Department of Mathematics
University of Colorado at Boulder
Graduate Program Descriptions and Requirements
The Department of Mathematics offers programs leading to the degrees Master of Arts in Mathematics, Master of Science in Applied Mathematics, and Doctor of Philosophy in Mathematics. A student wishing to pursue graduate work in mathematics should carefully read the section of the University of Colorado catalog describing university requirements. The student is responsible for satisfying these requirements at the proper time.
PREREQUISITES FOR GRADUATE STUDY IN MATHEMATICS
To begin graduate work toward one of the degrees named above, a student should have at least 30 semester hours in mathematics, including, with grades of "B" or better, two semesters of analysis beyond lower division calculus, a semester of linear algebra, and either a semester of modern algebra or a semester of differential equations.
The general Graduate Record Examinations (GRE) are required and the Mathematics subject examination is highly recommended for all entering students. The GRE scores will be included in the evaluation process. Therefore, it is advisable to take these examinations as early as possible before the application deadline.
REQUIREMENTS FOR THE MASTER OF ARTS DEGREE IN MATHEMATICS
The student should select a member of the Graduate Faculty to serve as advisor, and together they should develop a degree plan. If the student has no preferred choice of advisor, then any member of the graduate committee can serve that role.
Thirty hours of approved graduate credit are required, as is a grade point average of 3.0 or higher in these 30 hours. Two courses that are two-semester sequences in mathematics should be included in these 30 hours. No more than 6 hours of graduate seminars or independent study can be used as part of this 30-hour requirement for the M.A. degree.
Master of Arts students have four years to complete degree requirements.
There are two tracks that a student may follow to obtain a M.A. in mathematics, a thesis option and a no thesis option.
PLAN I: THESIS OPTION FOR THE M.A.
As part of the 30 required hours of graduate credit, the student must complete 3-6 hours of thesis work. The thesis topic is to be chosen in consultation with the advisor. It is usually of an expository character and may be in any field of mathematics. The student will give a 50 minute presentation on the thesis to a committee, including the advisor and two other graduate faculty members, and will answer questions on the thesis.
PLAN II: NO THESIS OPTION FOR THE M.A.
Together with the advisor, the student should decide on a relatively specific topic to prepare. The student will give a 50 minute presentation on that topic to a committee, including the advisor and two other graduate faculty members, and will answer questions on the topic being presented.
OPTIONAL MINOR FIELD OUTSIDE MATHEMATICS FOR THE M.A.
Students choosing a minor field outside of mathematics must complete 6 semester hours in an approved area. Proposed minor courses in a field other than mathematics must be approved by the advisor and by the minor department at least one semester before the degree is to be awarded.
REQUIREMENTS FOR THE MASTER OF SCIENCE DEGREE IN APPLIED MATHEMATICS
The student should select a member of the Graduate Faculty to serve as advisor, and together they should develop a degree plan. If the student has no preferred choice of advisor, then any member of the graduate committee can serve that role. Thirty hours of approved graduate credit are required, as is a grade point average of 3.0 or higher in these 30 hours. At least 18 of these hours must be obtained through graduate mathematics courses. No more than 6 hours of graduate seminars or independent study can be used as part of this 30-hour requirement for the M.S. degree.
Master of Science students have four years to complete degree requirements.
To earn an M.S. Degree in Applied Mathematics from the Mathematics Department, a student must meet the following additional requirements:
- The 18 hours in the Department of Mathematics must include 2 two-semester sequences, one of which must be MATH 5600-5610, Numerical Analysis.
- Of the 30 hours of graduate work, at least 6 hours must come from an area of application of mathematics, e.g. Physics, Biology, Economics, etc. The advisor must approve the outside area and courses. The student may count up to 12 hours of courses in such an outside area towards the required 30 hours. These courses must include a full-year sequence in the area of application.
- The student must pass a written or oral Master's examination, which will be prepared by the Graduate Committee in consultation with the student's advisor.
- The student may replace 3-6 hours of course work with the writing and defending of a Master's thesis, which may be expository in nature. The writing of a thesis does not otherwise reduce any of the requirements (1)-(3) above.
REQUIREMENTS FOR THE DOCTOR OF PHILOSOPHY DEGREE IN MATHEMATICS
General admission requirements for the doctoral program are outlined above; see the University of Colorado catalog for certain course, residency, and communication requirements.
ACADEMIC REQUIREMENTS
A minimum of 60 semester hours of graduate level course work and thesis credit (combined) is required for all doctoral degrees within the Department. Of that minimum, students must complete at least 30 hours of course work in mathematics at or above the 5000 level with a GPA of at least 3.0. No more than 6 of these 30 hours may be from independent study courses or graduate seminars. In addition to the course work the student must accumulate at least 30 hours of doctoral thesis (dissertation) credit. A maximum of 10 dissertation hours accumulated prior to the semester in which the comprehensive examination is passed may be counted toward the required 30 hours. See the University of Colorado Catalog for further explanation of academic requirements for the Graduate school.
COMPREHENSIVE EXAMINATION AND ADMISSION TO CANDIDACY
Before being admitted to candidacy for a Ph.D. degree in mathematics, a student must satisfy departmental course requirements and must pass the Comprehensive Examination.
Current specific course requirements are the following: A student must complete a semester each of Complex Analysis and Topology and a second semester of one of these before taking the third portion of the Comprehensive Examination (see below).
The comprehensive examination comprises three parts: (1) a written examination in Algebra, (2) a written examination in Analysis, and (3) a third examination in an area to be agreed upon by the student and a faculty member who agrees to serve as chair of the examining committee.
The examinations in Algebra and Analysis will be offered shortly before the beginning of each semester. Syllabi for the Algebra and the Analysis examinations will be prepared by those members of the Graduate Faculty of the Department of Mathematics interested in each area. Students will be examined on the specific content of these syllabi. The syllabi are intended to be somewhat stable, and will not be changed without due consideration for students who might be affected. The examination will be constructed by committees, appointed by the Chair of the Graduate Committee, consisting of three members of the Graduate Faculty of the Department of Mathematics. These examinations (together with written solutions) will be submitted to the Graduate Committee for approval. The number of points assigned to each problem will be clearly indicated. Each examination will be graded by each member of the committee that constructed it, and a consensus score of 0-100 will be submitted to the graduate committee. Although a score of more than 70 ordinarily is a passing grade, and a score of less than 55 is ordinarily a failing grade, the graduate committee is ultimately responsible for determining passing scores, and the committee may take into account other factors.
The purpose of the third part of the comprehensive examination is to test the student's mastery of advanced material and to assess the student's ability to do research in the area in question. A committee of at least five faculty members, one of whom serves as chair, administers this examination, which may be either written or oral. The student and the chair of the committee will decide on three graduate courses, other than MATH 6130-6140 (Modern Algebra), MATH 6310-6320 (Real Analysis), MATH 6210 (Topology I), and MATH 6350 (Complex Analysis I), and the examination will be based on material taken from these three courses.
Prior to, or upon completion of the Comprehensive Examination, the student should find a research advisor to direct a thesis. The student and thesis advisor should select a thesis committee. The thesis advisor and this committee may require examinations in the thesis area, and may require active participation in a seminar before agreeing to serve.
THE THESIS
Every candidate for the Ph.D. must write a thesis containing substantial original contributions to mathematics. A purely expository or historical thesis is not acceptable.
THE FINAL EXAMINATION AND THESIS DEFENSE
The final examination is an oral examination and will cover the subject of the thesis and closely related topics. This examination will be conducted by a committee consisting of the thesis advisor, the second reader, two other members of the Department of Mathematics, and one faculty member from outside of the Department of Mathematics. The student will give a lecture on the thesis, and then respond to questions from the committee.
LANGUAGE REQUIREMENTS
A Ph.D. student is required to make a written translation of a mathematical article in one of the following languages: French, German, Italian, and Russian. The graduate Committee will prepare the language exams each semester.
TIME LIMIT
All work, including the final examination, should be completed within six years from admission to candidacy. Work done earlier will not be accepted for the degree unless validated by special examinations. A student is expected to complete the work with reasonable continuity. Upon fulfillment of all these requirements, the candidate will be recommended to the Graduate School to receive the degree Doctor of Philosophy.
SATISFACTORY PROGRESS FOR FUNDED STUDENTS
Graduate students, who are supported as teaching assistants in the Mathematics Department, ordinarily will maintain their support as long as they are deemed to be making satisfactory progress. Specifically, funded students who keep to the schedule below will be considered to be making satisfactory progress toward the M.A., M.S., or Ph.D. degree.
M.A. OR M.S. DEGREE
Year I. The student should complete a minimum of 15 semester hours of graduate study with a grade point average of 3.3 (or higher) for the first semester and a cumulative 3.5 every semester thereafter.
Year II. The student should complete an additional 15 semester hours of graduate study with a grade point average of at least 3.5.
All requirements for the M.A. or M.S. degrees should be completed within five semesters.
PH.D. DEGREE
Year I. The student should complete a minimum of 15 semester hours with a grade point average of 3.3 (or higher) the first semester and at least 3.5 every semester thereafter. The course work should include anything needed as preparation for the written examinations in Algebra and Analysis. These examinations should be taken at the end of this first year (August).
Year II. The student should complete a minimum of 15 additional semester hours of graduate study with a grade point average of at least 3.5.
Year III. The student should complete a minimum of 12 additional semester hours of graduate study with a grade point average of at least 3.5. By the fall of this year the student should begin to prepare for the third portion of the comprehensive examination, which should be completed by the end of the third year.
Years IV-VI. The student should be working with an advisor on a thesis and continuing active participation in course work and seminars. After completing the Comprehensive Examination a student working as a Teaching Assistant for the Mathematics Department is expected to register for at least one course per semester. The course may be taken for no credit.
A continually funded graduate student should have completed all requirements for the Ph.D. degree by the end of the sixth year of study.
For a list of the research interests of the math dept faculty, click here.