**Qualifying Requirement for the Math PhD. **The requirement is meant to ensure that students have the foundational breath to develop their early career research program, and is meant to provide some guidance as they begin their graduate studies. Students are required to “qualify” in four graduate courses (from the list below) in (at least) three relatively distinct areas:

- Algebra & Number Theory
- Analysis & Probability
- Computational & Applied Mathematics
- Differential Equations
- Geometry & Topology

Example: Math 601, 620, 621, 631. To "qualify in Math XXX" a student must demonstrate **comprehensive proficiency** in the course material (see syllabi below). Whether or not this proficiency has been achieved will be assessed by the faculty instructor at the end of the course. (This assessment is distinct from the course grade.)

**Timeline.** Assuming that a student has the prerequisites for these courses, the expectation is that they will complete the requirement before the beginning of their second year (by July 31).

- Those students who are missing prerequisites will have an extended timeline. In such cases, the student will develop a plan (in consultation with their mentor) outlining their path to complete the qualifying requirement (no later than the end of the second year) and submit to the DGS before the first day of classes (mid-August).
- Faculty will meet August (after the first year, and before the second year begins) to discuss first-year progress and determine how to advise/direct any students who have not completed the requirement at that time.

**Testing-out of 601 and 631.** We offer a Qualifying Exam in these two courses in August (roughly the equivalent of the course final exam). If a student passes the exam, then they get credit for “qualifying” in the course.

**Oral option. ** Some students may be better served by having the option to qualify in a subject that is not amongst our QE courses. Such students could replace qualifying in a course with an oral exam on the subject. This should be a well-planned and thought-out part of the student’s first-year studies, and not become a path through which the student delays completing the requirement. To that end the student must make the following arrangements no later than January 31 of their first year:

- Form an exam committee of two faculty.
- Write an exam syllabus (in consultation with the exam committee), and submit to the DGS for approval.
- Schedule a (tentative) date for the oral exam. The exam must take place before the start of the student's second year (by July 31).

**Mentor's role. ** In consultation with their faculty mentor, the incoming students will develop a tentative plan/roadmap for completion of the qualifying requirement, and submit to the DGS/DGSA no later than August 31. This could be as simple as “I plan to qualify in Math 601, 620, 621, 631."

**Qualifying Eligible Courses & Syllabi. **The Qualifying Eligible courses have a set syllabus (see links below), regular, graded HW and a comprehensive final exam. (The parentheses indicate the area.) At the end of the term, the instructor will determine whether or not the enrolled Math PhD students "qualified in the course". This qualification is distinct from the course grade.

- 545: Introduction to Stochastic Calculus (Computational & Applied Maths)
- 553: Asymptotic and Perturbation Methods (Differential Equations)
- 555: Ordinary Differential Equations (Differential Equations)
- 557: Introduction to Partial Differential Equations (Differential Equations)
- 561: Numerical Linear Algebra, Optimization and Monte Carlo Simulation (Computational & Applied Maths)
- 563: Applied Computational Analysis (Computational & Applied Maths)
- 601: Groups, Rings and Fields (Algebra)
- 602: Introduction to Commutative Algebra & Algebraic Geometry (Algebra)
- 611: Algebraic Topology I (Geometry & Topology)
- 612: Algebraic Topology II (Geometry & Topology)
- 620: Smooth Manifolds (Geometry & Topology)
- 621: Differential Geometry (Geometry & Topology)
- 631: Measure & Integration (Analysis & Probability)
- 633: Complex Analysis (Analysis & Probability)
- 635: Functional Analysis (Analysis & Probability)
- 641: Probability (Analysis & Probability)

**Concurrent Masters.** The requirement for a concurrent Masters degree in Mathematics is:

- Ten mathematics courses at the 500/600 level.
- Qualify (as above) in three subjects in (at least) two distinct areas.