This stage begins with admission to the Doctoral Program and ends with the completion of the Qualifying Review. Students must complete this part of the program by the beginning of their sixth term, but the expectation if that it will be completed much earlier.

Students in Stage 1 are assigned a member of the Doctoral Committee as their counselor. They may also seek advice from any faculty member. However, only members of the Doctoral Committee can approve their course elections or changes.

Students meet with their counselors before each term to decide which courses they will take. Normally students should take two core courses in each of their first two terms, together with one or two other courses, which may be outside the core areas (for example, in the student's area of special interest).

### Qualifying Review Exams

The Qualifying Review Examination is a written examination, offered three times a year in early January, May and September, which covers applied analysis and each of the three major areas of pure mathematics---algebra, analysis, and geometry/topology. The syllabus for the Examination in each area is the same as that for the corresponding core courses:

- Algebra: Math 593, 594
- Analysis: Math 596, 597
- Applied Analysis: Math 556, 572
- Geometry/Topology: Math 591, 592

Students are responsible for the material in the syllabus even when it is not covered in a particular core course. Detailed syllabuses and copies of old exams are available in the Graduate Student Office.

#### Completing the QR Requirement

There are a few ways to satisfy the QR Requirement.

- Students who pass the Qualifying Review Exams in two of the pure areas, algebra, analysis, or geometry/topology, must also pass the two core courses in the third pure area, or one core course in each of the remaining two areas, with at least a B average and no grades C+ or lower. Such students must also successfully complete one further course in an area outside the pure core areas, for example, in algebraic geometry, applied mathematics, combinatorics, differential geometry, logic, mathematical physics, number theory, numerical analysis or probability. An applied core course may be used for this purpose. With the permission of the Chair of the Doctoral Committee, a student may substitute a more advanced course in the same immediate area for a core course. For example, the student may substitute a more advanced complex analysis course for the complex analysis core course, but not for the real analysis core course.
- Students who pass the Qualifying Review Exam in Applied Analysis and one other area must pass one core course in each of the remaining two areas, with at least a B average and no grades of C+ or lower. Such students must also successfully complete one additional course in algebra, pure analysis, geometry, topology, or discrete mathematics.
- Students who pass the Qualifying Review Exams in three of the four areas must also pass a course in an area outside of the pure core areas for example, in algebraic geometry, applied mathematics, combinatorics, differential geometry, logic, mathematical physics, number theory, numerical analysis or probability. An applied core course may be used for this purpose.

When the Qualifying Review Examination has been graded, the Doctoral Committee decides on the basis of the Examination results and the entire academic record whether a student should pass the requirement. Additional faculty input is often sought before making a final decision. For a student who has passed two examinations but has yet to complete the course work, the Committee may pass the student subject to successful completion of the courses.

**Retesting and Requirement Deadline:** Students are encouraged to take the Examination as soon as possible. One of the two written exams must be passed by the start of the fourth term. The Department expects the entire Qualifying Review to be completed by the start of the sixth term. For example, a student entering the Doctoral Program in the Fall of 2018 must pass the Examination in at least one area by early January 2020 and the whole Review by no later than early January 2021.

### Schedule of QR Exams 2018-19

#### September 2018

- Saturday, September 1 - Analysis
- Sunday, September 2 - Topology
- Monday, September 3 - Algebra [Labor Day]
- Monday, September 3* - Applied Analysis [Labor Day]
- Tuesday, September 4 - Classes Begin

- Thursday, September 6 - Results Meeting
- Monday, September 10 - Inform Students

#### January 2019

- Saturday, January 5 - Topology
- Monday, January 7 - Algebra

- Monday, January 7* - Applied Analysis
- Tuesday, January 8 - Analysis
- Wednesday, January 9 - Classes Begin
- Thursday, January 10 - Results Meeting
- Monday, January 14 - Inform Students

**May 2019**

- Thursday, May 2 - Last day of finals
- Wednesday, May 8 - Algebra
- Wednesday, May 8* - Applied Analysis
- Thursday, May 9 - Analysis
- Friday, May 10 - Topology
- Tuesday, May 14 - Results Meeting
- Friday, May 17 - Inform Students

*If a student wishes to take both algebra and applied analysis, alternate accommodations will be made