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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.

#### Timeline

PhD students in Mathematics are required to pass two written comprehensive exams (at least one of which is Algebra or Analysis) within their first two years in the program. Students who do not meet this requirement may receive a third year to take the exams if they are in otherwise good standing.

MS students who entered an MS program in the School of Math after the Fall 2023 incoming class are not eligible to take written comprehensive exams. MS students with home unit in the School of Math who matriculated in the Fall 2023 incoming class (or earlier) and wish to transition into the PhD Math program must pass the Algebra and Analysis written comprehensive exams within their first two years in the MS program. Current MS students who are beyond two years may request an extra attempt at the comprehensive exams by petitioning the Graduate Committee, and in that case the student should send the petition to dgs@math.gatech.edu.

#### Scope

The exams will each be 3 hours long and will be administered by the end of 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 can register for a maximum of 2 written comprehensive exams in a given semester. 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 they enter the program by emailing dgs@math.gatech.edu at least 5 days before the first comprehensive exam.

#### Spring 2025 Exam Schedule

The exams will occur in person, most likely in Skiles 005. We will have an alternate plan for students who are in isolation due to covid or other illness.

- Monday, January 13: Analysis
- Wednesday, January 15: Algebra and Topology
- Friday, January 17: Differential Equations, Discrete Mathematics, Numerical Analysis, and Probability

#### Past Exams

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