Department of Mathematics
Graduate Program Handbook
Revised March 2019
1. REQUIREMENTS FOR THE PH.D.
1.1. The student must be admitted to candidacy. (See Section 3.)
1.2. The student must qualify in the five core subjects and earn six additional qualification credits, subject to the distribution requirement. (See Section 8.)
1.3. The student must prepare for and successfully pass a Topics Examination on an advanced subject. (See Section 9.)
1.4. The student must complete a thesis which contains substantial new mathematics and which is approved by a committee of three faculty members (including possibly one mathematician from outside the department) 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 graduate education. All doctoral students in the Mathematics program are required to work 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. By university rule, 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 the student’s background, and is worked out in consultation with the Director of Graduate Studies. A full-time student without previous work equivalent to graduate work at Brown, and without serious weakness in background, will usually start with three or four of the five core courses: Differentiable Manifolds, Real Analysis, Complex Analysis, Algebra, and Topology, and continue to related courses in the second semester. In the second year, the student will complete the core requirement, while taking 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 core 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 qualification credit without taking the relevant course. A poor performance on this examination will not be held as a negative mark on the student's record.
2.3. Students are urged to devote considerable thought and energy to choosing an area of specialization; this should be under way as soon as possible, certainly by the second year. Taking special topics or reading courses, participating in seminars, organizing seminars in areas of interest, attending colloquia, and speaking with faculty are some ways of discovering what areas are of most interest to the student and which of these areas are active at Brown, and to gain a broader awareness of the scope of present-day mathematics. Such activities also give an opportunity for individual students and faculty members to have closer contact and to assess the prospects of continuing to work together. The Topics Examination (see Section 9) is an excellent way for a student to try out a possible field of research.
2.4. Advanced students, in consultation with their thesis advisors, often continue to take and audit courses outside their thesis areas.
2.5. 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. Some 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 with mathematical work outside of academia. Many graduates of the program have had successful careers in government, industry, consulting, etc.
3. ADMISSION TO CANDIDACY
3.1. Admission to candidacy is decided by the whole faculty of the department 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 the student 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), including the five core subjects, and should have passed a Topics Exam (Section 9 below), and should show promise for mathematical research. Occasionally a student may be conditionally admitted to candidacy, subject to completion of the last these requirements in the current semester.
3.3. As with other decisions (see Section 5), the decision regarding admission to candidacy is made on the basis of the overall quality of a student's work.
3.4. The student is normally expected to advance to candidacy by the end of the fifth semester.
4. THESIS ADVISOR
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 them on as an advisee. Students should take this into account in planning their programs in the second or third years.
5.1. The mathematics faculty normally meets each November and April to discuss the progress of each graduate student. Faculty members report what they know 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, deciding whether to obtain a Master's degree, deciding whether to continue after the present year, or making 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 any student 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 Director of Graduate Studies. A written notification of any formal decision on the student's status made at the evaluation meeting in November or April will be given to the students by December 15 or May. 15 respectively.
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.
6.1. A student who disagrees with a decision of the Department at any stage 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 their 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 their case. Students are also referred to the statement on the resolution of grievances (Faculty Rules and Regulations, July 1, 2017, pp. 122-123).
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 has every right to evaluate the performance of a student in total - as a comprehensive 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."
7.1. Graduate courses can be divided into several categories: core courses, elective courses, topics courses, and reading and research.
7.2. The five core courses, offered every year in the fall, are Differentiable Manifolds (2110), Real Function Theory (2210), Complex Function Theory (2250), Algebra (2510), and Topology (2410).
7.3. The following elective courses, related to the core courses, are offered in the spring:
Differential Geometry (2010), Real Function Theory (2120), either Riemann Surfaces (2260) or Complex Analysis (2261), Algebra (2520), and either Algebraic Topology (2420) or Geometric Topology (2421).
7.4. The following pairs of elective courses are also offered every year:
Algebraic Geometry (2050/2060), Number Theory (2530/2540), Partial Differential Equations (2370/2380), and Probability (2630/2640). These are generally intended more for the second year of study than for the first.
7.5. Every year there are some Topics Courses, numbered 2710 (fall) or 2720 (spring). The content is chosen by the instructor and may be either something specialized and related to current research or something of more general interest that is not treated in a regular course.
7.6. The course numbers 2910/2920 are used for reading and research. An advanced student not attending courses in the usual sense registers for 2910 or 2920, with the thesis advisor as instructor. These course numbers are also used when a faculty member agrees to supervise a student, or a group of students, in a reading course.
7.7. Among the courses mentioned above, each of the following is cross-listed with the Division of Applied Mathematics and sometimes taught by DAM: 2210/2220, 2370/2380, 2630/2640.
Whether taught by Mathematics or by DAM, it is considered the same course for purposes of these requirements.
8.1. The department does not have qualifying examinations in the usual sense. Instead, it has the rule mentioned in Section 1.2. Here are the details.
8.2. “Qualification credit” is an internal mathematics department concept. Earning qualification credit is also called “qualifying” in that subject.
8.3. In any core course or elective course (listed in 7.2-7.4) one may qualify by taking the course and doing sufficiently good work in the judgment of the instructor. A grade of “A” is always considered sufficient; in the case of a “B” grade the instructor must say whether the student has qualified.
8.4. For the five core topics (7.2) there is also the option of qualifying by Diagnostic Examination. A student who wishes to demonstrate mastery of the topic may request an exam, to be given in the first week of the fall semester, usually by the instructor of the relevant course.
8.5. With permission of the DGS, students may also seek qualification credit in 2000-level courses outside the Mathematics Department, usually in Applied Mathematics but occasionally in another subject, for example Computer Science or Physics. In such cases it is important for the student and the DGS to communicate with the instructor about qualification.
8.6. A Topics course or a reading course may also lead to qualification credit, with permission of the DGS, as long as the course involves work by the student comparable to the work in an ordinary (core or elective) course.
8.7. Of the six qualification credits counted toward the degree requirement (1.2) beyond the core topics, at least one must be in each of the three broad areas Algebra, Analysis, and Geometry. In some cases (such as Riemann surfaces) the same course may be reasonably counted toward either of two areas as needed. Such questions are at the discretion of the DGS, advised by the Graduate Committee.
8.8. In some cases where an instructor does not grant qualification to a student at the completion of a course, the student may be offered an opportunity to qualify by doing some additional work. Similarly, if the instructor of a core course judges a student’s performance on a Diagnostic Examination to be insufficient for qualification but does not feel that the student needs to enroll in the course, the student may be advised to attend portions of the course, or to do some of the work in the course, and to be reexamined at the end of the semester. Such plans are made in consultation with the DGS, advised by the Graduate Committee.
8.9. It is important for each student to plan carefully for qualification credits and for the DGS 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. TOPICS EXAMINATION
Each student must prepare for and successfully pass a topics examination on an advanced subject. The student initiates the examination process in conjunction with a willing faculty member, called a topics advisor. The faculty member determines the scope of the examination and the reading list in consultation with the student, and they agree on the date by which the examination will take place. An advanced topic is one that goes beyond the scope of the basic graduate courses in a single subject or area. The reading list may include one or several research papers, depending on length or difficulty. It might also include some portion of a text or monograph. The reading list should be designed so that it can be completed in a reading period of three to four months. Preparation by the student is expected to be a fairly independent process; consultation with the topics advisor is expected occasionally, not regularly. The preparation for this examination is not supposed to substitute for the regular coursework that is expected of the students.The topics examination is administered by the topics committee, which consists of the topics advisor and two other faculty members, who will have been invited to join the committee by the advisor. Other faculty members are welcome to attend.
The examination consists of two parts. First, there is a 45 minute presentation based on some part of the reading list. The subject of the presentation will have been determined during the reading period by the student and the topics advisor. Then there is a period of questions by the committee of three for the student on topics closely related to the reading list, which will last not more than 45 minutes. After the questioning, the committee will consult briefly in private and vote to determine whether the student has passed. The results of the decision are then recorded in the Department records. Students are encouraged to talk with faculty to discuss possible subjects of the topics examination. The end of the first year, or the first semester of the second year, is a good time to start. The student is normally expected to pass the topics examination before the end of the second year of study at Brown. In any case, the student should take the examination before November 15 of the third year. In case of failure at the first attempt, the student's second and last attempt should be made before February 15 of the third year of study. A faculty member may be the topics advisor to any graduate student. The topics advisor is not presupposed to be the eventual thesis advisor, and the reading topic is not necessarily the topic of the student's eventual dissertation, although either of these may become the case. A student retaking the topics examination may select a different subject with a new topics advisor.
10. LANGUAGE REQUIREMENT
Students are no longer required to demonstrate proficiency in a foreign language. Students wishing to have an official notation of their ability to read mathematics in a foreign language—typically French, German, or Russian—can arrange a competency exam with a faculty member of the department.
11. TRAINING FOR TEACHING ASSISTANTS
Students normally begin working as teaching assistants in their third semester. In their second semester they attend a mandatory weekly training program to prepare them for TA work, which includes reading, discussion, and simulated practice teaching.
12. ENGLISH PROFICIENCY FOR INTERNATIONAL TEACHING ASSISTANTS
Brown University requires that all international teaching assistants whose first language is not English must be evaluated and certified as proficient in English before they are allowed to teach. The English for International Teaching Assistants Program evaluates the spoken English proficiency of international teaching assistants prior to their assuming teaching duties at Brown. It also provides English language courses (EINT) for international graduate students who need to improve their spoken English before they begin teaching. International graduate students who have been awarded teaching assistant positions and whose native language is not English will be tested upon their arrival for English proficiency,so that those who need to improve their English language skills have time to do so.
For instructions on scheduling an English proficiency evaluation, test dates, course descriptions, and related information see the website of the English for International Teaching Assistants Program.
13. THE GRADUATE COMMITTEE
As of September 2018 the Department has a standing Graduate Committee chaired by the DGS. Its role is partly to consider recommending alterations in the graduate program—changes in the curriculum or the degree requirements or the advising system—and partly to assist the DGS in making decisions concerning particular students in the program.
14. ABOUT THIS DOCUMENT
14.1. The principle of continuous evaluation based on core areas was initiated in 1972 to replace the written qualifying examinations. The present document was written as a result of the revision of the graduate requirements approved by the Graduate Council in 1985, and again in 1995 and 2018. 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 Committee.
14.2. If a question arises as to the interpretation of any of the statements in this document, it is to be settled by the DGS and the Graduate Committee in consultation with other faculty members involved in the matter in question and, in truly exceptional cases, with the Department faculty as a whole.
14.3. The authoritative source for Graduate School policies is the Brown Graduate School Handbook, which can be found on the Graduate School website: https://www.brown.edu/academics/gradschool/academics/rules-regulations/graduate-school-handbook