## Ph.D. Degree Requirements

The following benchmarks have to be passed on a student's way to the Ph.D. degree.

- Passing of the Qualifying Exam. This is an exam in Real Analysis and Applied Linear Algebra. A student admitted directly into the Ph.D. program is expected to take this exam by the end of the first year at the latest.
- Completing 54 semester hours of courses. The grade of each course has at least to be B. The selection of courses must be approved by the student's supervisory committee and the Joint Program Committee which administers the program. At least 18 hours must be in a major area of concentration while at least 12 hours have to be in a minor area of study which is a subject outside mathematics.
- Passing a language or tool of research exam.
- Passing the Comprehensive Exam which consists of a written and an oral part.
- Preparing a dissertation which must be a genuine contribution to mathematics.
- Passing the Final Examination (thesis defense).

The coursework may be finished within two years after the Qualifying Exam and research should be started while coursework is still under way. Typically, work on the thesis itself takes 12 - 18 months. Therefore, depending on a student's background, one has to allow for four to six years to obtain both the M.S. and the Ph.D. degree.