decision regarding admission to candidacy is made on the basis of the overall quality of a student's work.

3.4. A student is normally expected to advance to candidacy by the end of the fifth semester if he or she is to continue toward the Ph.D.

**4. THESIS ADVISORS**

It is the student's responsibility to convince some faculty member to be his or her advisor. Most faculty need to know a student's work first-hand, either from a course, seminar, reading course, or topics examination, before taking on an advisee. Students should take this into account in planning their programs in the second or third years.

**5. EVALUATIONS**

5.1 The mathematics faculty normally meets each November and April to discuss the progress of each graduate student. In addition, special situations may be discussed at a meeting in January. Each faculty member reports what he or she knows about the mathematical activity of each student. This includes work in courses, reading courses or independent reading, participation in seminars, work on a topics examination or thesis, or work as a teaching assistant.

5.2. In addition to keeping the faculty informed, the meetings serve several purposes:

- To make recommendations and give advice to the student. This may include: planning a program and choosing courses, whether to obtain a Master's degree, whether to continue after the present year, or whether to make contingency plans if financial support for the next year appears unlikely.
- To decide which students are recommended for assistantships and other financial support. First priority goes to those in the first five years who are making good progress.
- To decide whether anyone who had not been admitted to candidacy should now be admitted; also to clarify any questions concerning qualification credits or other requirements.

5.3. Results of these evaluations will be conveyed to the students by the Graduate Advisor. A written notification of any formal decision on the student's status made at the evaluation meetings will be given to the students by December 14, January 30, or May 15.

5.4. It should be emphasized that the discussions and decisions are based on all aspects of a student's progress in mathematics. In order to encourage students to proceed, the Department needs evidence of motivation, mathematical ability, maturity, judgment, and originality, as well as satisfactory progress in learning the concepts and techniques of the main areas of mathematics.

**6. APPEALS**

6.1. If a student disagrees with a decision of the Department at any stage, he or she can and should feel free to appeal that decision. This could be the case especially if the student feels that a decision was made without full knowledge of his or her work. In this case a request can be made to present additional evidence, or to take an examination to demonstrate the quality of that work, or the student may ask for a hearing to present his or her case. The students are also referred to the statement on the resolution of grievances (Faculty Rules and Regulations, {date}, pp. {page}).

6.2. The following paragraph, taken from a statement "Due Process for Graduate Students" of September 1972 by then Dean Brennan on behalf of the Graduate Council, still represents the sentiments of the department:

"In addition to grades in courses or any one other indicator, a faculty hasevery right to evaluate the performance of a student in total - as acomprehensive unit so to speak. Long-standing precedent dictates that a departmental faculty may decide and (many times over) has decided that a student should be dismissed from the Graduate School even though, say, the student's grades in courses meet the requirements for course credit toward a degree as stated in the catalog. For instance, students whose course grades are satisfactory, and who have passed the preliminary examination have been denied the Ph.D. degree on grounds that they could not initiate the kind of independent research needed to complete a Ph.D. dissertation. In general, then, a faculty is entitled to view the work of a graduate student as a comprehensive whole, taking account of all relevant factors, and to find a student lacking overall in spite of satisfactory performance in one or more facets of the whole."

**7. COURSES**

7.1. Graduate courses, other than 2910, 2920, are divided into two categories: qualification courses and seminar courses. A qualification course involves some mechanism for evaluating students: specifically, homework assignments, plus either a final examination or a written project.

7.2. Each year, the qualification courses will include the first year courses in the five main areas, namely Differentiable Manifolds (2110), Real Function Theory (2210, 2220), Complex Function Theory (2250, 2260), Algebra (2510, 2520), Topology (2410, 2420), and courses such as Algebraic Geometry (2050), Differential Geometry (2010), Partial Differential Equations (2370), Probability (2640, not 2630), Number Theory (2530), Lie Theory (the last course under 2710 or some new number). Other graduate courses offered will be designated as qualification or seminar.

7.3. At the conclusion of a qualification course, the instructor will communicate to the Graduate Advisor which students should receive a qualification credit (as distinguished from a course credit) for that course. This result will then be communicated to the student.

**8. QUALIFICATION CREDITS**

8.1. To be considered for admission to candidacy, a student must have earned 8 qualification credits in the following manner:

- The credits must include 2110, 2210, 2250, 2410, and 2510 and three out of the four basic second-semester courses (2220, 2260, 2420, and 2520).
- Some qualification courses mentioned in 7.2 will be considered relevant to the main areas as follows:

For Algebra: 2050, 2530, and possibly Lie Theory;

For Real Analysis: 2370, 2640;

For Complex Analysis: 2370 and possibly 2050;

For Topology: 2010 and possibly Lie Theory.

- A Diagnostic Examination will be given in each of the four main areas during the first week of classes each year. There are three standard outcomes of a diagnostic test: a student can pass outright (thus earning two qualification credits and satisfying the distribution requirement in that area); or obtain a one-semester pass (thus earning one qualification credit and satisfying the first-semester requirement in that area); or fail outright.
- A Diagnostic Examination will be given in each of the four main areas during the first week of classes each year. There are three standard outcomes of a diagnostic test: a student can pass outright (thus earning two qualification credits and satisfying the distribution requirement in that area); or obtain a one-semester pass (thus earning one qualification credit and satisfying the first-semester requirement in that area); or fail outright.
- In the event that a student's work for qualification purposes is incomplete at the end of a first-semester course, the instructor may give permission to make it up by the end of January of that academic year. For a second-semester course, incomplete work may be made up before the start of the next academic year, if permission is given. A student may choose to take a Diagnostic Examination instead.
- In the event that a student fails to get a qualification credit in 2210, 2250, 2510, or 2410, he or she may continue to take 2220, 2260, 2520, or 2420, respectively, and be allowed to make up of the first semester depending on his or her achievement. Normally, this can lead only to one qualification credit (and completion of the first-semester requirement in that area) but not two qualification credits.

8.2. In the beginning of each semester a student must plan the work necessary for qualification credits in consultation with the Graduate Advisor, and, in each qualification course, notify the instructor whether he or she intends to have the work evaluated for the purpose of qualification within a reasonable time limit (set by the instructor with the consent of the Graduate Advisor). A subsequent change of plan should be reported.

8.3. Beyond the eight qualification credits on which admission to candidacy is based, a student must earn three more qualification credits before he or she graduates with a Ph.D., as already stated in 1.1. One of them at most may be earned by successfully completing a written project in connection with a reading course (see the end of 2.1).

8.4. It is important for each student to plan carefully for qualification credits and for the Graduate Advisor to monitor and record each student's progress. The full cooperation by all the faculty is also important to ensure the success of our evaluation system.

**9. LANGUAGE REQUIREMENTS**

9.1. Doctoral students must demonstrate ability to read mathematics in one of the three foreign languages: French, German, Russian, normally before November 1 of the fifth semester. This may be accomplished by passing the departmental examination, or by getting a B at the 2R level of the appropriate course at Brown. A student who wishes an extension of time should make an appeal in writing to the Graduate Advisor by the end of the fifth semester.

9.2. It is possible that another language may be approved at the discretion of the Graduate Advisor.

**10. ABOUT THIS DOCUMENT**

10.1. In no sense is satisfying the requirements equivalent to doing good work. The requirements should not be regarded as goals for the students to aim for; rather they reflect the department's belief in the standard for the solid foundation of a mathematician of today.

10.2. If a question arises as to the interpretation of any of the statements in this document, it is to be settled by the Graduate Advisor in consultation with other faculty members involved in the matter in question and, for a wider or deeper issue, with the consent of the Department. A truly exceptional case will be handled in this manner.

10.3. The principle of continuous evaluation based on four main areas was initiated in 1972 to replace the written qualifying examinations. A document "The Ph.D. Program in Mathematics at Brown" was written and revised to reflect the evolution of the evaluation system and requirements. The present document was written as a result of the revision of the graduate requirements approved by the Graduate Council in 1985, and further amendments in 1995. It incorporates, in some detail, the recent practice of the Department which is concordant with that of the Graduate School. It is hoped that the document serves as the basic guide to our Ph.D. program not only for the graduate students but also for the faculty in the Department, and in particular for the Graduate Advisor.