The goal of Stage 1 is to demonstrate mastery of the core curriculum in mathematics by passing the Qualifying Review.

## The Qualifying Review

The entire Qualifying Review should be completed by the start of the sixth term. How quickly a student passes the Review depends a lot on their undergraduate preparation, but 1-2 years is typical.

To pass the Qualifying Review, a student must demonstrate proficiency in core material from each of the three major areas of pure mathematics (algebra, analysis, and geometry/topology) or for students interested in applied mathematics, there is an option to substitute some applied analysis. The department offers eight “alpha” or core courses designed to help PhD students achieve this mastery:

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

Students demonstrate proficiency through a combination of coursework in the above listed courses, and Qualifying Review Examinations based on the same material.

The doctoral committee meets three times per year to review Stage I students' progress toward satisfying the QR requirement and promote those they deem ready to Stage II. Additional faculty input is often sought before making a final decision.

### Qualifying Review (QR) Exams

Qualifying Review (QR) Examinations are offered in each of the four core areas listed above (algebra, geometry/topology, analysis, and applied analysis.) Students can demonstrate proficiency in a core area by passing the Qualifying Exam for that subject, rather than by taking the course, although this is recommended only for students with very strong undergraduate preparation. The “alpha” courses are designed to prepare students for these exams. However, the courses cover the material at a deeper level than required for the exams, and depending on the instructor, may cover more or even slightly different material.

Each QR exam is a six-hour written examination taken on one day, with a break for lunch. The QR exams are offered three times per year in early January, May, and September. Students are encouraged to study the old QR exams in each area and take the exams when they feel ready. *However, unless they have a good reason and have discussed the situation with their academic counselor, students should not skip the alpha courses until they have passed the corresponding exams.*

**Important Deadline: **Students must pass at least one QR exam by the start of their fourth term. For example, a student entering the doctoral program in the Fall of 2019 must pass the QR examination in at least one area by early January 2021.

**Retesting:** Students are encouraged to take each QR examination as early and as often as possible. For example, QR scores can provide a useful calibration point in selecting courses. Taking QR exams early often moves students more quickly into Stage II, which allows more time to explore interests and/or begin research. Practice may increase students’ chances of passing the exams sooner. There is no penalty for failing QR exams, provided they are passed by the required deadlines.

### Completing the Qualifying Review Requirement:

There are a few ways to satisfy the QR Requirement:

(1). Students can pass all three Qualifying Review Exams in three of the four core areas. In this case, at least one additional course, not a core course in the three chosen areas of examination, must be passed with a grade of B- or higher. That course should be one that evaluates and provides feedback to students on their work. Courses in algebraic geometry, applied mathematics, combinatorics, differential geometry, logic, mathematical physics, number theory, numerical analysis or probability may be used for this purpose.

(2). Students can pass the QR Exams in two of the pure areas (algebra, analysis, or geometry/topology), and demonstrate mastery of a third core area by passing both ("alpha") courses in the corresponding subject with a B average (but neither can be a C+ or lower). As in the previous case, such students must also successfully complete one further course in an area outside their chosen three core areas. This is the most typical way students satisfy the QR requirement.

(2'). Students can pass the Qualifying Review Exams in two of the pure areas (algebra, analysis, or geometry/topology), and then pass one core course (with a B- or higher) in each of the remaining two core areas with at least a B average. Such students must also successfully complete one further course in an area outside the core areas.

(3) Students can pass the Qualifying Review Exam in Applied Analysis and another core area (algebra, analysis, or geometry/topology), but in this case they must pass (with a B- or higher) one core ("alpha") course in each of the two remaining core areas with a B average. Such students must also successfully complete one additional course in pure mathematics, such as algebra, pure analysis, geometry, topology, or discrete mathematics.

In satisfying the coursework requirement for the QR, a student may substitute a more advanced course in the same immediate area for a core course, provided permission is granted by the Chair of the Doctoral Committee. For example, the student may be able to substitute a more advanced Complex Analysis course for Math 596. However, a student may not substitute a more advanced Complex Analysis course in place of Real Analysis (Math 597). Such substitutions should be courses with problem sets and/or exams, providing evaluation and feedback; topics courses with no graded work are generally not appropriate.

Whether or not a particular course is acceptable for meeting the QR requirements in a particular semester is at the discretion of the Chair of the Doctoral Committee and requires his or her approval.

In all cases, the decision as to whether or not a student has passed the Qualifying Review is made by the Doctoral Committee on the basis of the QR examination results and the entire academic record. Additional faculty input is often sought before making a final decision.

**Important Deadlines:** Students must pass at least one QR exam by the start of their fourth term. For example, a student entering the doctoral program in the Fall of 2019 must pass the QR examination in at least one area by early January 2021. The Qualifying Review must be completed by the start of the 6th term. For example, the same student entering the doctoral program in the Fall of 2019 must pass the QR examination in at least one area by early January 2021, and complete the remaining requirements for the Qualifying Review by early January 2022.

### Advising in Stage 1

Students in Stage 1 are assigned a member of the Doctoral Committee as their counselor, who serves as their primary academic counselor, upon arrival in Ann Arbor. Students meet with their academic counselor before each term to decide what courses to take. Beginning students typically take three courses, carefully selected to progress the student toward completing the Qualifying Review, although students who are employed as GSIs need only register for 6 credits (typically 2 courses), which may be advisable in many cases.

Students may seek advice from any faculty member, and in fact are encouraged to get to know their professors and other faculty and seek advice especially from faculty in their areas of interest.

Students also meet with the Chair of the Doctoral Committee after each attempt at a QR exam to discuss the student’s progress and plan of study.

The staff in the Graduate Student Services Office are also an important resource for students at every stage of the program. It is important to pay attention to communication from the office which can contain information about deadlines, funding, and other time-sensitive and crucial issues. Staff often understand program rules, funding issues, and which professors might be good for students to talk for advice in different situations.