**Table of Contents:**

### General Information

The Ph.D. in Mathematics is a comprehensive program requiring considerable coursework and an original dissertation in Mathematics consisting of publishable research. Each student’s course of Ph.D. study is planned individually by the student, the student’s adviser, and the student’s graduate committee. This planning allows the student and adviser to develop specific programs in any of a large number of areas of mathematics.

The Ph.D. degree requires at least 72 semester credits beyond the bachelor’s degree. Additional requirements for the doctorate consist of a series of milestone examinations including a two part Qualifying Examination, a preliminary Examination and a Final Dissertation Defense. The Ph.D. preliminary examination, administered by the student’s Ph.D. committee, is a written and/or oral examination taken near the end of the course work and at the beginning of the work on the dissertation. All dissertations must be successfully defended in an open final oral examination.

The official Program Code for the Ph.D. degree is: **MATH-PhD**

Entering students will have a diagnostic and advising interview with the Graduate Director to plan an appropriate program of study.

- A minimum of 72 credit hours beyond the bachelor’s degree – all at the 300 level or above.
- Two thirds of the course work should be in mathematics.
- Course work in mathematics must be at the 400 level or above in order to be used towards the 72 credit hour requirement.
- 3.0 GPA must be maintained in all course work.
- 3.0 GPA must be maintained in all mathematics courses at the 400 level or above, excluding work in seminars and doctoral research.
- No more than 24 credit hours of research/thesis may be counted towards the 72 credit hour requirement.
- The Seminar Requirement must be satisfied – at most two additional credits beyond the two allowed for the MS may be counted towards the 72 hour requirement.
- At most 1 hour of MATH 584 (Supervised College Teaching) may be counted towards the 72 credit hour requirement.
- MATH 530 may not be counted toward the 72 hour requirement.

Students entering with a master’s degree in mathematics may receive up to 30 credit hours towards the 72 credit hour requirements. For these students, a minimum of 42 credits must be earned at CSU after admission to a doctoral program, and at least 21 credits beyond the master’s degree must be earned in courses numbered 500 or above. For students enrolled in a continuous master’s/Ph.D. program at Colorado State University, all courses taken during the master’s program may be applied to the doctoral degree, even if the total master’s degree credits exceed 30.

It must be stated on the Ph.D. program of study that all credits earned on the MS will be counted toward the PhD; furthermore, this must be approved by the student’s advisory committee, the Department of Mathematics, and the Graduate School. For more detailed information on university-wide Ph.D. requirements, please consult the Graduate School website:

To fulfill the breadth requirement for the Ph.D., all students will be required to take and pass, with a grade of B- or higher, at least 14 mathematics courses (each three or more credits) at the 500 level or above excluding:

- MATH 530
- MATH 695
- MATH 699
- MATH 798
- MATH 799

Students entering with a master’s degree in mathematics from another institution may use courses approved to count towards the 72 credit hour requirement to also count towards this requirement.

At least two courses must be chosen from each of the lists below. This breadth requirement may be completed over the course of the student’s graduate program. Credit earned for comparable courses taken at other institutions may be evaluated by the Graduate Director, at the student’s request, to determine suitability for substitution for particular course(s) in the lists below.

List I |
List II |
List III |

MATH 501 | MATH 517 | MATH 510 |

MATH 502 | MATH 519 | MATH 520 |

MATH 566 | MATH 617 | MATH 535 |

MATH 567 | MATH 618 | MATH 540 |

MATH 570 | MATH 619 | MATH 545 |

MATH 571 | MATH 717 | MATH 546 |

MATH 601 | MATH 718 | MATH 560 |

MATH 602 | MATH 561 | |

MATH 666 | MATH 620 | |

MATH 667 | MATH 621 | |

MATH 672 | MATH 640 | |

MATH 673 | MATH 645 | |

MATH 646 |

**Purpose of Part I of the Qualifying Examination **

The student will complete Part I of the Qualifying Examination in order to present evidence of performance at an appropriate level to the faculty. This evidence will be used as a basis for judging the candidate’s qualifications for continuing in the Ph.D. program in mathematics.

**The Suite of Qualifying Courses: Overview **

Each student seeking to pursue a Ph.D. in Mathematics will propose a Suite of Qualifying

Courses. The Graduate Committee will review each proposal. There are Standard Suites of courses that are automatically approved. A student may alternatively propose a Special

Suite of courses.

**What is a Proposal for a Suite of Qualifying Courses?**

A Suite proposal always consists of

- A list (Suite) of specified courses that the student has completed, or will complete, during his or her graduate studies,

AND

- For each course on the list, a specification of the documentation that will be provided by the student for assessment purposes. If the course is a QE course*, this documentation MUST be the student’s completed final examination in the course.

*A QE course (Qualifying Examination course) is any course in a Standard Suite.

There are two types of Suites of Qualifying Courses (described below): Standard Suites and Special Suites.

**How is a Suite proposal made? **

- The student, together with a proposal mentor, will complete the Suite Proposal form and return it to the Graduate Coordinator. Note that the Graduate Director will act as the default proposal mentor for all Standard Suite proposals.
- The signature of the proposal mentor, indicating support for the proposal, is necessary to complete each Suite Proposal.
- The Graduate Committee will review the proposal for approval.
- Each approved Special Suite Proposal will be announced to the faculty.

More details on policies and procedures for the review of Suite proposals are included in the departmental policies and procedures manual.

**What is a Standard Suite of Qualifying Courses? **

The following four groups of Suites of courses are those recognized as Standard Suites through which a graduate student may complete Part I of the Qualifying Examination for the PhD program.

**Analysis/Algebra/Geometry Suites **

MATH 517, MATH 566

plus two of MATH 519, MATH 567, MATH 570, MATH 617

**Applied, Computational, Interdisciplinary Suites in Algebra/Geometry**

MATH 517, MATH 566

plus one of MATH 519, MATH 560

plus one of MATH 519, MATH 540, MATH 560, MATH 561, MATH 567

**Applied, Computational, Interdisciplinary Suites in Analysis, Differential Equations and Dynamical Systems**

MATH 517, MATH 560

plus one of MATH 540, MATH 545, MATH 617

plus one of MATH 510, MATH 519, MATH 540, MATH 545, MATH 546, MATH 561, MATH 617,MATH 618, MATH 640

**Combinatorics/Algebra Suites**

MATH 501, MATH 517, MATH 566

plus one of MATH 502, MATH 560, MATH 567

Using one of the Standard Suites as Part I of the Qualifying Examination is the appropriate choice for most students in the program.

The Graduate Director will act as the default proposal mentor for all Standard Suite proposals. The Graduate Director will automatically approve all Standard Suite proposals, if properly made.

**What is a Special Suite of Qualifying Courses? **

Any other sort of Suite is a Special Suite. Special Suites may include QE courses.

For example,

A student who has successfully completed a graduate-level course, comparable to one of

the QE courses at another university, may request that this course be included in a

Special Suite, OR that a more advanced CSU course in the same area, or a closely related one, be included in a Special Suite.

A student’s CSU proposal mentor may recommend the inclusion of a particular graduate level course (not necessarily a CSU course) in a Special Suite to the student. Such a course does not have to be a mathematics course. It must be a graduate-level course.

A proposal for a Special Suite must specify the documentation that the student will submit to allow assessment of performance in each non-QE course in the Suite.

**How does a student complete Part I of the Qualifying Examination?**

Once a student’s Suite of Qualifying Courses is approved by the Graduate Committee, the student must provide the assessment documents specified in the student’s Suite Proposal to a Qualifying Examination Committee (The role of the QEC is defined in the department policies and procedures manual ) after each course in the Suite is completed. A student must receive a Pass- rating or higher from a QEC for each course in his or her Suite in order to complete that component of his or her Part I Examination. For more details on policies and procedures for assessment and evaluation of courses in Suites, consult the department policies and procedures manual.

Each student must complete Part I of the Qualifying Examination by the end of the fifth semester after admittance to the Ph.D. program. Students who enter the Ph.D. program with an M.S. degree are encouraged to complete Part I by the end of the third semester after admittance to the Ph.D. program.

**Qualifying Examination (QE) course syllabi:**

The Faculty approved QE course syllabi can be found in appendix D below.

After the selection of a permanent advisor, the student together with the Ph.D. advisor, and sometimes in consultation with additional faculty, will devise a second evaluation to be completed by the student before the preliminary examination, and after the completion of Part I of the qualifying examination. This evaluation must be completed within two academic years following the passing of Part I of the qualifying examination. The student should demonstrate proficiency on material approximately at the 600-level. An MS completed at CSU will fulfill Part II, but students coming in with an MS need to fulfill this requirement locally.

At some time after a Ph.D. advisor and committee have been assigned, the student will be required to pass an oral preliminary examination prepared and administered by the student’s committee.

This examination must be passed at least one academic year (two semesters; the summer session may be included) before defense of the dissertation. The purpose of this oral examination is to evaluate the student’s proposed research project.

Two weeks prior to the examination, the student must provide to the Ph.D. committee a written copy of the dissertation proposal, including a survey of the relevant literature, a description of the problem to be investigated, and an outline of methodology to be considered. The student either passes, passes provisionally (certain additional conditions must be met), or fails. In case the student fails the examination, it may be repeated once subject to Graduate School regulations. After this examination has been passed, the student becomes a Ph.D. candidate.

Doctoral students in the mathematics department at CSU are considered to achieve “candidacy” for the degree upon passage of preliminary examinations.

A written report of the examination will become part of the student’s file.

Each Ph.D. candidate must prepare a dissertation containing original mathematical research that is acceptable to the students committee. The student must successfully defend this dissertation in an open oral examination before the committee as well as other interested parties. This examination can be given no sooner than two semesters (including summer) after the student becomes a Ph.D. candidate.

The Dissertation:

- All dissertations will be submitted electronically through ProQuest. Instruction for submitting a thesis are detailed on the Graduate School’s website

http://graduateschool.colostate.edu/current-students/thesis-dissertation/index.aspx

- The format of the dissertation must strictly adhere to the requirements of the Graduate School. A format Requirements Checklist and sample pages of the Title page, Copyright page, Signature page, and Abstract page are available on the Graduate School website:

http://graduateschool.colostate.edu/current-students/thesis-dissertation/index.aspx

- The dissertation must be submitted to the Graduate School by the published deadline.

http://graduateschool.colostate.edu/for-current-students/

- PhD students must also submit the Survey of Earned Doctorates confirmation certificate with the Thesis/Dissertation Submission form.

The Graduate Coordinator will assist with room scheduling once the date and time for the defense have been established with your committee.

The student’s committee must have adequate time to review the thesis before the final examination. Therefore, the committee must have the thesis at least two weeks (including two weekends) before the final examination which meets the following standards:

- The Thesis must be in final form
- The Print is letter quality, and a uniform style has been used throughout.
- Figures, printouts, etc. are clear and easy to read.
- Text is double-spaced and printed on only one side of the paper.
- Spiral bound

Results of the final oral examination will be reported on the GS24.

Voting is limited to the members of the student’s committee, and a majority vote is necessary to pass the examination. A tie vote is interpreted as failure to pass the examination. All committee members must vote to either Pass or Fail the student; there is no option to withhold a signature. Committee members who are not academic faculty do not have a vote on the final examination.

If significant revisions to the thesis are required by the committee after the final oral examination is complete, a reasonable amount of time must be given for committee members to re-review the thesis once the revisions have been made.

A candidate who fails the final examination may be reexamined once and, for the reexamination, may be required to complete further work. The reexamination must be held not later than 12 months after the first examination. The examination must not be held earlier than two months after the first examination unless the student agrees to a shorter time period. Failure to pass the second exam results in dismissal from the Graduate School.