Mathematics (Ph.D.)

The goal of the doctoral program in Mathematics at Kansas State University is to provide each student with the tools needed to become a productive member of the mathematical community. To reach this goal, candidates must demonstrate to the faculty that they have met criteria for both breadth and depth in their studies. Specifically, each student will be required first to pass a written Qualifying Examination covering a broad range of mathematical subjects, and then, a Specialty Examination demonstrating the ability to pursue original and independent research in a chosen field of study. These will be followed by original research culminating in the Ph.D. dissertation. In addition, each student in the doctoral program in mathematics must demonstrate reading proficiency in either French, German or Russian. Since the vast majority of mathematics published in languages other than English are in one of the above languages, the graduate faculty feels that having at least a reading knowledge in one of the above languages is an important research tool.

The breadth requirement consists of two components: required coursework and the Qualifying Exam (QE). The Qualifying Exam is intended to measure the student’s ability to understand and to communicate with specialists from a broad range of mathematics. The depth criterion is met by the Specialty Examination (SE) and the dissertation. To pass the SE, a student must show a high level of mastery of the major ideas in the chosen area of specialization and thereby demonstrate the potential for making tangible contributions. The particular format of the Specialty Examination will be left to the discretion of the student’s Ph.D. Supervisory Committee. The Qualifying and Specialty Examinations together make up the Graduate School’s Preliminary Examination.

Passing the Qualifying Exam.
Passing the Preliminary Exam.
Demonstrating reading proficiency of mathematical exposition in French, German, or Russian.
Writing and defending the Ph.D. dissertation.

It is the Department’s philosophy that the Qualifying Examination tests the student’s mathematical breadth, whereas the Preliminary Examination tests the student’s depth.

After passing the Qualifying Exam, the graduate student will, within one year, select a thesis advisor and form the Ph.D. supervisory committee. This committee is responsible for administering the Preliminary Examination, which tests the student’s mathematical depth, and therefore his or her readiness to undertake serious mathematical research. The Ph.D. dissertation resulting from the student’s independent and original research represents the successful conclusion of the time spent as a graduate student. The student’s supervisory committee will schedule a public defense of the dissertation, during which the student will defend not only the mathematical correctness of the work, but also its originality and importance.

Course Requirements

At least 90 credit hours of mathematics, all at the 700 level and above, of which at least 24 credit hours are taken in residence at Kansas State University, and which includes at least 30 credit hours of MATH 999. Eligibility to enroll in MATH 999 is restricted to students having passed the Speciality Exam (see below). A master’s degree from a reputable graduate program in mathematics can be used in lieu of 30 credit hours in mathematics.

800-LEVEL COURSE-REQUIREMENT: The above coursework must also include a one-year sequence from the list below. To be able to fullfill this 800-level course requirement a minimum grade of B must be obtained in both semesters. Such a one-year sequence may not count if it corresponds to one of the subdisciplinary topics that the students has passed in the Qualifying exam (see paragraph B. below), e.g. if a student passes the Qualifying Examination by taking Analysis and Algebra, then the sequences Math 821-822 and Math 810-811 cannot be counted as fullfilling the 800-level course requirement. On the other hand, a student may opt to take a third qualifying exam in lieu of the 800-level course requirement.

1.)
MATH 810 - Higher Algebra I Credits: 3
MATH 811 - Higher Algebra II Credits: 3
2.)
MATH 821 - Real Analysis Credits: 3
MATH 822 - Complex Analysis Credits: 3
3.)
MATH 840 - Differential Equations I Credits: 3
MATH 841 - Differential Equations II Credits: 3
4.)
MATH 861 - Numerical Analysis I Credits: 3
MATH 862 - Numerical Analysis II Credits: 3
5.)
MATH 875 - Algebraic Topology I Credits: 3
MATH 876 - Algebraic Topology II Credits: 3

MATH 999 Coursework

It is the intention of the Department that a Ph.D. student enrolled in MATH 999 is actively doing research directly relevant to the writing of the doctoral dissertation. Since a student not having passed the Specialty Exam has not yet demonstrated sufficient depth to embark on Ph.D. level research, enrolling in MATH 999 coursework is restricted to only those students having passed the Specialty Exam. Students enrolling in MATH 999 for the first time (only) will need to complete the Request for Permission to take Math 999, Ph.D. Research.

Qualifying Examination

Beginning Fall 2003, the QE will consist of two written Subject Area Examinations and will constitute part of the Graduate School’s Preliminary Examination. The Subject Area Examinations are to be chosen by the student from the following list:

Algebra (based on MATH 810-811)
Analysis (based on MATH 821-822)
Geometry/Topology (based on MATH 701-875-876)

The individual subdisciplinary examinations are taken individually (possibly in different semesters); passing the Qualifying Examination is tantamount to passing each of the Subject Area Examinations.

Students who wish to obtain the Ph.D. at Kansas State University will be required to pass the QE by the end of the third year from the time that they originally enter the graduate program in mathematics. (Students who enter the graduate program in a Spring semester will be regarded, for the purpose of meeting the deadline for successful completion of the QE, as having entered the program at the beginning of the following Fall semester.) Furthermore, students entering with weak backgrounds but showing good progress may petition for an extra year within which to pass the Qualifying Exam. The QE Subject Area Examinations will be offered early in the Fall and Spring semesters of each year. Students who intend to take a QE Subject Area Examination in a given semester should declare their intention to the Director of Graduate Studies before the end of the preceding semester. Each Subject Area Examination is based on material covered in a one-year 800-level course in the subject area (as indicated above), and tests the student’s general knowledge of that subject area. These examinations are roughly equivalent to final examinations in the corresponding 800-level courses. Each student will receive a grade of “pass” or “fail” on each Subject Area Examination; passing QE as a whole requires passing grades on each of the Subject Area Examinations.

Specialty Examination

The Specialty Exam is the second part of the Graduate School requirement for the Preliminary Examination and must be passed in order to become a doctoral candidate. Each student must take the SE, but can do so only after having passed the QE, formed the Supervisory Committee, and submitted a Program of Study to the Graduate School. The structure and content of the SE may vary, depending on the characteristics of the student’s chosen field of study, the Program of Study, and the discretion of the Supervisory Committee. For example, the SE may include one or more of the following components: a written and/or oral examination, a prepared colloquium talk, a computer program, or a publishable paper. The Supervisory Committee will determine the date for the SE, which should be no later than two years after successful completion of the QE and at least seven months before the final oral examination. The SE shall be scheduled through the Graduate School, using the Request for Preliminary Examination Ballot form. Shortly before the examination date, the Graduate School will issue a ballot for the Preliminary Examination to the Major Professor on which the results of both the Qualifying Examination and the Specialty Examination are to be recorded. Failure of the SE is tantamount to failing the Preliminary Exam, and will be grounds for termination of financial support by the Department of Mathematics (if applicable), and failure of the second attempt will be grounds for dismissal from the Ph.D. program by the Graduate School. Upon satisfactory completion of the Preliminary Exam, the doctoral student is automatically advanced to candidacy for the Ph.D. degree.

Foreign Language Examination

In addition to the QE and SE, each Ph.D. candidate will be required to pass a language examination in French, German or Russian. No substitute foreign languages will be allowed. The requirement must be completed at least seven months prior to the final examination and may be met in one of two ways:

1. Pass the Graduate Foreign Language Proficiency Exam for that language administered by the Modern Language Department or,

2. Within two years prior to entering the Graduate School here at KSU, (or during one’s studies while a graduate student at Kansas State University) receive a grade of at least B for fourth-semester (or higher) study in the elected language at the college level.

Ph.D. Dissertation

After successful completion of the SE, students will continue their research under the supervision of their doctoral advisors. In order to receive the Ph. D. degree, the student will be required to write a Ph.D. thesis containing non-trivial, new results of publishable quality, and successfully defend this thesis in a final oral examination administered by the Supervisory Committee.

After the student’s dissertation has been deemed acceptable by the doctoral advisor, a final oral examination will be scheduled through the Graduate School and held in accordance with Graduate School requirements.