The Ph.D. in Mathematical Sciences program with concentrations in Mathematics and Statistics is an interdisciplinary program designed to ensure that the student acquires knowledge in a broad spectrum of the mathematical sciences in addition to expertise in a chosen field of concentration. Programs of study are structured to reflect the belief that a student in the mathematical sciences should be proficient in a specialized area, and understand how it relates to other areas and be able to apply this knowledge to solve real-world problems. As a result, the Ph.D. graduate is prepared to work in industry, government, or academia. Besides opportunities for consulting experience through the Center for Statistical and Mathematical Services, students in the Ph.D. program may have opportunities for participation in research projects through other facilities on campus, such as the Engineering Research Center for Computational Field Simulation.

**Breadth Requirements for the Ph.D.**

In keeping with the emphasis in the doctoral program that each student acquires breadth in the mathematical sciences, each student in the program will demonstrate a satisfactory grasp of each of the areas listed below.

- A Programming Language;
- Matrices/Linear Algebra;
- Probability/Statistics;
- Advanced Calculus;
- Computational Mathematics/Statistics.

The Graduate Coordinating Committee members will determine a student's satisfactory grasp of the areas listed above.

# Examination Procedures

**Comprehensive Examinations**

To ensure that the skills and basic knowledge have been acquired to carry out the research necessary for the dissertation, the student must demonstrate competence in the three areas, two of which are chosen from the list below. Competence will be demonstrated by comprehensive examinations, which shall consist of written examinations over each of the chosen areas. The third area shall be related to the student's concentration and the advisor's area of expertise. This graduate-level course fulfills the program of study of the student’s concentration and will be related to the research area for the Ph.D. dissertation. The graduate course will be suggested by the major advisor, which requires the approval of the Graduate Coordinating Committee. Passing this graduate-level course with a grade of at least B will be considered for the third area of the comprehensive examinations.

Comprehensive examinations will normally be scheduled at the beginning of the spring semester of the student's second year. To show satisfactory progress in their graduate studies, students are normally expected to complete their comprehensive examinations by the end of the second full academic year of their Ph.D. work. A student will be allowed to retake each exam only once should they fail it.

For a Ph.D. with a mathematics concentration, the student chooses two sequences one from A-C and one from D-F from the list below:

- Complex Analysis: MA 8713 - Complex Analysis I and MA 8723 - Complex Analysis II;
- Functional Analysis: MA 8663- Functional Analysis I and MA 8673- Functional Analysis II;
- Real Analysis Analysis: MA 8633 Real Analysis I and MA 8643 Real Analysis II;
- Ordinary Differential Equations: MA 8313 - Ordinary Differential Equations I and MA 8323 - Ordinary Differential Equations II;
- Partial Differential Equations: MA 8333 - Partial Differential Equations I and MA 8343 - Partial Differential Equations II;
- Applied Mathematics: MA 8203- Foundations of Applied Mathematics I and MA 8213 Foundations of Applied Mathematics II.

For a Ph.D. with a statistics concentration, the student chooses two sequences one from A-B and one from C-E from the list below:

- Statistical Inference: ST 8733 – Advanced Statistical Inference I and ST 8743 - Advanced Statistical Inference II;
- Linear Models: ST 8613 - Linear Models I and ST 8633 - Linear Models II;
- Applied Probability: ST 8533 Applied Probability and ST 8553 Advanced Probability Theory;
- Design of Experiments: ST 8853 Advanced Design of Experiments I and ST 8863 Advanced Design of Experiments II;
- Multivariate Statistics: ST 8413 Multivariate Statistics Methods and ST 8433 Multivariate Statistical Analysis.

**The Dissertation**

After the comprehensive examinations have been passed, the student's doctoral committee will be reconstituted to form the dissertation committee, which normally consists of the same committee members. In a rare case, the student and the doctoral committee's major professor will select the student's dissertation committee, subject to the departmental Graduate Coordinating Committee's approval. The dissertation committee will consist of at least five graduate faculty members, including a major professor and at least three additional graduate faculty members from the Department of Mathematics and Statistics. The committee's primary responsibility will be to supervise the student's research and writing of a dissertation in the area of specialization, and its members should be chosen with this mission in mind.

The student's major professor will enlist a person from outside the Department of Mathematics and Statistics who has expertise in the dissertation area to serve as an external examiner. This person will read the dissertation and submit written comments regarding its quality and significance to the student's committee. The external reviewer's letter will be sent to the Graduate Coordinator for their record.

**Preliminary Examination**

When the comprehensive examinations have been passed, the breadth requirements have been met, and all course work on the program of study has been completed, the student may request the preliminary examination to be scheduled. In the early stages of the research effort, the student will make a formal dissertation proposal to the dissertation committee. The dissertation will be an original work that makes a significant contribution to the student's area of specialization. This examination will be an examination in the allied areas as well as an in-depth examination in the area of specialization. It will be administered by the student's doctoral committee and must contain an oral component. The majority vote of the dissertation committee will determine pass or fail. The oral component of the examination is open to the general members of the faculty and students.

**Admission to Candidacy: **A Ph.D. student who has passed the preliminary examination successfully will also pass the Admission to Candidacy.

**Final Examination**

Once all other examinations and the dissertation have been completed, the student's committee will schedule the student's final examination. This examination will consist of an oral defense of the dissertation and will be open to the public. After consultation with the Graduate Coordinator, the major professor will publicize the time and place that the examination will be held. This announcement should be made at least one week prior to the scheduled date of the examination. The major advisor will notify the Graduate Coordinating Committee on meeting the requirement(s) and provide a copy of the external reviewer's letter of support at least one week before the final examination.

A pass or fail on this examination will be determined by a majority vote of the student's committee. In making its decision, the committee will give due consideration to the external examiner's dissertation assessment.

Note:

- To set the final examination, from work completed during the Ph.D. studies, students are expected to prepare at least one manuscript suitable for submission to peer‐reviewed journals. This ready-to-submit manuscript shall be acceptable to the major professor.
- By the Graduate Coordinating Committee's recommendation and Head's approval, a Ph.D. candidate will be rewarded for each accepted paper in a qualified refereed journal related to the Ph.D. dissertation's research area prior to his/her graduation.