The typical student takes courses in the first and second year, at the same time exploring some more advanced mathematics with the goal of finding a research area. Ideally, by around the beginning of the third year, the student has found an area to specialize in and a thesis advisor. (Of course, this timeline varies for each student.)
The department has no “qualifying exam” or “preliminary exam”. We do insist that every student develop a basic broad mastery of mathematics, but we insure this more by courses than by exams; see the section on degree requirements below.
There is no expectation that students will be “weeded out”. In fact, the program has very little attrition.
Another distinctive feature of the program is the Topics Exam. This is nothing like a Qualifying Exam. It can be thought of as a useful exercise rather than a hurdle. At some point, ideally before the beginning of the third year, the student identifies some mathematics that they would like to learn, makes a reading list in consultation with a faculty member, and studies the material for some months. This culminates in an exam, which consists of a presentation by the student to a committee of three faculty followed by questions from the committee. The purposes are to encourage the transition from course work to research, to let the student demonstrate a readiness to do independent work, and to provide a framework for trying out a possible research area and a possible thesis advisor.
The following five two-semester sequences are intended primarily for beginning graduate students. A typical student will begin with four of these.
- Real Analysis 1: Measure and integration
- Real Analysis 2: Functional analysis
- Complex Analysis 1: Introduction
- Complex Analysis 2: Either Riemann surfaces or other topics
- Algebra 1: Introduction
- Algebra 2: Mostly commutative algebra
- Topology 1: Introduction to algebraic topology
- Topology 2: Either cohomology, Poincare’ duality, and related topics, or else some topics in geometric topology
- Differentiable Manifolds: Introduction to manifolds
- Differential Geometry: Introduction to Riemannian geometry
The other regular courses (courses offered every year) are two-semester sequences in Partial Differential Equations, Probability, Number theory, and Algebraic geometry. Second-year students typically take some of these as well as some more of the courses listed above.
In addition to the regular courses, we offer one or two Topics courses in any semester. These are one-time courses that either address an advanced topic of current research interest or cover a more basic subject that is not covered in regular courses.
Many of our students also take advantage of course offerings in the Division of Applied Mathematics, or occasionally in Computer Science or Physics. Some of the analysis courses listed above are taught by Mathematics and Applied Mathematics in alternate years.
Another option is to pursue a topic by taking a reading course, either alone or as a group. This can count for course credit. Faculty members are generally very happy to be asked to supervise a reading course.
Admission comes with a promise of five years of support. First-year support is in the form of a fellowship (no teaching duties). Thereafter, the student can expect to be working as a Teaching Assistant or Teaching Fellow in most semesters. We do our best to arrange at least one more non-teaching semester for each student.
Students who choose to spend the summer on study/research toward the PhD receive a stipend. There are also some summer teaching opportunities at Brown.
Sixth-year support is decided on a case by case basis.
Students are required to “qualify” in eleven subjects: five required and six elective.
The required subjects are the first-semester courses Real 1, Complex 1, Algebra 1, Topology 1, and Manifolds. In these subjects “qualifying” ordinarily means passing the course at a sufficiently high level in the judgment of the instructor. There is also the option of qualifying by passing a “diagnostic exam” at the beginning of the semester.
Elective courses may include second halves of those five sequences, other regular courses, topics courses, courses in other departments as appropriate, and up to one reading course. They are subject to a mild distribution requirement.
The topics exam is described above.
The program no longer has a foreign language requirement.
Of course, the main requirement is to write an original thesis (also called a dissertation) and to successfully defend it to a committee of three faculty readers.
In the second semester of the first year, students attend a weekly session to help prepare them for future work as a TA and later as a TF. (A TF is a graduate student teaching a section of a course.) This includes some simulated practice teaching.