# Doctor of Philosophy in Mathematics

## Admission

Applicant must meet Graduate School requirements plus departmental requirements as given for admission to the master's program. A master's degree is not a prerequisite for admission to this PhD program.

## Reapplication

A student who receives the master's degree must formally reapply for admission to the Graduate School before continuing studies toward the PhD.

## Major Professor as Advisor

The student must have a major professor to advise and supervise the student's studies as specified in Graduate School regulations. The entering graduate student is assigned a temporary advisor by the Department Graduate Program Coordinator.

## Course of Study

Minimum degree requirement is 54 graduate credits beyond the bachelor's degree, at least 27 of which must be earned in residence at UWM. The student, in consultation with the major professor, must select both a primary and a secondary area of specialization. The primary area may be chosen from one of the following seven fields with minimum credit requirements as shown. The secondary area may be chosen from another of these fields or may be chosen from another appropriate department. Minimum course requirements for all work in both areas of specialization require approximately two full years of study.

- Algebra field
- 12 credits in algebra
- 3 in complex analysis
- 3 in real analysis
- 3 in topology
- 3 in applied mathematics
- 3 outside the field

- Analysis field
- 3 credits in algebra
- 6 in complex analysis
- 6 in real analysis
- 3 in topology
- 3 in applied mathematics
- 3 outside the field

- Applied mathematics field
- 3 credits in algebra
- 6 in complex analysis
- 3 in real analysis
- 12 in applied mathematics
- 3 outside the field

- Probability and statistics
- 3 credits in complex analysis
- 6 in real analysis
- 12 in probability and statistics
- 3 in applied mathematics

- Topology field
- 3 credits in algebra
- 3 in complex analysis
- 3 in real analysis
- 12 in topology
- 3 in applied mathematics
- 3 outside the field

- Industrial mathematics field
- 3 credits in algebra or topology
- 3 in complex analysis
- 3 in real analysis
- 9 in applied mathematics
- 6 in probability and statistics
- 6 in approved credits outside the 600 and 601 curricular codes

- Atmospheric sciences field
- 12 credits in atmospheric sciences (synoptic, dynamic, cloud/radiative, air pollution)
- 9 in applied mathematics or probability and statistics
- 3 in analysis

## Foreign Language

Except for students in the atmospheric sciences field, each student must pass a written examination in one foreign language; the examination is administered by the Department's Language Committee. Acceptable languages are French, German, and Russian; exceptions may be permitted upon written request of advisor.

## Computer Proficiency

The student shall pass an examination on a higher programming language and/or other appropriate advanced computer skills; the examinations administered by the Department's Computer Committee. The Computer Committee may accept advanced computer science course work in lieu of the examination.

## Residence

The student must meet minimum Graduate School residence requirements.

## Doctoral Preliminary Examination

When the student is sufficiently prepared — normally when the student has earned 24 credits in specified areas above the 700 level — a doctoral preliminary examination to determine the student's knowledge and achievement is taken. For students in mathematics, the exam evaluates the student's general knowledge of mathematics, as well as the student's knowledge of the major area of concentration. Students in atmospheric sciences are examined in three areas: at least one from the 190 curricular area code and at least one from the 600/601 curricular area codes. Students must pass this examination to continue in the program. With permission of the examination committee, the student may repeat this examination once. If the student does not have a master's degree in mathematics before this examination, the committee will determine whether the student's performance is sufficient to qualify for the master's degree.

## Doctoral Dissertation Proposal Hearing

After passing the language requirements and the doctoral preliminary examination, the student participates in a doctoral dissertation proposal hearing. At this hearing, the student is examined on the student's chosen area of research and a dissertation topic is approved.

## Dissertation

The primary requirement for the PhD in mathematics is the candidate's completion, under the supervision of the Department advisor, of an original and significant mathematical investigation presented in the form of a dissertation. The investigation is to be in the field of algebra, analysis, applied mathematics, probability and statistics, topology, or atmospheric sciences. A dissertation for the industrial mathematics field must involve an industrial problem requiring a mathematical solution.

## Dissertation Defense

The candidate must, as the final step toward the degree, present a colloquium based on the dissertation and must pass an oral examination in defense of the dissertation. If the candidate does not successfully defend a thesis within five years of admission to candidacy, the candidate may be required to take another doctoral preliminary examination and be readmitted to candidacy.

## Time Limit

All degree requirements must be completed within ten years from the date of initial enrollment in the doctoral program.

## Minor Area for Other PhD Majors

A doctoral student planning a physical science major other than mathematics may fulfill requirements for mathematics as the minor area of concentration by completing 12 credits of approved mathematics courses with a grade of B or better, at least 6 credits of which must be in courses 700 or above.

A doctoral student planning a non-physical science major may fulfill requirements for mathematics as the minor area of concentration by completing 12 credits with a grade of B or better in approved mathematics courses 300 or above.

For additional information on the PhD, see the Graduate School Doctoral Requirements page.