# Ph.D. in Mathematics

There are two options that lead to the Doctor of Philosophy in Mathematics. They are **Mathematics** and **Mathematics with an emphasis in statistics**.

#### I. General Requirements

Students are formally admitted to the Ph.D. program upon the completion of the requirements for the Masters degree in mathematics via the mathematics track and passing the Preliminary Examinations.

All the candidates for the Ph.D. Degree in mathematics must satisfy the same general requirements. They are

- Completion of the requirements for the Masters degree in mathematics via the mathematics track.
- Completion of a total of at least 90 hours of graduate course work of which up to 30 hours may be dissertation credits. Credits taken toward the completion of the Master's Degree may be included with the approval of the Mathematics Graduate Committee. Dissertation credits can be taken only after students pass the full Qualifying Examinations.
- Maintaining grade point average of at least 3.6 in the courses covered by the Preliminary and Qualifying Examinations and in the courses in their field of specialization. The Mathematics Graduate Committee may recommend that a student in the Ph.D. program who fails to maintain the required grade point average not be continued in the Ph. D. program.

#### II. The Preliminary Examination Requirement

The preliminary examination covers the two foundational course sequences, MAT 601-602 and MAT 631-632. It consists of two written examinations, each lasting three hours. These may be taken separately. There are only two outcomes on each examination, Pass or Fail. Both examinations must be passed to pass the Preliminary Examination. These examinations will be given twice a year, before or near the beginning of each semester. The exact dates will be announced at least a month ahead of time.

Masters students who wish to continue into the Ph.D. program must take these examinations no later than the August preceding their second year of graduate study and pass these examinations no later the January during their second year of graduate study. Students who pass one, but not both of these examinations on their first attempt are required to retake only the examination that is not passed. Only two attempts at each of these examinations are permitted.

With the approval of the Mathematics Graduate Committee, students who enter the Mathematics Graduate Program with extensive preparation in mathematics may attempt the Preliminary Examinations upon entering the Program. This attempt will not be counted as one of the two attempts at the Preliminary Examinations.

With the approval of the Mathematics Graduate Committee, students who enter the Mathematics Graduate Program with limited preparation in mathematics may postpone the Preliminary Examinations until their third year of graduate study.

#### III. Course Requirements in the Mathematics Option

Students following the mathematics option to the Ph.D. Degree in Mathematics are required to

- Complete two breadth courses in each of algebra/topology and analysis. The choices of breadth courses in these areas are
- Algebra/Topology - MAT 661 and either MAT 761 or MAT 731
- Analysis - MAT 701 and either MAT 712 or MAT 721

- Demonstrate breadth in applicable mathematics by completing a two-course sequence chosen from the following list:
- MAT 651-652
- MAT 653-654
- MAT 645-646
- MAT 682-683
- MAT 683-684
- MAT 683-704
- MAT 684-704

- Complete three additional mathematics courses numbered 700 or above. These courses may not include reading courses or courses taken to satisfy earlier requirements.

#### IV. Course Requirements in the Mathematics with an Emphasis in Statistics Option

Students following the option in mathematics with an emphasis in statistics leading to the Ph.D. Degree in Mathematics are required to

- Complete two breadth courses in each of analysis and statistics. These courses must be chosen from the options listed in III 1 and 2 above.
- Complete either MAT 682 or MAT 683.
- Complete three additional mathematics courses numbered 600 or above. These courses may not include reading courses or courses taken to satisfy earlier requirements. These courses must be approved by the student's Ph.D. advisor and the Graduate Committee.
- Complete one course in applied statistics taken from a list of courses not offered by the Mathematics Department but approved by the Graduate Committee. The list of electives in applied statistics consists of the courses in item III 3 b) above of the requirements for the Masters of Science degree.

#### V. The Qualifying Examination Requirement

The Qualifying Examination consists of two written tests, each covering one of the following two-course sequences.

- Algebra: MAT 731, 732
- Analysis I: MAT 701, 712
- Analysis II: MAT 701, 721
- Combinatorics: MAT 645, 646
- Numerical Analysis: MAT 683, 684
- Statistics: MAT 651, 652
- Topology: MAT 661, 761

One of these exams is designated by the student as the Major Examination and should be chosen in consultation with his/her advisor or anticipated PhD advisor. The Major Examination may be any of the seven options above.

The other exam is called the Minor Examination. The Minor Examinations permitted depend on the Major Examination taken, as follows:

- If the Major Exam is Numerical Analysis, the Minor Exam must be Analysis I: MAT 701, 712.
- If the Major Exam is Statistics, the Minor Exam must be Analysis II: MAT 701, 721.
- Otherwise, the Minor Exam must be one of Algebra, Analysis I, or Topology, and must be on a course sequence disjoint from the Major Exam.

Each part of the Qualifying Examination lasts four hours. The parts may be taken separately. There are only two outcomes on each part, Pass or Fail. These examinations will be given twice a year, before or near the beginning of each semester. The exact dates will be announced at least a month ahead of time.

Students must pass both the Major and Minor Examinations. Only two attempts at each of the Major and Minor examinations are permitted. The students' first attempt at the Major examination must be no later than the August preceding their third year of graduate study and their second attempt no later than the August preceding their fourth year of graduate study. The students' last try at the Minor examination must be taken no later than the January during their fourth year of graduate study.

With the approval of the Mathematics Graduate Committee, students who postpone taking one or both of their Preliminary Examinations until their third year of graduate study may also postpone the Qualifying Examinations, in the same area, until their fourth year of graduate study.

Students who wish to change the area in which they plan to write their thesis may be required to retake all or part of the Qualifying Examination.

#### Old version of V. The Qualifying Examination Requirement (available only to students who entered the graduate program in Fall 2013 or earlier)

#### Old version of V. The Qualifying Examination Requirement (available only to students who entered the graduate program in 2011/12 or earlier)

#### VI. The Specialty Examination

The Specialty Examination serves at least two purposes:

- To acquaint faculty members with the research potential and interests of the student, and the student with the research interests of faculty members.
- To determine whether the student can read and assimilate advanced material in a specialized field.

The Specialty Examination is an oral examination. After a student has passed the Major Examination in his/her field of interest, he/she asks a faculty member to direct his/her preparation for the Specialty Examination and to chair a committee to conduct his/her Specialty Examination. The chair, in consultation with the student, selects at least two other faculty members to comprise the committee.

The committee members, in consultation with each other and the student, assign topics and a reading list to the student. To pass the examination, the student must demonstrate competence in and an understanding of the assigned topics and prerequisite material. The examination may include a short presentation by the student, but will also include direct examination by the committee members.

Only two attempts are permitted at the Specialty Examination and it must be passed by the end of a student's fourth year. The mathematics Graduate Committee may recommend non-continuance in the Ph.D. program for a student in the Ph.D. program who fails to pass the Specialty Examination by this time.

#### VII. The Language Requirement

All students must demonstrate mastery of English. Generally, admission to graduate study implies proficiency in English, but in some cases, as determined by the chair, associate chair, or by the student's advisor, remedial work may be required.

#### VIII. The Dissertation Requirement

The most important part of the Ph.D. program is the writing of a dissertation that demonstrates the candidate's ability to carry out an independent investigation that makes an original, scholarly contribution to mathematics.

Each Ph.D. candidate has a dissertation advisor who provides guidance in the final part of the course program and during the writing of the dissertation. A student should discuss this program with the dissertation advisor each semester before meeting with an academic advisor as part of registration.

#### IX. The Oral Examination (Dissertation Defense) Requirement

When the dissertation meets with the approval of the student's dissertation advisor and the student has acquired a total of 90 hours of graduate credit, the candidate is given a final oral examination of the dissertation and the immediately related field. The committee for this examination will consist of a chair appointed by the Graduate School, the student's dissertation advisor, and four other members appointed by the Mathematics Graduate Committee with advice from the student and the dissertation advisor. Upon final approval of this committee, a copy of the dissertation will be placed in the mathematics library.