Ph.D. Programs

Students who enrolled in the Ph.D. program in Fall 2025 or later must pass two qualifying exams by the end of their second year in the program. Each exam consists of two parts chosen by the student from three possible subjects. These subjects are based on the following one-semester courses:

• For Exam 1, students choose two subjects out of the following three options: Introduction to Analysis (MA 512), Functional Analysis (MA 515), Topology (MA 551).
• For Exam 2, students choose two subjects out of the following three options: Linear Algebra (MA 520), Abstract Algebra I (MA 521), Numerical Analysis I: Discrete Approximations (MA 780).
• Each exam is a 3-hour long written exam. Students can work on both parts of the exam during the whole time.
• Exams are offered every year in August before classes start and in May after the final exams.
• Incoming students may attempt the exams in August before they enroll in their first year. If they do not pass an exam, they have to register for a course from the corresponding group. Additionally, if they lack an undergraduate Analysis II course, they will have to take an analog of MA 426, either as MA 591-001 or starting from Spring 2026 as the new course MA 512.
• It is recommended that students attempt both qualifying exams by the end of their first year. All Ph.D. students must pass both exams by May of their second year.
• Each exam part is written and graded by two faculty members. In order to pass an exam, both parts of the exam must receive a passing grade.
• At the end of the Spring semester, the two exam writers for a particular subject will come up with a study guide for the qualifying exams that year. That study guide will be given to the students in the course, and will be given to the DGP so that any students who want to take the exam and who were not in the course will be able to prepare for the exam.
• Retakes can be done both in August and May, and are limited only by the time deadline that students must pass two exams by the May of their second year.
• When an exam is retaken, the student must take both parts of the exam, even if one part has previously received a passing grade. Retakes do not have to be in the same subjects as those initially failed.
• It is possible to “drop” an exam one has registered for any time up to two weeks before the exam (no questions asked). After that, we recommend students talk to the DGP.
• For part-time students, years will be counted using credit hours, with one year equal to 18 credit hours. For full-time students, calendar years are used. Students who start the graduate program in the spring semester should have a statement from the DGP put into their file specifying the date by which their exams must be taken.

Students who enrolled in the Ph.D. program in Fall 2024 or earlier must pass three exams in subjects selected by the student from a list of twelve possibilities. Each subject is represented by a two-semester sequence of courses from among the following:

• For PhD students without a prior Master’s degree in Math or Applied Math: Such students must attempt as many exams as necessary to complete the requirement by August before the start of the third year. If they do not pass all exams at that time, they have until January of the third year to pass all remaining exams.
• For PhD students with a prior Master’s degree in Math or Applied Math: Such students must attempt as many exams as necessary to complete the requirement by August before the start of the second year. If they do not pass all exams at that time, they have until January of the second year to pass all remaining exams. A student with a prior Master’s degree who does not pass any exams when entering the program should enroll in 3 qualifying exam sequences in the first year so that they can attempt 3 exams in the August before the start of the second year.
• For all PhD students: Each exam is written and graded by two faculty members.
• At the end of the Spring semester, the two exam writers for a qualifying exam will come up with a study guide for the qualifying exams that year. That study guide will be given to the students in the course, and will be given to the DGP so that any students who want to take the exam and who were not in the course will be able to prepare for the exam.
• Each exam is a 3-hour long written exam.
• Retakes can be done both in August and January, and are limited only by the time deadline that students must pass all exams by the January of the student’s third year (second year for students with a previous MS degree in Math or Applied Math).
• Retakes do not have to be in the same subjects as those initially failed.
• The number of examinations taken at any given exam period cannot exceed the remaining number of passes need to reach a total of three.
• It is recommended that students attempt at least one to two exams by the end of their first year. By the end of their second year (first year for students entering with an MS in Math or Applied Math), students have to attempt a number of exams equal to the number of exams they have yet to pass.
• It is possible to “drop” an exam one has registered for any time up to two weeks before the exam (no questions asked). After that, we recommend students talk to the DGP. Dropping an exam only applies to students who do have the option to “wait”, i.e., are not required to take the exam at that exam session.
• For part-time students, years will be counted using credit hours, with one year equal to 18 credit hours. For full-time students, calendar years are used. Students who start the graduate program in the spring semester should have a statement from the DGP put into their file specifying the date by which their exams must be taken.

Faculty have provided these “suggested paths” of courses that students can take, depending on their area of interest.  Note, however, that the particular courses most relevant for an individual student will vary depending on which faculty the student works with, and what project the student works on.  Hence, it’s important to talk to faculty early to figure out the best courses that are right for each student.