Overview
The general aim of the doctoral program in mathematics is to prepare qualified graduate students to become productive workers in industry or government, or academic research scholars capable of communicating their knowledge to students and to the mathematical community. The program is planned to develop in the student a fundamental understanding of certain basic fields of mathematics, a deep understanding of a major field of interest, the ability to formulate and recognize significant research problems, and the ability to analyze problems, reach solutions, and transmit ideas to others.
Requirements
A total of 72 units are required. Students must complete at least two semesters of full-time study in residence (both semesters must be in the same academic year or in consecutive years). Whenever possible, it is strongly recommended that all of the graduate work be done in full-time resident study. A student who is well prepared can expect to complete the Ph.D. program in four to five years of full-time study or the equivalent, but some students will find it necessary to take longer than this.
Upon arrival, students are given a more detailed description of the regulations and procedures governing the administration of the graduate program in mathematics. Students are strongly encouraged to obtain good working knowledge of computer programming, and knowledge at least equivalent to a year’s course in applied analysis and a year’s course in probability and statistics.
Written Qualifying Examination
All doctoral students in mathematics are required to pass the written qualifying examination, which is administered at the end of the spring term. This examination tests the students’ mastery of the following topics from their undergraduate studies: advanced calculus, complex variables (including contour integration), real and complex analysis, linear algebra, probability, and differential equations (ordinary and elementary partial). A reading list and past exams are available upon request. Ph.D. students must take this examination after the completion of their first academic year of coursework. Students who do not pass the examination within two attempts will not be allowed to continue in the Ph.D. program in Mathematics.
Written Preliminary Examinations
During their second or third year of study, all doctoral students in mathematics are required to pass two written preliminary examinations in advanced topics from the graduate mathematics curriculum. Each preliminary examination covers roughly two semesters of advanced coursework, and the students choose these topics with the approval from their academic advisor and the program chairman.
Research Tools
Shortly after the completion of the qualifying and preliminary exams, doctoral students in mathematics must demonstrate proficiency with two research tools. Acceptable research tools include: foreign languages (French, Russian, or German), a computer programming language relevant to mathematics, a database language, and statistical research methods. Proficiency is usually determined by written examination, but may also be demonstrated by relevant coursework.
Oral Qualifying Examination
To be advanced to candidacy and begin the dissertation research, doctoral students in mathematics must make an oral presentation of their dissertation research proposal. In preparation for this examination, the student is expected to spend at least one semester in advanced graduate or directed reading courses where exposure to research material is emphasized, and the student is expected to gain some facility in the appropriate research techniques.
The student, in consultation with the student’s qualifying examination committee and with the approval of the faculty, must prepare a written dissertation proposal containing an outline of the research to be undertaken, references to relevant source materials, and a list of mathematical support areas appropriate to the research topic. The qualifying examination will test the student’s knowledge and understanding of the basic mathematics and the research approaches, and will assess the soundness of the dissertation proposal.
Before taking the oral qualifying examination, the student must have completed satisfactorily the written qualifying examination, the two written preliminary examinations, the research tools requirements, and two years of full-time graduate study or the equivalent (at least one year at CGU). The student may take the qualifying examination at any time thereafter, when prepared to demonstrate breadth of knowledge at an advanced level and some acquaintance with an area of research. The qualifying examination may not be attempted more than twice without the approval of the faculty.
Dissertation and Oral Defense
The final requirements for the Ph.D. in Mathematics are the publication of the dissertation and the oral defense of the dissertation.
Other Requirements
General Ph.D. requirements at CGU may be found in the Bulletin under “Degree Regulations.”