Stage 1

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.

Change of Requirements

The rules below were modified in the summer of 2020 and apply starting in the Fall of 2020. However, students who started in the Fall of 2019 or earlier can choose to use the old system to pass the Qualifying Review. Individual cases will be considered by the Doctoral Committee.

Overview

To pass the Qualifying Review, a student must demonstrate proficiency in six out of eight subjects of core material from the following areas: algebra, analysis, geometry/topology, and applied analysis. The department offers eight “alpha” or core courses designed to help Ph.D. students achieve mastery of the corresponding subjects:

• Algebra:  Math 593 (Algebra I), 594(Algebra II)
• Geometry/Topology: Math 591 (General and Differential Topology), 592 (Algebraic Topology)
• Analysis: Math 596 (Complex Analysis), 597 (Real Analysis)
• Applied Analysis: Math 556 (Applied Functional Analysis), 572 (Numerical Methods for Differential Equations)

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 for Stage II. Additional faculty input is often sought before making a final decision.

Qualifying Review (QR) Exams

A Qualifying Review (QR) Examination is offered for each of the subjects above. Students can demonstrate proficiency in a subject by passing the QR 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 may 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 three-hour written examination. 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 two QR exams by the start of their fourth term. For example, a student entering the doctoral program in the Fall of 2022 must pass the QR examination in at least two subjects by early January 2024.

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.

Exam Protocols & Procedures

There are eight QR subjects, grouped by two into the following subjects: Algebra, Analysis, Applied Analysis, and Topology. There are two Examiners for each of the four groups: they are appointed by the Chair of the Mathematics Department. Each pair of QR Examiners writes the corresponding two QR exams in consultation with recent instructors of the corresponding alpha/core courses. 

Each exam consists of 5 questions with a 3 hour time limit.  Exams are administered by one of the exam writers and in accordance with any accommodations granted and requested by a student’s Student Services Department Verified Individualized Services and Accommodations (VISA). Students put a 3-digit secret number on the exam as an identifier instead of their name.

The exams are graded separately by the 2 examiners and the scores are then combined. All the scores are discussed by the Doctoral Committee, who determines who passes each QR subject based mainly on the results of the exam, but occasionally also on other factors.

The final results by the Doctoral Committee are shared with the faculty, communicated to the student by letter, then discussed in a personal meeting with the Doctoral Chair.  Multiple attempts to pass the Qualifying Review exams are encouraged with the following deadlines: Math Ph.D. students must pass at least two QR exams by the start of their fourth term, and the whole Qualifying Review must be completed by the start of the 6th term. 

Completing the Qualifying Review Requirement:

There are a few ways to satisfy the QR Requirement:

(1). Students can pass six Qualifying Review Exams. In this case, at least one additional graduate course, not a core course in the chosen subjects 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 five QR Exams and demonstrate mastery of a sixth subject by passing the "alpha" course in the corresponding subject with a grade of B or higher. As in the previous case, such students must also successfully complete one further course in an area outside their chosen subjects of examination.

(3). Students can pass four QR Exams and demonstrate mastery of two more subjects by passing the corresponding "alpha" courses with a grade of B or higher. As in the previous cases, such students must also successfully complete one further course in an area outside their chosen subjects of examination.

(4). Students can pass three QR Exams and demonstrate mastery of three more subjects by passing the corresponding "alpha" courses with a grade of B or higher. As in the previous cases, such students must also successfully complete one further course in an area outside their chosen subjects of examination.

(5). Students can pass two QR Exams, demonstrate mastery of one additional subject by earning an A or A+ in the corresponding “alpha” course, and the remaining three subjects by passing the corresponding "alpha" courses with a grade of B or higher.

(6). Students can pass one QR Exam, demonstrate mastery of two additional subjects by earning an A or A+ in the corresponding “alpha” courses, and the remaining three subjects by passing the corresponding "alpha" courses with a grade of B or higher.

(7). Students can demonstrate mastery by earning an A or A+ in the “alpha” courses of three subjects, and the remaining three subjects by passing the corresponding "alpha" courses with a grade of B or higher.

In satisfying the coursework requirement for the QR, a student may substitute a more advanced course in the same subject as the 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. 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.

Deadlines:
Students must pass at least two QR exams by the start of their fourth term. Any grade of A or A+ in an “alpha” course counts towards this deadline.  The Qualifying Review must be completed by the start of the sixth term. For example, a student entering the doctoral program in the Fall of 2023 must pass the QR examination in at least two subjects by early January 2025, and complete the remaining requirements for the Qualifying Review by early January 2026.

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.