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## PhD 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)
- Numerical Analysis: Numerical Analysis I & II (MATH:5800-MATH:5810)

Each PhD student is required to demonstrate competence in 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 PhD Qualifying Exam with a mark of "PhD 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. PhD qualifying examinations (PhD qual exams)

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

The PhD Qual Exams consist of examinations in core areas (see Section I) chosen by the PhD 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 "PhD level pass", "Master's level pass" or "fail." An honorific grade of "PhD level pass with Distinction" will be given for exceptional performance. To pass the PhD Qual Exams, a PhD student must pass three different examination areas with the mark of "PhD 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 PhD student is considered as having passed the PhD 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 "PhD level passes" and one "Master's level pass". Note that an examination area passed with a "PhD level pass" cannot be repeated.

#### Sample timelines

- Entering PhD students with exceptional preparation the opportunity to pass the PhD Qual Exams at the start of their graduate study, and move directly to research-related activities, finish both core course requirements and PhD Comp Examination by the beginning of their second year;
- Entering PhD 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 PhD 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 PhD 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 PhD 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 PhD 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 PhD Comp Examination preparation, finish both core course requirements and PhD Comp Examination by the end of their third year.

The two-year timeframe also allows the flexibility for any PhD student to switch the areas of the PhD 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. PhD comprehensive examination (PhD comp exam)

Each PhD student is required to pass the PhD Comp Exam in their chosen area within the first three and a half years of graduate study.

The preparation for the PhD Comp Exam is under the supervision of a faculty. Typically the PhD 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 PhD 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 PhD 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

Important notice: 5000-level courses counted towards the Core Areas and Core Courses requirement in part I do not count for any of the breadth requirements below.

#### 33/18-hours breadth requirements

A PhD student must pass (grade of C- and higher or S or P) at least 33 credit hours of graduate math courses

- Either numbered from MATH:6000 to MATH:7900 plus MATH:5800-5810 except Seminars;
- Or in the fifth 5000-level area that was not counted towards the Core Areas and Core Courses requirement in part I.

At least 18 credit hours must be from MATH:6000-level courses.

#### Three areas breadth requirement

Among the following MATH courses, a PhD student must cover at least three areas from the following five areas:

- Algebra: (MATH:5000 - MATH:5010); Algebra I-II (MATH:6000 - MATH:6010); Homological Algebra (MATH:7000)
- Analysis: (MATH:5200 - MATH:5210); Analysis I-II (MATH:6200 - MATH:6210); Functional Analysis I-II (MATH:7200 - MATH:7210)
- Topology: (MATH:5400 - MATH:5410); 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: (MATH:5600-MATH:5700); Ordinary Differential Equations I-II (MATH:6600 - MATH:6610); Partial Differential Equations I-II (MATH:6700 - MATH:6710)
- Numerical Analysis: (MATH:5800-MATH:5810); Theoretical Numerical Analysis I & II (MATH:6850 - MATH:6860)

A PhD student must pass (grade of C- and higher) at least two courses in each of three areas. Only the fifth 5000-level course sequence that was not counted towards the Core Areas and Core Courses requirement in part I can count for this breadth requirement.

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 they earn their degree.

View also the College of Liberal Arts and Sciences Academic Policies Handbook.

### VI. PhD dissertation

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

### Notes relative to the MS in Mathematics

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

PhD students are generally automatically registered as a Secondary Program of Study to the Program I or III of the Math MS. To be conferred the MS 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 MS 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.