Ph.D. Program in Mathematics

The Ph.D. program in Mathematics at UAF involves intensive work directly with a faculty supervisor in order to prepare the student for a career in mathematical research and teaching. As the program is small, there is little formal coursework. Students prepare for their written Ph.D. qualifying exams in three broad areas primarily through directed study. An oral exam ensures both breadth and depth in the general area of the student's research field, while the thesis work provides entry into conducting mathematical research.

Admission to the Ph.D. program requires that the candidate already hold a Masters degree, or equivalent. Prospective applicants should contact faculty members directly about potential research supervision before applying. Research areas must of course align with current faculty interests.

Teaching and research assistantships are available. Application materials are the same as outlined elsewhere on this webpage for the Masters program .

Examinations in the Mathematics Ph.D. Program

The Mathematics Ph.D. program seeks to produce mathematicians with both a solid knowledge of core mathematical fields and expertise in a more focused area of research. Because of constraints on curricular offerings in a small department, achieving these goals requires the program be structured so that much of the necessary learning is undertaken independently, with faculty guidance. A system of examinations both encourages appropriate study and ensures adequate student performance.

In progressing toward the Ph.D. degree students will undergo three sets of examinations, with different purposes.          

1. Written comprehensive exams, to ensure a broad doctoral-level understanding of mathematics in at least three areas. The breadth of these exams will be defined by the mathematics graduate faculty following national norms, and can be expected to span multiple courses and/or exceed formal curricular offerings at UAF.
2. An oral comprehensive exam, to ensure a deeper understanding of those parts of mathematics broadly relevant to the student’s area of research focus. The breadth of this exam will be defined by the student’s graduate committee, and will typically require the student undertake substantial independent study.
3. An oral thesis defense, at which the student is examined on his or her own research and mathematics directly related to it.

Normally, these shall occur in the second, fourth, and fifth years following admission to the Ph.D. program.

A. Written comprehensive exams:

1. The exams will consist of three two-hour exams in different topic areas. The three exams need not be taken in the same semester. An exam period, set by the Mathematics graduate committee, will occur once each semester, and normally fall in September and February.
2. The specific choice of the exam areas will be made by the student in consultation with his or her graduate committee. Two of the areas must be chosen from Algebra, Real Analysis, and Topology, and those exams will cover material as described on departmentally-approved study syllabi. The third area need not be in this list, but the material tested must attain a depth comparable to that of the other exams. A study syllabus for this exam will be set by two faculty members who agree to author and grade the exam.
3. Each exam will be jointly authored and independently graded by two faculty members. Grades of ‘Pass’ and ‘Fail’ will be assigned by  consensus of the graders, and student will be informed of results within two weeks of the end of the exam period.
4. To pass the comprehensive written exams, a student must pass all three exams. A student who attempts and fails an exam will be allowed only one additional opportunity to pass that exam. (If a student substitutes a different topic area for this additional exam opportunity, only one attempt will be allowed on the new topic.)
5. A student who has not passed the exams by the end the third year in the Ph.D. program, may not remain in the program.

B. Oral comprehensive exam:

1. The oral exam will be scheduled and organized by the student’s advisor, but conducted by the student’s entire graduate committee. Each committee member is expected to prepare and ask several questions. Other departmental faculty may participate if they wish.
2. A study syllabus for the oral exam will be set by the student’s graduate committee. This syllabus is created when the student `advances to candidacy’ for the Ph.D., but modifications can occur with committee approval.
3. As required by UAF policy, an outside examiner must be present for the oral exam. The advisor will be responsible for arranging an examiner through the graduate school.
4. A ‘Pass’ or ‘Fail’ grade will be assigned by consensus of the student’s committee at the conclusion of the exam, and immediately communicated to the student.
5. A student who fails the oral exam may make one additional attempt to pass within one year of the failure. A student who has failed twice, or who has not passed the exam by the end of the sixth year in the Ph.D. program, may not remain in the program.

C. Thesis defense:

1. The thesis defense will begin with a publicly-announced presentation by the student at which the student’s graduate committee is present. After a period for audience questions, the committee and other departmental faculty who wish will further question the student privately. Questions should be relevant to the student’s research.
2. As required by UAF policy, an outside examiner must be present for the thesis defense. The advisor will be responsible for arranging an examiner through the graduate school.
3. A grade of ‘Pass,’ ‘Conditional Pass,’ or ‘Fail’ will be assigned by consensus of the committee, and will include both an assessment of performance during the defense and of the written thesis document. The grade will be immediately communicated to the student. Conditions for converting a ‘Conditional Pass’ to a ‘Pass’ will be set by the committee.

Syllabi for Written Comprehensive Exams

• Algebra
• Analysis
• Topology

Faculty