Doctoral Program (Ph.D.)
Prerequisites: To enter the Ph.D. program a student should hold at least a Bachelor's degree in mathematics. The academic record of a student applying to the Ph.D. program should contain substantial evidence that the student will succeed in the doctoral program. In reviewing an applicant's folder, the Graduate Committee gives substantial weight to the applicant's transcripts, letters of recommendation, and GRE scores.
Requirements: The Ph.D. degree has no rigid course requirement beyond the residency requirement (however, breadth and depth of knowledge are strongly encouraged). It does require (1) knowledge of two "languages" as discussed below, (2) passing written and oral qualifying examinations, (3) writing a dissertation embodying the results of original research which is acceptable to the student's dissertation committee, and (4) a final oral defense of the dissertation. A student's progress towards the Ph.D. degree is initially supervised by a three person committee, increasing to four or five members following the written qualifying exams. The student's faculty advisor chooses this committee, and is its chair.
The Ph.D. Qualifying Examination System consists of two parts. The first part consists of three Written Quals and the second consists of an Oral Qual.
Qualifying Exams are offered in Algebra, Real Analysis, Complex Analysis, Topology, Numerical Analysis and Probability. The 8000-level sequences in each area are designed to prepare the student for Qualifying Exams. Written Qualifying Exams are offered every year in August, during the week before the start of classes, and in January. Syllabi, and copies of old Exams, are available from the department office for students to use in studying for Exams.
Each Ph.D. candidate is required to pass four Qualifying Exams, including both Analysis Qualifying Exams and either the Topology or Algebra Qualifying Exam.
The final determination of pass or fail on a written examination lies with the student's examining committee. The committee may elect to reverse the decision of the examiner (with a unanimous vote) or may administer its own examination in addition.
The Oral Qualifying Exam is based on the student's anticipated area of specialization. In it, the student is expected to present material from a research paper and to answer general questions about his or her area of specialization. It is to be taken within 9 months of the time the student passes his or her last Written Qual. (A student who passes Written Quals early will be allowed additional time to pass the Oral Qual.) To begin preparing for the Oral Qual, a committee of five is chosen (including the student's thesis advisor). The student, advisor, and committee agree upon a body of material for which the student will be responsible. The student reads research papers in the area: in general, in the examination, the student presents a 30-minute lecture on those prepared papers, followed by a question period of at least one hour on the paper and background material.
Ph.D. Language Requirements: A student must either demonstrate: a reading knowledge of two foreign languages with significant mathematical literature, chosen from French, German, and Russian; OR a reading knowledge of one of the above languages, and sufficient competence with computers to do mathematical research; OR a reading knowledge of one of the above languages, together with sufficient improvement in English, if the student is an international student whose English is initially inadequate.
Language/Research Skills: A student can satisfy the language requirements by passing an appropriate course (French, German, or Russian 2001 or above, or CSCI 7010) with a B- or better; OR by translating an unfamiliar mathematical paper, using a dictionary, in a reasonable length of time (3-4 hours for a 4 page paper), to the satisfaction of a qualified examiner from the Mathematics faculty; OR by having native proficiency in one of the languages above, as certified by a qualified examiner; OR doing a computer project.
See the Graduate Guidebook for full details.