College of Liberal Arts & Sciences

# Grad Student Handbook

## Ph.D. in Mathematics Degree Requirements

Check also the section Doctor of Philosophy in the Mathematics Graduate Programs of Study.

## I. Core Areas and Core Courses

The core areas and core course sequences are:

- Algebra: Abstract Algebra I-II (MATH:5000 - MATH:5010)
- Analysis: Introduction to Analysis I-II (MATH:5200 - MATH:5210)
- Topology: General Topology (MATH:5400) - Introduction to Smooth Manifolds (MATH:5410)
- Differential Equations with Numerical Methods: Nonlinear Dynamics with Numerical Methods (MATH:5600) - Partial Differential Equations with Numerical Methods (MATH:5700)

These courses should be accessible to students who have completed the equivalent of MATH:1850, MATH:1860, MATH:2700, MATH:2850, MATH:3720 and MATH:3770.

Each Ph.D. student is required to demonstrate competence in each of the four core areas within the first two years of graduate study, either by passing (grade of C- and higher) each of the MATH:5000-level corresponding courses or by passing the relevant area of the Ph.D. Qual Exam with a mark of "Ph.D. level pass" (see Section II).

Graduate-level courses transferred from other universities may be used to satisfy part of these requirements, subject to approval of the DGS and the Graduate Committee.

## II. Ph.D. Qualifying Examinations (Ph.D. Qual Exams)

Each Ph.D. student is required to pass the Ph.D. Qual Exams within the first two years of graduate study. The four examination areas are based on the four MATH:5000-level core course sequences listed in Section I. The Ph.D. Qual Exams will be offered in all areas (as needed by student registration) at the beginning of each fall and spring semester. A Ph.D. student must register for the Ph.D. Qual Exams by the announced deadline, usually about a month before the start of the semester. A Ph.D. student cancelling his/her registration must do so at least one week prior to the Ph.D. Qual Exams session. After this date, any examination area not taken will be marked as "fail."

The Ph.D. Qual Exams consist of examinations in core areas (see Section I) chosen by the Ph.D. student. Each examination area is three hours long and can be taken during different sessions. For each examination area a student will receive a grade of "Ph.D. level pass", "Master's level pass" or "fail." An honorific grade of "Ph.D. level pass with Distinction" will be given for exceptional performance. To pass the Ph.D. Qual Exams a Ph.D. student must pass three different examination areas with the mark of "Ph.D. level pass". Each examination area can be attempted at most three times. For a repeated examination area the best result counts, i.e., a mark of "fail" does not replace a mark of "Master's level pass". It is not mandatory for a student to retake an examination area with a mark of "fail" or "Master's level pass" in a subsequent session, another area can be substituted. As an exception to the above rules, a Ph.D. student is considered as having passed the Ph.D. Qual Exams in the situation that at least three examination areas are taken during the same session and the result of three of those examinations consists at least of two "Ph.D. level passes" and one "Master's level pass". Note that an examination area passed with a "Ph.D. level pass" cannot be repeated.

### Sample Timelines:

The core course requirements and Ph.D. Qual Exams system allow for example:

- Entering Ph.D. students with exceptional preparation the opportunity to pass the Ph.D. Qual Exams at the start of their graduate study, and move directly to research related activities, finish both core course requirements and Ph.D. Comp Examination by the beginning of their second year;
- Entering Ph.D. students with very strong preparation the opportunity to pass some of the area exams at the start of their graduate study, and concentrate on the remaining areas, and possibly pass the Ph.D. Qual Exams at the beginning of their second year of graduate study and thus again move quickly to research related activities as soon as their second year, finish both core course requirements and Ph.D. Comp Exam by the end of their second year;
- Entering students with adequate preparation the opportunity to start three core course sequences in the first year and pass the Ph.D. Qual Exams at the beginning of their second year and thus again move quickly to research related activities as soon as their second year, finish both core course requirements and Ph.D. Comp Exam by the end of their second year;
- Entering students with weaker preparation the opportunity to start two core course sequences in the first year and another two in the second year, pass Examinations during these two years and be finished at the end of their second year, while in their third year they start on Ph.D. Comp Examination preparation, finish both core course requirements and Ph.D. Comp Examination by the end of their third year.

The two years timeframe also allows the flexibility for any Ph.D. student to switch the areas of the Ph.D. Qual. Examinations during their first two years.

These new rules will start being applied to the new class of students entering in Fall 2016 and also to current first year students opting for these new rules before August 15, 2016.

## III. Ph.D. Comprehensive Examination (Ph.D. Comp Exam)

Each Ph.D. student is required to pass the Ph.D. Comp Exam in his/her chosen area within the first three and a half years of graduate study.

The preparation for the Ph.D. Comp Exam is under the supervision of a faculty. Typically the Ph.D. Comp Exam preparation would build on one of the MATH:6000-level sequences with additional readings from books and research papers. Each student must submit a Ph.D. Comp Exam proposal to the DGS for approval by the Graduate Committee at least one month before a presentation is scheduled. The proposal must list the examining committee consisting of at least five people including the chair, and be first approved by the chair of the examining committee. The student would give an oral presentation of the material (usually about one hour) and be questioned over the material by the examining committee. Note that the Ph.D. Comp Exam is not open to the public (except for special requests made to the DGS).

Waivers, or modifications of rules, for students with weak backgrounds, or other special circumstances may deal with time restrictions, but may not weaken content requirements.

## IV. Breadth Requirements

### 18-Hour Breadth Requirement:

A Ph.D. student must pass (grade of C- and higher) at least 33 credit hours of graduate math courses numbered from MATH:6000 to MATH:7900 plus MATH:5800-5810 with the exception of Seminars, such that at least 18 credit hours must be from MATH:6000-level courses.

### Three Areas Breadth Requirement:

Among the following MATH courses, a Ph.D. student must choose three areas from the following five areas:

- Algebra: Algebra I-II (MATH:6000 - MATH:6010); Homological Algebra (MATH:7000)
- Analysis: Analysis I-II (MATH:6200 - MATH:6210); Functional Analysis I-II (MATH:7200 - MATH:7210)
- Topology: Introduction to Algebraic Topology (MATH:6400); Introduction to Differential Topology (MATH:6410); Differential Geometry I-II (MATH:6500 - MATH:6510); Topology of Manifolds (MATH:7400)
- Differential Equations: Ordinary Differential Equations I-II (MATH:6600 - MATH:6610); Partial Differential Equations I-II (MATH:6700 - MATH:6710)
- Numerical Analysis: Numerical Analysis I & II (MATH:5800-MATH:5810); Theoretical Numerical Analysis I & II (MATH:6850 - MATH:6860)

A Ph.D. student must pass (grade of C- and higher) at least two courses in each of the three chosen areas.

Graduate-level courses transferred from other universities may be used to satisfy part of the breadth requirements, subject to approval of the DGS and the Graduate Committee.

## V. Academic Registration

According to Section XII. Doctor's Degrees, C. Academic Registration Requirement of the Graduate College Manual of Rules and Regulations : All doctoral programs will contain a minimum of 72 semester hours of graduate work. Of those 72 semester hours, at least 39 must be earned while registered in The University of Iowa Graduate College. After completing 21 semester hours of graduate work under Graduate College registration and in compliance with the Graduate College policy for time limits on academic credit, i.e., courses ten years or older may not be counted toward the degree, students must complete an additional 18 semester hours to be taken as follows: (1) enrollment as a full-time student (9 semester hours minimum) in each of two semesters, or (2) enrollment for a minimum of 6 semester hours in each of three semesters. A student must be registered in the semester in which (s)he earns her/his degree.

See also the College of Liberal Arts & Sciences Academic Policies Handbook.

## VI. Ph.D. Dissertation

See Section XII. Doctor's Degrees, M. Dissertation for the Doctoral Degree of the Graduate College Manual of Rules and Regulations.

## Notes relative to the M.S. in Mathematics

As mentioned above an entering Ph.D. student may register for one or two area exam(s) before the beginning of graduate study without being registered for the Ph.D. Qual Exam. The Ph.D. student will be able to carry forward the area exam mark(s) with a pass ("Ph.D. level pass" or "Master's level pass") for the Master Comprehensive Examination.

Ph.D. students are generally automatically registered as a Secondary Program of Study to the Program I or III of the Math M.S. To be conferred the M.S. Math degree please be aware of the following rule in Section X. Master's Degrees, J. Final Examination of the Graduate College Manual of Rules and Regulations: A student must graduate within one calendar year after passing the final examination for a master's degree; failure to meet this deadline will require reexamination of the student. For example if you pass the M.S. Comp. Exam in the fall, the deadline to apply to be conferred the MS degree is March 1 of the following calendar year.

Check also the section Master of Science in the Mathematics Graduate Programs of Study.