Degree Programs & Requirements

Note: The information and policies contained on this website is for informational purposes only. The Undergraduate Record and Graduate Record represent the official repository for academic program requirements. These publications may be found at

Official requirements for our graduate degrees are listed in the Mathematics section of the Graduate Record.

The general rules and requirements governed by the Graduate School of Arts and Sciences can also be found in the Graduate Record under Policies.

PhD Requirements

Course Requirements

Courses and research amounting to 72 credit hours. Courses must include satisfactory work in two semesters of analysis (MATH 7310, 7340), algebra (MATH 7751, 7752), and topology (MATH 7820, 7800), or the equivalent. Students may take approved courses from other departments. Students entering our program with a masters degree from another institution may get credit for up to 24 hours of coursework.

General Examinations

Passing grades on two General Examinations, chosen from Algebra, Analysis, and Topology. These written exams cover the following material: Algebra (MATH 7751, 7752), Analysis (MATH 7310, 7340), and Topology (MATH 7820, 7800). The Analysis exam is split into two parts: Complex and Real, which are taken and graded independently.

These exams are typically offered twice a year, January and August. Students are expected to attempt two different general examinations by August of their second year, to pass at least one general examination by January of their second year, and to pass both exams by January of the student's third year. A student may repeat the examinations as many times as needed in order to satisfy these requirements.

Current syllabi for the Algebra, Analysis, and Topology General Examinations:

Examples of past general exams can be found here.

Second-Year Proficiency Examination

The purpose of the oral exam is to assess whether a student is prepared to begin doing research in their chosen area with a thesis advisor. The exam format is flexible and can be tailored to the students’ and examiners’ preferences. For example, it can consist of the student giving a presentation, followed by a question-and-answer period with the examiners, but it could also simply take the form of a question-and-answer exam. Typically these exams should be expected to take 1.5-2 hours.

A student should develop an exam syllabus together with the intended thesis advisor and one additional permanent faculty member. The nature of the syllabus can be flexible, depending on the research area and preferences of the student and examiners. It could still take the form of a list of topics in a given area, drawn from a course or from standard textbooks, but could also take the form of a review of relevant papers, or a suitable combination. The quantity of material should be roughly comparable to the content of two second-year courses. While the exam format can be flexible, in the ways discussed above, it should be specified in the syllabus for approval.

The syllabus should be submitted to the Graduate Committee at least four weeks before the intended exam to allow for adequate time for review and revision, if necessary. After the exam, regardless of the outcome, a form should be submitted to the department, including signatures of the two faculty examiners, indicating which of the two is the intended advisor, as well as the Director of Graduate Studies, to confirm that the syllabus was approved by the Graduate Committee.

Students already enrolled in Fall 2022 or earlier, before this format was approved, may opt to take the oral exam in the previous format of two faculty examining a student on the material from two semester courses on material beyond the first-year courses.

Teaching Requirements

Successfully teaching and directing discussion sections of undergraduate courses, and participating in the professional training seminars MATH 7000 and MATH 7010. Teaching is evaluated, and a good record is an asset to graduates seeking academic employment.

Dissertation and Defense

Written under the supervision of the major advisor, the PhD dissertation must contain original contributions to the field of mathematics. The main results of the dissertation are presented at a public oral defense. A committee consisting of the major advisor and three other faculty members (two from within the Department and one from outside) must approve the dissertation and defense in order for the dissertation to be accepted by the faculty.

Masters Degree Requirements

Please note that Department of Mathematics does not normally admit students seeking only a Masters degree; we are primarily a PhD program.

Information on the The 4+1 BA/MA Program for current UVA undergraduate students