Doctoral Degree Program in Mathematics
The department offers a Ph.D. degree with tracks in either pure or applied mathematics. The pure mathematics track is primarily in algebra, analysis, and topology. The applied mathematics track covers a broad spectrum of research areas, including dynamical systems, partial and ordinary differential equations, integral equations, optimal control theory, numerical analysis, computational harmonic analysis, statistics, stochastic processes, coding theory, and mathematical biology.
Recent graduates have written dissertations on the theory of non-commutative rings and modules, linear algebra, group theory, optimization theory, general and set theoretic topology, real analysis, boundary value problems, KdV equations, and nonlinear differential and integral equations, and algebraic coding theory.
Each student is encouraged to design a program of study suited to his or her needs, for close association between students and faculty members is a major strength of our department.
A student is admitted to candidacy for the Ph.D. degree upon completion of examinations and recommendation of his or her advisor. The Ph.D. candidate is expected to write a dissertation that is a scholarly work demonstrating the ability to understand, organize, improve, and present mathematical ideas of outstanding importance, depth, or interest. It should include original mathematical research and be worthy of publication.
Admission to Candidacy
Before a student is admitted to candidacy for the Ph.D. in mathematics, he or she must pass Comprehensive Examinations, which consist of two parts: a general examination and a specialty examination.
- Comprehensive Examination Syllabi (general part)
The General Examination
The purpose of this examination is to determine whether the student has assimilated a reasonably broad spectrum of mathematics. The examination consists of three separate tests. Students have a choice between two separate tracks for their general examination; these will be referred to as the "pure mathematics track" and the "applied mathematics track."
Students pursuing the pure mathematics track must take at least two of their three tests in the areas of real analysis, algebra, or topology. These three subject areas will be referred to below as "group A areas." The third exam may be taken in complex analysis, differential equations, functional analysis, numerical analysis, statistics, set theory or other subjects depending on the interests of the student and faculty, subject to the approval of the Graduate Committee. A student who wants to select a subject not explicitly mentioned in this document should make the request in writing one year in advance of the test. If only two of the tests are taken in a subject from group A, the student must demonstrate proficiency in the third subject of group A. If all three tests are taken in group A subjects, the student must demonstrate proficiency in a subject different from those in group A. Proficiency may be demonstrated by completing a two-quarter O.U. course sequence at or above the 600-level in the subject in question, with a grade of at least B in each course.
Students pursuing the applied mathematics track must take tests in real analysis, differential equations, and one of the following "group B" subjects: complex analysis, functional analysis, numerical analysis, or statistics. In addition, the student must demonstrate proficiency in a fourth subject, which may be another subject in group B, algebra, topology, set theory, or another subject depending on the interests of the student and faculty, subject to the approval of the Graduate Committee. Proficiency may be demonstrated either by passing a suitable examination or by completing a two-quarter O.U. course sequence at or above the 600-level in the subject in question, with a grade of at least B in each course.
Each of the three tests will be a "closed book" test and will be designed so that it can be reasonably completed in four hours, but students will be given eight hours per test. The three tests will be given in three consecutive weeks. There will be two "comps sessions" per academic year; a regular one in August and a make-up session in December/January. The student will schedule his or her first attempt in each of the three chosen subjects in the first or second summer after enrolling the Ph.D. program at Ohio University. It is possible to schedule the first attempt in some of the three subjects for the first summer and the first attempt in the remaining subject(s) for the second summer. If a student fails a test in the first attempt, he or she will be required to write a make-up test in this subject. These make-up tests will be scheduled in the next comps session after the original tests were taken. The student can take one make-up test in each of the three subjects. In addition, a second make-up test is allowed in at most one subject and has to be taken one year after the first attempt at passing the comprehensive exam in this subject has been made.
Each of the separate tests will be prepared and graded by an appropriate "examining committee" appointed by the Graduate Chairman with the approval of the Graduate Committee. After grading the test, each examining committee will report to the Graduate Committee whether the student has passed or failed the test. For passing the general examination, it is required that the student passes the tests (or make-ups, if applicable) in all three of the chosen subject areas. A student who does not pass the tests in all three subjects within the framework for possible make-up tests outlined above will be excluded from the Ph.D. program of Ohio University.
All Ph.D. students who have passed the
first part of their general examination will be required to sign up each
quarter for one credit hour of seminar (MATH 891). It will be up to the
Ph.D. advisor to specify the requirements for each advisee to obtain a
passing grade for the one credit hour course. Appropriate requirements
might be to regularly attend a specified research seminar and giving at
least one talk. It is recommended that the Ph.D. advisor consult the
organizer of the relevant seminar concerning appropriate criteria for
obtaining a passing grade. Instead or in addition, the student could also be required to attend a
specified number of departmental colloquia.
The Specialty Examination
After passing the general examination, the student, in consultation with the Graduate Committee, will choose a dissertation advisor. The advisor will initiate the formation of an Advisory Committee, consisting of the advisor and at least one other faculty member. The Advisory Committee will be in charge of the student's study plan, and will administer the specialty examination part of the Comprehensive Examinations no later than two years after the general examination part has been completed. At any time during this period, the Advisory Committee may recommend that the student be withdrawn from the Ph.D. program for unsatisfactory progress. The purpose of the specialty examination is to establish that the student is prepared to undertake the independent research necessary for a dissertation. Its nature will be determined by the Advisory Committee in concert with the Graduate Committee.
The Dissertation Committee
Shortly after the completion of the specialty examination part of the Comprehensive Examinations, a Dissertation Committee will be formed. (Usually the Advisory Committee will be a subset of the Dissertation Committee. The latter must include a College Representative.) The student will be required to present a research plan and a thesis proposal to the Dissertation Committee for approval, within three months from the date the Dissertation Committee has been formed. Then the student will officially be admitted to candidacy for the Ph.D.
