graduate degree information

GRADUATE STUDY IN MATHEMATICS AT THE UNIVERSITY OF COLORADO AT BOULDER

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

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:

1. The 18 hours in the Department of Mathematics must include 2
two-semester sequences, one of which must be MATH 5600-5610, Numerical
Analysis.

2. 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.

3. 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.

4. 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.