First Year Courses

Courses taken during the first year are usually chosen from the following list, which is designed to prepare students for the PhD qualifying examinations. Master’s students will typically take these courses as well in the first year before focusing in the second year on the requirements for a Master’s degree.

653, 656, 661, 662, 668, 669, 676, 677, 680, 681.

Qualifying Exam Requirements

There are five qualifying exams:

• Algebra: Math 676 and 677
• Analysis: Math 653 and 656
• Geometry-Topology: Math 680 and 681
• Methods of Applied Mathematics: Math 668 and 669
• Scientific Computation: Math 661 and 662

PhD qualifying exams are given twice each year, near the beginning of classes in August and January. PhD students are encouraged to take the PhD qualifying exams early and in principle would be ready to take them in August after the first year. The qualifying exams, sometimes called comps, are subject to the following rules:

1. A PhD student can pass either the Pure Math option for the for the qualifying examination or the Applied Math option.
2. To pass the Pure Math Option, a student is required to pass three of the five qualifying exams by the beginning of the sixth semester. Any three of the five can be passed.
3. To pass the Applied Math option, a student is required to pass Methods of Applied Math and Scientific Computation by the beginning of the sixth semester.
4. In each examination period a student can take any number of qualifying examinations, from one to five.
5. A graduate student who passes a PhD qualifying examination, as determined by the Graduate Committee, will receive credit for that examination toward the PhD requirements.
6. If a student does not pass at least one PhD exam by the beginning of the fourth semester, the department will not guarantee financial support past the end of the fourth semester. However, if such a student passes one exam by the beginning of the fifth semester, then reinstatement of financial support will be considered.
7. To remain in the PhD program, a graduate student must pass the written qualifying exam, with either the Pure Math or the Applied Math option, by the beginning of the sixth semester. A student who does not do this will be out of the PhD program.
8. Any exceptional cases will be decided by the graduate committee.

Course Requirements

All Ph.D. students must take and receive a grade of P or higher in at least six courses from the following two lists:

1. Comprehensive courses that are not basic courses for any of the three comprehensive exams passed by the student
2. Second tier courses
   • Algebra: Math 641, 643, 771, 774, 775
   • Analysis: Math 657, 751, 753, 754, 857
   • Applied Mathematics: Math 635, 761, 762, 768, 769, 892
   • Geometry-Topology: Math 781, 782, 773, 775, 776

Furthermore, of these six courses every PhD student must take and receive a grade of P or higher in three courses numbered over 700 from the second tier list

A student taking a pure math option, respectively an applied math option, may replace one, respectively one or more, course/s in the second tier requirement with other graduate level courses inside or outside of the department. For each replacement course, the student must obtain permission, prior to registration, from both the adviser and the graduate director.

Any incomplete grades must be resolved within one year (ideally within one semester).

Oral Examination Requirement

The doctoral oral exam assesses a student’s preparation and plans for their thesis research. It provides an opportunity for students to present their thesis plan, outline their expected contributions to the research area, and demonstrate their ability to contribute to their field of study.

Students should consult with their thesis advisor to determine the timing of their exam. The Department recommends that the oral exam be completed by the end of the third year of the program to ensure that students have time to receive and act on feedback provided by their exam committee.

Students are required to pass their oral exam by the end of their seventh semester in the program. Requests for exceptions will be evaluated by the Graduate Committee. The written comprehensive exams must be completed before the oral exam is scheduled.

The exam has two primary objectives:

• To confirm the student’s mastery of their specialized area.
• To provide feedback from the committee on the proposed research plan.

Students are expected to:

• Demonstrate a solid understanding of their specialization and the relevant
  literature.
• Present a clear and feasible thesis plan, defining the research problem, its
  significance, proposed methods, and an anticipated project timeline.
• Engage in a critical discussion with the committee about their research plan.

The oral exam typically lasts about one hour. The student will give a presentation on their research (about 45–50 minutes), addressing the key elements of their thesis plan. Questions and comments may occur during, after, or throughout the presentation.

The exam is conducted by the student’s PhD committee, consisting of five faculty members, and chaired by the student’s advisor. The general public is allowed to attend the presentation portion and may ask questions. Afterward, the committee will have time for private questions and comments with the student. Following this, the committee will deliberate in private to assess the student’s performance and vote on the outcome of the exam.

Note: Students who are currently (as of 10/08/2024) in their 6th semester and beyond are given a year to pass the exam.

Dissertation Defense

After finishing the dissertation the candidate gives a public oral presentation of the thesis and is examined on it by the PhD committee. For information about paper and electronic submission of dissertations consult the Thesis and Dissertation guide.

Computer Language Requirement

A Master’s or PhD student must pass a computer language requirement by demonstrating a certain level of programming ability. Please note the following guidelines:

• Computer language requirements are the same for PhD and Master’s students.
• Passing the following courses at UNC will be sufficient to satisfy the computer language requirement:
  MATH 565, MATH 566, MATH 661, MATH 662, MATH 761, MATH 762, COMP 110, COMP 116, COMP 121, COMP 401
• Any computer science course which lists one of these courses as a prerequisite is also acceptable.
• A Master’s or PhD student may also satisfy the computer language requirement by passing an approved one semester undergraduate course on computer programming at any university.
• If the course title title on the transcript is not self-explanatory, then a syllabus, text or other information may be required. In all cases where there is some question about whether a course fulfills the requirement the Graduate Director will decide in consultation with the Graduate Committee.

Teaching Requirement

Students are required to take and successfully pass the TA Teaching Seminar, a special section of Math 920, during their first fall semester of their program. Students are also required to perform a minimum of two semesters of instructional service.

A semester of instructional service can be satisfied by any of the following:

1. Teach one course: 12 hours
2. Lead 4 recitations (includes 110L and 231L): 3 hours each
3. Lead 2 labs (383L, 528L, 529L): 6 hours each

Recitations and labs can be completed in different semesters and combined to satisfy the requirement.  For example, if a student leads two recitations in a fall semester, and two recitations in a spring semester, these duties can be combined to count for one semester of instructional service.