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# PhD Requirements

## Contact

The basic requirements for the PhD degree in mathematics include demonstrating a mastery of a broad area of mathematics and writing a dissertation making an original and substantial contribution to mathematics. Specific requirements for the PhD degree in mathematics consist of:

### Coursework

- Completion of at least 54 semester hours of graduate credit.
- Completion of the Common Core Courses: Algebra I (Math 7210), Real Analysis (Math 7311), and Topology I (Math 7510).
- One hour of Communicating Mathematics (Math 7001, 7002) is required each semester of the first year.
- Completion of the Breadth Requirement, which consists of taking at least four courses from lists A and B below, at least two of which must come from List A. Each of the four breadth courses must be passed with at least a grade of B, and at least one must be passed with a grade of A. It is expected that 3 of the breadth classes will be taken in the Spring semester of the first year of study, with the 4th class being taken in the second year. In order to complete Ph.D. qualifying, it is necessary to pass the PhD. Qualifying Exam at the PhD qualifying level and complete the breadth requirement.
List A List B Math 7211 (Algebra II) Math 7320 (Ordinary Differential Equations) Math 7330 (Functional Analysis) Math 7360 (Probability) Math 7350 (Complex Variables) Math 7410 (Graph Theory) Math 7512 (Topology II) Math 7710 (Numerical Linear Algebra) Math 7550 (Differential Geometry)

### PhD Qualifying Exam

The PhD Qualifying Examination consists of three sub-tests, each of three hours duration. These sub-tests cover the material commonly found in the Core courses: Algebra I (7210), Real Analysis (7311), and Topology I (7510). Passing each of the three sub-tests at the PhD Qualifying level completes the examination component of PhD Qualifying, with completion of the breadth requirement being the second component of PhD Qualifying. A PhD student is encouraged to take the PhD Qualifying Exam as early as feasible, but must pass it no later than the beginning of the fourth regular semester of study, except by permission of the Graduate Committee. Meeting this requirement is part of the definition of satisfactory progress---a requirement for holding a graduate assistantship or fellowship. The PhD Qualifying Exams are scheduled twice per year: normally the week before the beginning of the fall semester, and the week before the beginning of the spring semester.

Qualifying Exam policy is that at least 50% of the credit on each exam will come from the test problem banks below. There will normally be approximately 6 to 8 problems offered on each exam, and students will typically need to turn in approximately 5 of these.

Syllabi and test bank problems for each of the Qualifying Examinations:

- Algebra Syllabus (Math 7210-7211); Algebra Test Bank
- Analysis Syllabus and Test Bank
- Topology Syllabus and Test Bank

Copies of past Qualifying Exams can be found in the Exam Archive.

### Oral General Exam

The General Examination, an oral exam for Ph.D. students, tests if a student is prepared to begin dissertation research. The General Exam must be passed prior to the start of the fourth year of study, unless a postponement is allowed by the Graduate Committee. The exam is conducted by the student's Advisory Committee, which will consist of at least three Graduate Faculty Members plus a Dean's Representative appointed by the Dean of the Graduate School. The committee is to be chosen in consultation with the intended dissertation advisor, who will serve as Chair of the Advisory Committee, and must be approved by the Graduate Director.

The specific content of the general examination is established by the individual student's Advisory Committee in consultation with the student. It should consist of material that is appropriate for advanced graduate courses, and it should not seriously overlap the Comprehensive exam material. The Exam Syllabus will be the material chosen for the exam. A copy of the syllabus should be provided to the student, to each member of the Advisory Committee, and to the Graduate Director for inclusion in the student's file. The Exam Syllabus should be completed at least four weeks before the date of the exam, and before the Exam is scheduled with the Graduate School.

The format of the exam will consist of an optional, at the discretion of the advisor, short presentation by the student (of at most 20 - 30 minutes), and an oral exam on the content of the exam syllabus. The exam should last at most two hours.

The Advisory committee shall determine whether the student passes the General Exam. If the student fails, the Advisory Committee will determine whether the exam can be retaken. The exam can be taken at most one additional time.

General Exam forms:

- General Exam Syllabus Proposal Form
- Doctoral Degree Audit and Request for General Exam from the Graduate School Enrolled Students Forms page.

### Original dissertation and defense

Writing a dissertation and passing the Final Examination which is primarily a doctoral dissertation defense.