### Preliminary Examinations and Coursework Policy - "3+2+2" Requirement

The department offers two-semester course sequences in six core areas: Algebra, Analysis (real and complex), Applied Mathematics (functional analysis), Numerical Analysis, Probability, and Topology (algebraic and differential). Examinations (“prelim exams”) covering these areas are administered twice per academic year: in August before the start of the Fall semester, and in January before the start of the Spring semester.

Two 90-minute exams are administered for every area, each covering one semester’s worth of material from the corresponding prelim course sequence. (The course syllabi can be found here.) Whenever possible, exams covering different areas are administered on different days. The two exams covering a single area are administered sequentially on the same day, with a brief rest break in between. Exams are graded separately, so that a student’s proficiency on each semester’s worth of material is assessed independently.

Of the 12 Prelim segments, students must pass at least 3 Prelim Exams and 4 Prelim Courses, all distinct. A passing grade in a course is a "B". To remain in good academic standing within the Mathematics Department, students are required to:

- By the beginning of their 2nd Semester: Pass 1 Prelim Exam

- "3+2" - By the beginning of their 4th Semester: Pass 3 Prelim Exams and 2 Prelim courses, all distinct

- A student becomes eligible to take a candidacy oral exam once the above requirements are completed

- "3+2+2" - By the beginning of their 8th semester (or before the beginning of the semester of their PhD thesis defense, whichever comes first), students are required to pass two additional prelim courses, distinct from the "3+2"

There are three ways in which students may skip some or all of the four required prelim classes: (1) Students may choose to pass prelim exams in the corresponding areas. (2) Students with prior graduate coursework may appeal to the Graduate Adviser for waivers of one or more prelim course requirements. (3) Interdisciplinary students, with the advice of an academic supervisor and permission of the Mathematics ASGSC (Administrative Subcommittee of the Graduate Studies Committee) may be allowed to substitute courses in their specialty for some of the four required prelim courses. There are no waivers for Prelim Exam requirements.