Within two years of entering the Ph.D. program in Math, a student must pass at least 2 written comprehensive exams, one of which must be from Area I described below, and the other of which may be from Area I or Area II. Each exam a student passes may be used to meet the breadth requirements for coursework.

MS students may only take the written comprehensive exams if their home unit is the School of Math. Moreover, MS students may only register for the Analysis and Algebra exams.

#### Scope

The exams will each be 3 hours long and will be administered in the first 2 weeks of the Fall and Spring semesters. There is a detailed syllabus for each exam that correlates with graduate course offerings, but the exam syllabus need not completely match any course syllabus. The graduate coursework should help a student prepare for the corresponding exam, but it is each student’s responsibility to master the appropriate material before taking an exam.

#### Subjects and Syllabi

The following links lead you to the syllabi for the exams.

**Area I:**

**Area II:**

- Differential Equations
- Discrete Mathematics
- Topology (corresponds to Geometry and Topology subject area)
- Numerical Analysis
- Probability (corresponds to Probability and Mathematical Statistics subject area)

#### Exam Creation and Grading

Each exam will be created by a committee of 2 faculty members chosen by the Graduate Director. Usually at least one of the people on the committee will have taught related courses within the prior two semesters. The exams will be graded anonymously. Each student’s exam will be assigned a number, and that number will be the only identifying information on the exam.

Each exam will consist of 6 to 10 questions, of which a student may attempt up to 5. A student will pass the exam if 3 problems are worked “almost perfectly” and some progress is made on a fourth problem. (This is the “3 + ε rule.”) The Graduate Committee will consider the recommendations of the graders and make the final decision as to who passes an exam. A student is entitled to see his or her exam, ask questions of the graders, and appeal its grading to the Graduate Committee.

#### Administration

A student wishing to take an exam must register for the exam no later than the Week Preceding Final Exams in the semester before the exam is administered. Exams from Area I will be given every Fall and Spring semester, but exams from Area II will only be given if a student registers for it. An entering Ph.D. student may take any exam given in the semester he or she enters the program. MS students may sign up only for the Algebra and Analysis exams.

#### Spring 2022 Exam Schedule

In the Spring 2022 semester, we plan to hold the Math PhD written comprehensive exams online. The exams will occur at the dates and times below. If a registered student prefers to take it in person rather than online, they may do so. We will post details for a room location as the date approaches.

- Friday, January 14, 2:00-5:00 PM: Algebra
- Wednesday, January 19, 2:00-5:00 PM: Analysis
- Friday, January 21, 2:00-5:00 PM: Discrete Mathematics, Numerical Analysis, and Topology

#### Past Exams

There is an extensive library of **past comprehensive exams** which can help students prepare for the exams.