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

#### General Requirements

- A minimum of 72 credit hours beyond the bachelor’s degree – all at the 300 level or above
- 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: http://graduateschool.colostate.edu/

The faculty has passed a new breadth requirement for the PhD that is effective for incoming students Fall 2021. Students who entered the program before Fall 2021 can also opt into this breadth requirement. The policy is that students opting into this requirement must fulfill the whole requirement. ** There will be no hybrid options**.

** **

**New Breadth Requirement: **

**Math Course Requirement: **every graduate student must pass, with a grade of B- or higher, at least 10 courses in the math department of 3+ credits each, at the 500+ level.** **

**Breadth seminar requirement: **every graduate student must pass the following two seminars within their first two years at CSU. ** **

**Seminar on applications of math**. 1-credit. Fall.**Seminar on diversity of mathematics and mathematicians in the 20th century.**1-credit. Spring

**Old Breadth Requirement: **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: **MATH 501, 502, 566, 567, 570, 571, 601, 602, 666, 667, 672, 673

**List II: **MATH 517, 519, 617, 618, 619, 717, 718

**List III: **MATH 510, 520, 535, 540, 545, 546, 560, 561, 620, 621, 640, 645, 646

**The faculty has passed a new Ph.D. Qualifying System effective for incoming students Fall 2022. **

**For Ph.D. students admitted before fall 2022:**

#### Departmental Qualifying Examination

The departmental Ph.D. qualifying exam consists of two examinations, parts I and II.

**PART I OF THE QUALIFYING EXAMINATION:**

**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 faculty advisor, 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 all Special Suite proposal for approval (or not). The Graduate Director will automatically approve all Standard Suite proposals , if properly made.
- 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 department 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 faculty advisor 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.

**PART II of the qualifying examination:**

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.

#### Departmental Preliminary Examination:

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.

#### PhD Dissertation and Final Examination

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:

- Dissertations will be submitted electronically through ProQuest/UMI though the Colorado State University libraries ETD submission website. Instruction for submitting a thesis are detailed on the Graduate School’s website and can be found here.
- The format of the thesis must strictly adhere to the requirements of the Graduate School. A requirements checklist, formatting guide and sample pagesare available on the Graduate School website and can be found here.
- The thesis must be submitted to the Graduate School by published deadlines.
- 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.

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.

**Departmental Qualifying System for Ph.D. students admitted fall 2022 or after:**

The qualifying exam consists of Parts (A) and (B). It is followed by the Research Prelude, which is described below after Parts (A) and (B). The Departmental Preliminary Exam and the PhD Dissertation and Final Examination are as described above.

Policies for Part (A) and (B) are listed in detail following this brief summary.

**Time-Line: **recommended timeline; required timeline.

QS Part (A): | Recommended year 1; required by end of academic year 2. |

QS Part (B): | Recommended year 2; required by end of academic year 2.5. |

Research Prelude: | Recommended year 2-3; required by end of academic year 3.5 |

Preliminary Exam: | Recommended year 3-4.5; required by end of academic year 4.5 |

Defense: | Recommended year 5-6; financial support is not guaranteed after year 5 |

** ****Part (A):**

Each student must pass the final exam in 3 of the qualifying system (QS) courses, subject to requirement R1.

The QS course are:

- MATH 501
- MATH 517
- MATH 519
- MATH 545
- MATH 560
- MATH 566
- MATH 570
- MATH 574

Requirement R1: each student must pass the final exam in at least one of MATH 517 or MATH 566.

**Part (B):**** **

Every student must demonstrate conceptual understanding and technical mastery of a topic which builds on but is not covered in their courses. To do this, a student must provide written and verbal explanation of material contained within 1-2 chapters of a graduate level textbook or 1-2 research papers, or an alternative source.

A graduate student will qualify for the doctoral program by completing parts (A) and (B). The qualifying exam committee will meet every semester to determine which graduate students have completed parts (A) and (B) successfully.

**Exceptions: **

Part (A): if a graduate student has already succeeded in a course that can be considered equivalent to one of these 8 courses at another institution, they can request to use the final exam from that course to fulfill a portion of the part (A) requirement. This request can be made for up to 3 courses and may include either or both of 517 and 566. The graduate committee will evaluate whether or not to approve each such request, on a course-by-course basis.

Part (B): no exceptions.** **

**Policies for Part (A):**

For each of the QS courses:

There will be a syllabus of recommended topics and a list of recommended textbooks, which will be considered current if it was approved by the math faculty in the last 10 years. If the syllabus is current, the faculty member is required to cover the recommended topics for at least 10 weeks of the course.

The faculty member will choose another faculty member as a course consultant, preferably someone who previously taught the course. The course consultant will be responsible for approving the following material developed by the faculty member: the syllabus, the choice of textbook, and the structure, sources, and content for the final exam.

Students will be assessed on the material from the syllabus of recommended topics at the end of the course. For each QS course except MATH 574, the qualifying exam will be an in-class 2-hour exam at the time scheduled for the course final by the university and the final exam will only cover topics from the syllabus of recommended topics. Students may be expected to complete problems about material discussed in class, material from homework, or new material, as long as it is closely related to the material on the syllabus of recommended topics.

The faculty member will assign one of the following grades to each final assessment: **strong pass, pass, weak pass, or fail**. The course consultant will be responsible for approving the cut-off between a passing and failing grade on the qualifying exam.

To determine grades for the course, the faculty member may choose to evaluate students’ understanding of material not on the list of recommended topics and outside of the course final exam. The faculty member has complete decision-making authority for the content and method of this evaluation, as long as it is clearly communicated in the syllabus.

The department will encourage faculty to invest in these courses and will enable students to meet potential PhD advisors by rotating a broad range of faculty through the QS courses.

**Policies for Part (B):**

A student will ask 2 faculty members from the math department to be on their part (B) evaluation committee. The two faculty members are responsible for evaluating the student’s part (B) written work and presentation, but are not committed to being the student’s future advisor.

The student and faculty members will fill out the part (B) agreement form. In particular, they will decide on the content, references, and time-line for the part (B) evaluation. The expectation is that the material in the part (B) project should be approximately equivalent to five weeks of a graduate course.

They will decide on the length, content, and format of the written assessment. For the alternative sources, valuable ways to provide written explanation might include an analysis of technical reports and real data, instructional materials for new techniques, a survey about a series of interviews with an expert, or documentation for new code.

They will decide on the audience, length, content, and focus of the verbal assessment. The part (B) presentations will not be announced by the department. The student and faculty members may choose to invite several people, or to have a private meeting.

The faculty members will evaluate the student’s part (B) written and verbal material and decide whether the student passed or failed. In either case, the faculty will provide the student with written feedback, for example, by filling out the questions on the part (B) evaluation form. A student who attempts but does not pass part (B) may try again, with either the same or with different faculty members, but must pass part (B) by the second attempt.

The committee recommends that Part (2) of the current qualifying system be renamed and re-interpreted as a requirement for the doctoral degree process.

**Research Prelude:**

Every doctoral student must propose a research area and general topic to their doctoral committee. With this proposal, the student should demonstrate the ability to synthesize research concepts with a written paper and presentation, including examples, data, or proofs. This may involve either an analysis of an important reference, an analysis of a recent publication, or new material generated by the student.

A doctoral advisor may decide that a masters defense completed at CSU can fulfill a student’s research prelude requirement. If a student switches advisers, it is up to the discretion of the new advisor whether the student should complete another research prelude.