## Additional Requirements

#### Transfer of Credit

No transfer of credit will be considered until the screening examination is passed. Normally a maximum of 30 units of graduate work at another institution may be applied toward the course requirements for the PhD A grade of B- or lower will not be accepted, and, at most, two grades of B will be accepted. A PhD candidate may petition the department for transfer of additional credit after passing the qualifying examination.

#### Foreign Language Requirement

The student must demonstrate a reading comprehension of mathematics in one language (other than English) in which there is a significant body of research mathematics (such as Chinese, French, German, Japanese and Russian) by passing a written examination, administered by the department, in translation of mathematical content.

#### Qualifying Examination

The written portion of the qualifying examination is comprehensive, consisting of two, two-hour examinations administered by the department. These examinations cover two out of the following five options, excluding the option already selected for the preliminary examination: algebra, analysis, geometry/topology, probability/statistics, differential equations. Each option approximately covers the content of two, one-semester graduate courses, with the precise list of possible topics made available to the students by the department. The selection of options must be approved by the qualifying exam committee.

The oral portion of the qualifying examination covers one topic selected from department research areas in mathematics and approved by the qualifying exam committee. The student must demonstrate research potential in this field. A dissertation proposal (10 pages minimum) must be submitted to the department at least 1 week before the oral qualifying examination.

#### Dissertation

Following passage of the qualifying examination and approval of a dissertation topic by the qualifying exam committee, the student begins research toward the dissertation under the supervision of the dissertation committee. The primary requirement for the PhD is an acceptable dissertation which is based on a substantial amount of original research conducted by the student.

#### Research Areas

Opportunities for research are offered in the areas of algebraic geometry, arithmetic geometry, combinatorics, complex geometry, control theory, differential equations, differential geometry, dynamical systems, functional analysis, geometric analysis, group theory, K-theory, nonlinear analysis, number theory, numerical analysis, optimization, probability, representation theory, ring theory and topology.