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 the applicant's transcripts, letters of recommendation, and GRE scores.

Requirements: The PhD degree has no rigid course requirement beyond the residency requirement (however, breadth and depth of knowledge are strongly encouraged).  It does require (1) passing written and oral qualifying examinations, (2) writing a dissertation embodying the results of original research which is acceptable to the student's dissertation committee, (3) a final oral defense of the dissertation, and (4) a language/research skills requirement.

Language Research Skills Requirement: The student must demonstrate competence in two areas:  either two natural languages, or one natural language and computer science, or one natural language together with sufficient improvement in English, if the student is an international student whose English is initially inadequate.  See the section on Language/Research Skills Requirements in the Graduate Guide Book for detailed requirements.

A student's progress towards the PhD degree is initially supervised by a three-person Preliminary Advisory Committee, formed at the beginning of his or her graduate program.  The student's faculty advisor chooses this committee, and is its chair.  After the student has passed the Written Qualifying Exams, and before taking the Oral Qualifying Exam, the Advisory Committee is increased to five members.

The PhD Qualifying Examination System consists of two parts.  The first part consists of four Written Qualifying Exams and the second consists of an Oral Qualifying Exam. 

Written Qualifying Exams are offered every year in August before the start of Fall semester classes and in January before the start of Spring semester classes.  Study guides and copies of previous qualifying exams are available on the Graduate Program website for students to use in preparing for their Written Qualifying Exams.

The Written Qualifying Exams are divided into three groups:

Group 1:  Complex Analysis, Real Analysis

Group 2:  Algebra; Topology

Group 3:  Probability; Numerical Analysis

Each PhD candidate is required to pass four Written Qualifying Exams, including both exams from Group 1 and at least one exam from Group 2.  The exams in Group 1 are two hours long, and the other exams are three hours long.  Each of the six introductory 8000-level courses (MATH 8000, 8100, 8150, 8200, 8500, and 8600, along with the associated 8xx5 problem session) is designed to help prepare students for the written qualifying exam in the corresponding subject area. 

The Written Qualifying Exams may be taken in any order, and more than one exam may be taken at a time.  An exam may be repeated until passed; however, timely completion of the Written Qualifying Exams is expected according to the Progress Guidelines.  For each written qualifying exam taken by a student, an examining committee decides on a pass/fail recommendation communicated to the student’s advisory committee.  The final determination of pass or fail on a written examination lies with the student's committee.  The student’s committee may reverse the examining committee’s decision or may choose to 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 the area of specialization. It is to be taken within 9 months of the time the student passes his or her last Written Qualifying Exam.  (A student who passes Written Qualifying Exams early will be allowed additional time to pass the Oral Qualifying Exam.)  To begin preparing for the Oral Qualifying Exam, the student decides upon a thesis advisor.  At this time the student's committee will increase from 3 to 5 members.  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.