# Doctoral Degree Requirements

The PhD in Mathematics requires completing at least 36 credits of graduate coursework beyond the bachelor’s degree, preliminary written and oral examinations, 24 thesis credits, and a doctoral thesis with final defense.

PhD students in Mathematics are expected to earn an MS degree upon completion of their preliminary oral examination.

## Coursework

### Entering students

- Required to have completed an undergraduate degree prior to matriculation into the program.
- Some students enter with previous graduate coursework which can include having completed a Master’s degree.

### Course requirements for a doctoral degree

**36 credits of coursework**— Students normally complete the coursework for the PhD at the University of Minnesota. Students may transfer up to 12 credits of previously completed graduate coursework toward their PhD in consultation with their advisor and the Director of Graduate Studies upon submitting their degree plan.**MATH 8001: Preparation for College Teaching**— All PhD students must complete MATH 8001 in the Fall semester of their first year.

### Tracks

There are two coursework tracks to earn a PhD in Mathematics.

#### Mathematics track

Students completing the Mathematics track are required to pass the following four course sequences with a grade of “B” or better both semesters. The requirement for any of these courses may be completed by passing the corresponding preliminary written examination.

- Math 8201/02 (Algebra)
- Math 8301/02 (Manifolds/Topology)
- Math 8601/02 (Real Analysis)
- Math 8701/02 (Complex Analysis)

#### Applied Mathematics track

Students completing the Applied Mathematics track are required to complete the following two course sequences with a grade of “B'' or better both semesters.

- Math 8401/02 (Math Modeling)
- Math 8441/42 (Numerical Analysis)

Applied Mathematics track students are also required to:

- Complete at least six credits of coursework outside of Mathematics or complete a formal minor in another program.

## Credits

**24 major credits**— Courses in Mathematics that support the student’s area of study.**12 “outside” credits**— Courses that support the student’s program and are outside their primary area of study. This may be a formal minor in another program or Mathematics coursework that is outside of the primary area of study.**24 thesis credits**— Taken upon the completion of the preliminary oral examination. Students are expected to complete their thesis credits no more than two semesters after successfully completing their preliminary oral examination.

## Preliminary written exam

Students must demonstrate proficiency in basic areas of mathematics by passing the preliminary written exams.

### Timing

Students are expected to complete their preliminary written exams by the end of the second year of study. Changes to this timing may be approved by meeting with the advisor and the Director of Graduate Studies.

### What’s required

Students must pass two written examinations in:

- Real Analysis or Complex Analysis
- Algebra or Manifolds/Topology

### Two ways to complete

Complete the preliminary exam requirements by either:

#### 1. Passing the written examinations.

**Preliminary written examinations are given in both Fall and Spring semesters, typically before the beginning of Fall semester and in the later half of Spring semester.**

#### 2. Completing the corresponding coursework.

**Students must receive a “B” grade or higher in both semesters of the course. Courses are offered each academic year:**

- Real Analysis (Math 8601/02)
- Complex Analysis (Math 8701/02)
- Algebra (Math 8201/02)
- Manifolds/Topology (Math 8301/02)

## Preliminary oral exam

Students complete a preliminary oral examination to demonstrate proficiency in their primary area of study and supporting program or minor program. The exam also serves as a final exam for a Plan B Master’s in Mathematics.

### Timing

- Typically completed after the third year of study.
- Must be completed prior to the last day of Spring semester in the fourth year.
- Changes to this timing may be approved by meeting with the advisor and the Director of Graduate Studies.

### What to expect

The preliminary oral examination concentrates on the primary area of study and the supporting program or minor.

**Examination committee**— The examination committee of four faculty members consists of the major area advisor and three additional faculty members. Students who have declared a formal minor are required to have a committee member from the minor program.**Advisor**— The advisor recommends coursework in the basic material, as well as expository and research papers for individual study.**Expository paper**— The study, including the recommended research papers, will be summarized in an expository paper (approximately 10 pages), with a substantial bibliography. This will demonstrate a knowledge of the definitions and results in the area, and indicate open problems which may form the basis for the PhD thesis.**Registration and credits**— Students taking their preliminary oral examinations during Fall or Spring semester are expected to register for six pre-thesis credits (MATH 8666).

### Timing

- The student should choose their primary area and decide on an advisor who will support them through these exams by the end of fall semester in the second year.
- In the semester prior to the oral exam, students should work with their advisor to determine a committee and expectations for their exam.
- Students are expected to have begun work on their expository paper by the beginning of the semester in which they are planning to complete this exam.

### Process

See the PhD degree (steps 1-7) and the Plan B Master’s degree (the preliminary oral examination serves as a Final Exam for a Plan B Master’s in Mathematics) completion steps:

## Final exam/dissertation defense

Mathematics PhD students typically complete their degree in the sixth year of their program. The examination is a defense of the thesis that is the culmination of the student’s work in the program.

### Committee

- The doctoral final exam committee consists of four members, including the advisor(s).
- The chair of this committee must be a full mathematics faculty member who is not the student's advisor.
- At least two major field thesis reviewers are required.
- One committee member must represent a field outside of mathematics. If there is a formal minor, this committee member must represent that program. In the case of multiple minors, there must be a separate thesis reviewer for each minor.

### Format

The doctoral final exam must include a public presentation of the candidate’s dissertation to the doctoral final oral examination committee and the invited scholarly community, followed by a closed session for questions by the examiners.

### Process

See the PhD degree completion steps (steps 8-15):