# View Entire Graduate Handbook

## Handbook for Graduate Studies in Mathematics West Virginia University 2016

## 1. General Information

**(1.1) **The Department of Mathematics offers the following degrees: Master of Science (M.S.) and Doctor of Philosophy (Ph.D.).

**(1.2)** This handbook provides information on both the M.S. program and the Ph.D. program in Mathematics. It is intended for the use of faculty as well as prospective and current graduate students. Other information and regulations concerning graduate study are contained in the University's Graduate Catalog.

**(1.3) **Students in any of the degree programs are expected to remain in good standing. In addition to meeting University requirements, to remain in good standing, and as a condition of satisfactory progress, students must, by the end of the first year and each semester thereafter, maintain a GPA of at least 3.0 in course work included in their program of study and, separately, in such course work taken in the Mathematics Department. Students must also maintain satisfactory progress toward their degree as discussed in (2.5.1) and (3.10) and meet program deadlines. Students are also expected to receive at least a B in any course taken to remedy deficiencies in their background, usually in connection with their admission as a provisional student.

**(1.4) ** Graduate Faculty List: A list of current graduate faculty is maintained by the Eberly College of Arts and Sciences. Students who are being supervised for an M.S. or Ph.D. thesis must be supervised by a regular member of the graduate faculty.

**(1.5) ** Course work: When used below “course work” and “courses” refers to standard three-credit courses that include Independent Study courses (Math X95), but exclude credit hours in connection with Research (Math X97) , Thesis (Math X98), Graduate Colloquium (Math X99), and one-credit seminar courses, including Math 696, Math 590, and Math 694.

**The Master's Program**

Option A (Pure Mathematics)

Option B (Industrial/Applied Mathematics)

Option C (Applied Mathematics)

Mathematics for Secondary Educators Option

### (2.0) Common Requirements

**(2.0.1) Admission Requirement**

For regular admission, a baccalaureate degree in mathematics or its equivalent is required, including an introductory course in real analysis. Students who have deficiencies in their background may be granted provisional admission to the M.S. program. Deficiencies must be completed within the first year.

**(2.0.2) The Basic Exam**

The Basic Exam in Advanced Calculus and Linear Algebra is given as a placement exam at the beginning of graduate study for M.S. students, as part of the Real Analysis and Linear Algebra Requirement (see section (2.0.3)). Further information on the contents of the Basic Exam can be found in Attachment 2.

**(2.0.3) Real Analysis and Linear Algebra Requirement**

The Department requires that all M.S. students demonstrate knowledge of introductory real analysis and linear algebra. In this regard, first-year M.S. students will be required to take Math 543 and receive at least a B, unless this requirement is waived in writing as a result of performance on the Basic Exam (administered on entering the graduate program). Knowledge of real analysis must be demonstrated by receiving at least a B in Math 551, or by taking both Math 451 and Math 452 with a minimum B average. Entering graduate students will be placed into Math 451 unless they show on the Basic Exam that they have sufficient background to take Math 551.

** (2.0.4) Grade Point Average**

To be awarded the M.S. degree, students must have a GPA of at least 3.0 in course work presented for the degree, and separately have a GPA of at least 3.0 in mathematics course work presented for the degree. A grade requirement of “B or better” applies to certain courses; students may repeat a course in which the requirement is not met, and the higher grade will be used in determining departmental eligibility for graduation. University GPA requirements, which typically apply to all courses taken, must also be met.

**(2.0.5) Graduate Seminar**

Each M.S. student must satisfactorily complete the one credit hour “Professional Tools Seminar”, offered in the spring under Math 694.

### (2.1) Option A (Pure Mathematics)

**(2.1.1) Introduction**

Option A is centered around a core curriculum of four courses covering the fundamental areas of mathematics. The requirements are flexible enough to accommodate students with a variety of mathematical interests and post-graduate plans, including doctoral study in mathematics or the mathematical sciences, and employment in government or industry.

**(2.1.2)** **Core Curriculum**

An M.S. student in Option A must complete these core courses: Math 541 (Algebra), Math 543 (Linear Algebra), Math 551 (Real Analysis) and Math 581(Topology) with a grade B or better in each of these courses.

**(2.1.3) Examinations**

The Basic Exam in Advanced Calculus and Linear Algebra is given as a placement exam at the beginning of graduate study.

Students in Option A must pass the M.S. Advanced Exam, as described in Attachment 2, by the end of the third year of their study. The Advanced Exam also serves as the Ph.D. Entrance Examination. See Attachment 2.

**(2.1.4) Other Requirements**

A plan of study must include a minimum of eighteen (18) hours of graduate course work within the Department of Mathematics. A student can complete the degree requirements in one of the following options:

A. Complete twenty-seven credits in course work plus a thesis representing three hours of credit.

B. Complete thirty credits in course work.

### (2.2) Option B (Industrial/Applied Mathematics)

**(2.2.1) Introduction**

Option B provides a broad background in areas of industrial and applied mathematics. It is aimed at students from undergraduate majors in mathematics, science, and engineering who have an interest in applications of mathematics, have succeeded at a high level in their mathematics and science courses, and have some undergraduate training in application disciplines. The curriculum includes the "classical" areas of applied mathematics (ordinary and partial differential equations), computational methods, modeling, applied probability and statistics, discrete mathematics, optimization, and basic mathematics (advanced calculus of one and several variables, complex variables, linear algebra). While self-contained and suitable as a degree that would prepare a student for employment, with a suitably chosen program of study this degree could also prepare a student to proceed on to doctoral work at WVU or elsewhere in mathematics or the mathematical sciences.

**(2.2.2) Entrance Prerequisites**

Students pursuing Option B should have, at a minimum, the following undergraduate background:

Three semesters calculus.

Linear Algebra (Math 343 or 441).

Elementary Differential Equations (Math 261).

(Students lacking Differential Equations or Linear Algebra may be admitted provisionally under some circumstances.)

**(2.2.3)** **Foundations**

These include areas basic to industrial/applied mathematics, at an upper-division undergraduate level. Each student graduating from the program should have some experience with the subject matter represented in the following areas (with applicable undergraduate courses listed):

Programming experience in a general programming language (e.g., C, Pascal, Basic, Fortran, MATLAB ).

Introductory Partial Differential Equations (Math 465).

Probability theory (Stat 215 or Stat 461).

Numerical Analysis (Math 420).

Mathematical Analysis (advanced calculus Math 451 and complex analysis Math 456).

Algebra/Discrete Math (Math 341, Math 375, or Math 378).

Students will be expected to demonstrate proficiency in all these areas at the level indicated, through examination or course work. More specifically, proficiency may be demonstrated in the following ways:

(1) By examination.

(2) By completion of an appropriate undergraduate course with a grade of A or B, either as an undergraduate, or as a graduate student in the Department.

(3) By completion with a grade of A or B of an appropriate graduate-level course which covers substantially similar material at a more advanced level. The graduate courses (some of which are required) which can be used to satisfy foundation requirements are as follows:

Math 521 will satisfy the Numerical Analysis foundation requirement.

Stat 561 will satisfy the Probability requirement.

Math 567-568, Math 551, or Math 555 will satisfy the mathematical analysis requirement.

Math 571,671,573,545,645,541, or 543 will satisfy the Algebra/Discrete Math requirement.

**(2.2.4) Course Requirements**

Note: Students who intend to pursue doctoral study in mathematics or the mathematical sciences should carefully plan their course work so as to meet the prerequisites of their intended field of study and degree program.

A total of at least 33 credit hours of course work is required, distributed as follows.

__Prerequisite/Foundation Courses__

At most 3 hours of credit toward the 33 hours will be given for one of Math 465, Math 441, Math 456, Math 451, provided these courses (or equivalents) have not been previously taken at WVU or elsewhere.

__Required Core Graduate Courses (12-15 hours)__

Math 567 and either Math 568 or Math 555 (Advanced Calculus/Complex Variables).

The Math 567-568 sequence will include multivariable and vector calculus in Math 567, applied complex variables in Math 568, and other topics important to applied mathematics as time allows. Students may take Math 555 in lieu of Math 568. Students with a grade of B or better in Math 456 may be exempted from the Math 568 or Math 555 requirement.

Math 564 (Differential Equations).

Math 521 (Numerical Analysis).

Math 563 (Modeling).

__Distribution Requirements (at least 15 hours)__

Students must complete at least two courses from Math 452, Math 541, Math 543, Math 551, Math 555 and Math 581, with a grade at least B on these courses.

Including the core courses listed below (Math 521 and Math 563), at least one course from 4 different groups must be included in the program of study. Within two different groups, there must be a sequence of two suitably linked courses, one of which may be a core course listed. Once these conditions are met, the remainder of the 33 hours may be made up of any of the courses below or other mathematics electives, subject to approval by the advisor. Overall, at most 4 courses may be from outside the Mathematics Department.

I. Computation/optimization: Math 521, Math 522 Numerical Solution of PDE's.

II. Probability theory/mathematical statistics: Stat 561-562; (other approved Stat courses provided Stat 562 is also taken).

III. Algebra/Discrete Math: Math 541; Math 543;Math 571,671; Math 573;Math 545,645; other suitable math electives in this area, or CS courses approved by the advisor and the Graduate Director.

IV. Modeling: Math 563; a second semester of modeling.

V. Mathematical Analysis: Math 452, Math 551, Math 651, Math 555.

VI. Graduate courses (one 400-level possible with permission) from outside Math/Stat/CS (students are responsible for meeting prerequisites). These are to be approved by the advisor and the Graduate Director.

**(2.2.5) Examinations**

The Basic Exam in Advanced Calculus and Linear Algebra is given as a placement exam at the beginning of graduate study.

**(2.2.6) Research/Internship/Project** **(1 hour)**

This is required for all students. The work represented should be equivalent to that associated with at least one credit hour of course work. If warranted, more credit can be granted with permission, but course work requirements will not be reduced. A presentation and approval by the advisory committee are required.

Successful completion of the above requirements (2.2.2)-(2.2.6) with a minimum GPA of 3.0 in course work presented will suffice for the granting of the M.S. degree.

### (2.3) Option C (Applied Mathematics)

**(2.3.1)** **Introduction**

Option C provides additional flexibility for those students who wish to specialize in applied analysis/differential equations or in discrete mathematics.

**(2.3.2)** **Core Curriculum**

An M.S. student selecting Option C must complete at least 3 courses from Math 541, Math 543, Math 551, and Math 581 with a grade B or better, and must take at least one additional math course either from the differential equation and analysis group (Math 564, Math 551 (if not taken) and Math 555); or from the algebra, discrete math and number theory group (Math 571, Math 573, Math 545, Math 677, Math 578, Math 541 (if not taken) and Math 543 (if not taken)).

**(2.3.3) Examinations**

The Basic Exam in Advanced Calculus and Linear Algebra is given as a placement exam at the beginning of graduate study.

Students in Option C must pass the Advanced Exam, as described in Attachment 2, by the end of the second year of their study. The Advanced Exam also serves as the Ph.D. entrance examination. See Attachment 2.

**(2.3.4) Other Requirements**

A plan of study must include a minimum of eighteen (18) hours of graduate course work within the Department of Mathematics. A student can complete the degree requirement in one of the following options:

A. Complete twenty-seven credits in course work plus a thesis representing three hours of credit.

B. Complete thirty credits in course work.

### (2.4) Mathematics for Secondary Educators Option

**(2.4.1)** **Introduction**

This option provides an appropriate mathematical background for students who are currently, or will become, teachers of mathematics at the secondary school or community college level. A plan of study must include a total of 33 hours of graduate work, with at least 18 hours in graduate courses in mathematics. The remaining hours may be taken, for example, in mathematics, mathematics education, statistics, or computer science upon prior approval of the Advisory Committee. A maximum of 9 hours from any one discipline outside of mathematics may be used to meet the 33 hours required for this degree.

**(2.4.2)** **Core Curriculum**

15 - 18 hours

- Real Analysis requirement: Math 451/452 or Math 551 (Math 451 counts toward the 33 hours)
- Linear Algebra requirement: Math 543
- Math 535 – Foundations of Geometry
- Math 541 – Modern Algebra (or Math 533 – Modern Algebra for Teachers 1; not regularly offered.)
- Statistics 505 – Foundations of Probability/Stat or Stat 561 or Stat 511- 512 (Stat 511 does not count toward the 33 hours)

And a choice of one of the following core courses in applied/discrete mathematics:

3 hours

- Math 530 – Introduction to Applied Mathematics (not regularly offered)
- or Math 545 – Number Theory 1
- or Math 571 – Combinatorial Analysis 1
- or Math 573 – Graph Theory
- or Math 563 - Modeling
- or Math 521 – Numerical analysis
- or Math 564 – Intermediate Differential Equations

And a choice of one of the following:

3 hours

- Math 536 – Transformation Geometry
- or Math 641 – Modern Algebra 2
- or Math 534 – Modern Algebra for Teachers 2 (not regularly offered)

And the remaining hours to be approved by the student’s advisor:

A maximum of 9 hours may be taken from any one discipline outside of Mathematics

**(2.4.3)** **Examinations**

The Basic Exam in Advanced Calculus and Linear Algebra is given as a placement exam at the beginning of graduate study.

Students must pass a final written examination based on their course work, consisting of four sections: algebra, geometry, applied/discrete mathematics, and probability/statistics. With the approval of the Advisory Committee, the exam may be taken after completion of the required 18 hours of graduate mathematics courses and the core curriculum.

### (2.5) General M.S. Program Guidelines

**(2.5.1) Minimum Number of Courses/Independent Study**

As a condition of satisfactory progress, all M.S. students receiving full time financial support from the Department of Mathematics are expected to take three courses each semester (exclusive of seminar courses) that are either part of program requirements or represent approved or mandated prerequisite courses. In addition, they are expected to satisfactorily complete with a grade of B or better at least two graduate mathematics courses in the Department of Mathematics each semester toward fulfillment of their program requirements. These must be courses which count toward meeting their credit requirements, with the sole possible exception of Math 451. Courses taken for audit do not count toward maintenance of satisfactory progress or these minimum enrollments. Failure to maintain satisfactory progress may result in the withdrawal of financial assistance.

Independent Study cannot be used to fulfill program requirements until the student has successfully completed the applicable core course requirements.

**(2.5.2) The Academic Advising of M.S. Students**

Within the first year of study, each M.S. student will be assigned a faculty advisor, who is a member of the graduate faculty. The advisor's duties include offering the advisee academic advice to the best benefit of the advisee, signing the registration forms of the advisee, helping the advisee to prepare the plan of study, and reminding the advisee of the academic deadlines. An Advisory Committee, consisting of three faculty members, will be appointed. For students not writing a thesis or project, the Advisory Committee will normally consist of the faculty advisor, together with the graduate director and associate graduate director.

The advisee may consult with the graduate director directly should the advisee feel uncomfortable in discussing a problem with the current advisor. In the event that the advisor is the graduate director, the advisee should consult the chair person of the department under this situation.

Students will be advised by the graduate director before the appointment of their advisors.

**(2.5.3) Duration of Financial Support**

An M.S. student will be supported as a graduate assistant by the department of mathematics for up to two years, assuming that the student remains in good standing and is making satisfactory progress. Any extension must be approved by the Graduate Committee.

**(2.5.4) M.S. program course work**

Courses at the 400-level do not count toward meeting credit hour requirements except for Math 452, and under other conditions explicitly noted in this document. Credit hours in research, graduate seminar, and teaching practicum, also do not count toward meeting course work credit hour requirements. Certain graduate courses designed for secondary-school mathematics teachers may be used for credit only within the Mathematics for Educators option, and certain courses offered for inservice mathematics teachers normally do not count for M.S. program credit hour requirements.

**(2.5.5) Alternative to certain required courses/grades**

** **

The Graduate Committee will consider requests from students for alternatives to certain of the above course/grade requirements in the following special circumstances:

A. A student who receives a grade of C in Math 541, Math 551, or Math 581, when a grade of B is required, may petition the Graduate Committee to allow the student, as an alternative to that requirement, to obtain at least a B in the subsequent course in the sequence (Math 641,651,681 respectively), and to pass the corresponding M.S. Advanced Exam at the M.S. level. The exam may be taken as part of the normal examination requirements in Options A or C, or may be taken apart from, or in addition to, those requirements, for purposes of fulfilling the alternative requirement.

B. Advanced students who have a background equivalent to certain of the required M.S. courses, may, with approval of the Graduate Director, substitute a more advanced course in the same subject area, or pass the corresponding M.S. Advanced Exam at the M.S. level, or, with a sufficiently strong background, have the requirement waived on the basis of demonstrated expertise obtained from prior course work. Overall credit hour requirements, however, will not be reduced. This section is aimed primarily at more advanced students admitted to the Ph.D. program who subsequently wish to obtain an M.S. degree.

**(2.5.6) ****Exam policy for M.S. Students continuing on to the Ph.D. program**

An M.S. student who takes the M.S. Advanced Exam/Ph.D. Entrance Examination will have their examinations in each subject graded as Ph.D. pass, M.S. pass, or fail. If they achieve a pass at the Ph.D. level in a given subject area, this result may be used as part of the Ph.D. Entrance Exam if they subsequently enroll in the Ph.D. program within 5 years. Overall, any subject area exam may be attempted a maximum of three times while a graduate student.

In the awarding or continuation of graduate assistantships in the Ph.D. program, performance on the M.S./Ph.D. Exam will be a consideration. To be considered for, or to maintain such awards, M.S. students who proceed directly into the Ph.D. program are required to pass two subject areas at the Ph.D. level by no later than the second time the exam is offered after they enroll in the Ph.D. program, and to pass any subject area attempted at the Ph.D. level by the second attempt overall while a student in either graduate degree program. Cases in which these conditions are not met, or in which an M.S. student interrupts their studies for a year or more after graduation before enrolling in the Ph.D. program, can be considered by the Graduate Committee on an individual basis.

## 3. The Ph.D. Program

Admission Requirement

Dissertation Advisors

Advisory Committee and Dissertation Committee

Ph.D. Extrance Examinations

Course Requirements

PhD Qualifying Exam

Residency Requirement

Language Requirements

### (3.1) Admission Requirements

An M.S. degree in Mathematics or equivalent is normally required, but provisional admission can be granted in some circumstances for students with a very strong mathematics background acquired during their undergraduate study. Details are in Attachment 5. A student who has deficiencies in admission requirements may be granted entrance to the Ph.D. program as a provisional Ph.D. student. Deficiencies must be completed within the first year. Entrance examination requirements for Ph.D. students are the same for regular or provisional admission status.

### (3.2) Qualification of and Limitation on Dissertation Advisors

Any faculty member who is a regular member in the graduate faculty can serve as a dissertation advisor. There is no limit on the number of Ph.D. students a particular faculty member may supervise at a given time.

### (3.3) Ph.D. Advisory Committee and Ph.D. Dissertation Committee

These two committees will be called advisory committee and Ph.D. committee, respectively, in the rest of this document, when no confusion arises. The duty of the advisory committee is to advise students in work prior to the start of a dissertation. The duty of the Ph.D. committee is to oversee and direct work of a dissertation. See the Advising Guidelines in Attachment 1.

An advisory committee will be assigned to each Ph.D. student within the first year in the program. The formation of a Ph.D. advisory committee will be identical with that of an M.S. advisory committee. (See Section (2.5.2).) The advisory committee will be dissolved upon the formation of the Ph.D. committee for the student.

The Ph.D. committee will be appointed by the Graduate Programs Committee with consultation of the applicant and the applicant's advisory committee within one year after the applicant has successfully passed the Ph.D. entrance exams.

Members of a Ph.D. committee should be chosen so as to be in research areas closely related to the dissertation. The committee will serve not only to judge the final dissertation, but also as a research and advisory resource to the student and dissertation advisor.

Once the Ph.D. committee for a student is formed, the chairperson of this committee should submit an annual progress report for the Ph.D. candidate to other members of the committee and to the graduate director. The information on the report will be useful for decisions about further financial support for the candidate and for identifying students who may need additional help from their committee members.

The Ph.D. committee of a Ph.D. candidate consists of 4 regular graduate faculty members in the Department of Mathematics, including the chairperson of the committee, who is also the dissertation advisor, plus another faculty member who is a regular member of the WVU graduate faculty outside the Department of Mathematics, or a faculty member outside WVU who would otherwise qualify for regular member status.

### (3.4) Ph.D. Entrance Examinations

A common Ph.D. Entrance Exam/M.S. Advanced Exam structure is adopted. No later than the end of their second full academic year in the program, students must pass two subject area exams at the Ph.D. level, chosen from Algebra, Real Analysis, Differential Equations, and Topology. Failure to meet this requirement will result in the student being dropped from the program. The Examination procedures are fully described in Attachment 2.

Any delay in taking the examinations, or other exceptions to examination procedures, must be approved in advance by the Graduate Programs Committee.

### (3.5) Course Requirements

Regular Ph.D. students must complete a program that includes a minimum of eight graduate courses (exclusive of seminar and research requirements), subject to the requirements below. Additional course requirements generally apply to Ph.D. students admitted provisionally. Subject to the requirements below, a student’s program should be chosen in agreement with the Graduate Director and the student’s dissertation advisor so as to meet the educational objectives of the program and the needs of the student’s research area.

For students whose dissertation area is represented by one of the subject areas in groups A and B below, four courses in a major area must be taken at the 700-level. Two courses must be taken at the 700-level in each of two minor areas. As a way of ensuring a measure of breadth in mathematical background, the major and minor areas must be chosen from the two major groups A and B below, so that at least one subject from each of the two groups is selected.

Students whose dissertation area is Research in Undergraduate Mathematics Education must complete the four course RUME sequence and six additional mathematics courses, chosen from the areas listed below and including at least four courses at the 700-level. At least two courses should be chosen from each group A and B so as to fulfill the requirements of a minor area. The remaining two courses can be used, for instance, to acquire additional breadth in mathematics or to achieve research level depth in an area.

A. Algebra/Number Theory/Discrete Mathematics/Set Theory.

B. Analysis/Differential Equations/Numerical Analysis and Computational Mathematics/Topology.

With the prior approval of the Graduate Programs Committee, a total of at most 3 hours of course work in each of the two minor areas may be at 500-600 level.

With the prior approval of the student's advisor and the Graduate Director, one minor area may be chosen outside of the areas listed in groups A and B, and may include suitable course work outside the mathematics department (see Attachment 1, Item 7). If this minor area achieves, in a comparable way, the objective of ensuring breadth in the student’s program, an exception to the requirement of including both groups A and B among the major and minor areas may be made, with the approval of the Graduate Director.

A minimum of 6 courses at the 700-level in the Mathematics Department must be included in the plan of study. Using an independent study to fulfill any of the major or minor course requirements must be approved by the Graduate Programs Committee.

Students interested in interdisciplinary research in computer science and/or statistics may pursue the option in Combinatorial Computing and Discrete Mathematics (CCDM) which is described in a separate document.

Doctoral students will register for one credit of graduate seminar (Math 696) each semester they are in residence. Format and expectations associated with the graduate seminar are described in Attachment 4.

Each Ph.D. student must satisfactorily complete the one credit hour “Professional Tools Seminar”, offered in the spring under Math 694. This is taken in place of Math 696 during the semester of enrollment.

Any exceptions to the above program requirements must be approved in advance by the Graduate Programs Committee.

### (3.6) Ph.D. Qualifying Examinations

The Qualifying Exam (also referred to as the candidacy exam) is intended to establish that the student has mastered the subject matter in the major and minor areas, and has acquired the background needed to successfully complete the Ph.D. dissertation. Knowledge of the minor areas may be established by meeting a GPA requirement. The student’s knowledge of the major area, and other areas needed for dissertation research as determined by the student’s advisor and committee, is established through examination or through the completion of a research assignment (RUME option). The student’s ability to complete a thesis is established by the writing and presentation of a dissertation prospectus that demonstrates a suitable dissertation subject, knowledge of relevant literature, and a clear plan for the dissertation research.

In the minor areas, the student must complete the two courses in each of two minor areas with a GPA of at least 3.5. If a student fails to meet the GPA requirement on one or both of the minor area(s), a written examination for the area(s) will be given as Part 3 of the Qualifying Exam (see below).

The Qualifying Exam consists of three parts:

PART 1: Students whose dissertation area is represented by one of the subject areas in groups A and B above, take a three-hour written examination over the major area, followed by an oral exam. Students whose dissertation area is Research in Undergraduate Mathematics Education will be provided by their committee with a research assignment, based on content areas and research techniques in the field, to be completed over a four-week period. The results will be presented in written form and orally examined by the student’s committee.

PART 2: Presentation of a dissertation prospectus.

PART 3: This part is needed only when the student fails to meet the GPA requirement on one or both of the minor areas. One three-hour written examination is taken over each minor area in which the student fails to meet the GPA requirement.

### (3.7) Qualifying Exam Guidelines

A Ph.D. student is expected to pass the qualifying exam(s) within three (3) years after the first enrolling in the Ph.D. program. Any delay requires prior approval of the Graduate Programs Committee. The student passes the qualifying examination provided that four members of the Ph.D. committee vote that the student has passed. Voting is by secret ballot. The Qualifying Examination is not passed until all parts have been successfully completed. In the event of an unsatisfactory performance, the committee as a whole will decide on the time and content of an appropriate reexamination. After a second unsatisfactory performance, the student's doctoral program will be terminated, subject to appeal to the department Graduate Programs Committee. An informal hearing on the issues will be called; attendees are the members in the Graduate Programs Committee and the dissertation committee. The graduate director presides over the hearing. More details on Qualifying Exam procedures can be found in Attachment 3.

### (3.8) Residency Requirement

Doctoral students must satisfy University requirement of "one year in residence in full-time graduate study" while enrolled in the doctoral program.

### (3.9) Language Requirements

Each Ph.D. student must demonstrate a reading knowledge in either French, German, Russian or another foreign language approved by the Graduate Programs Committee. Upon the approval of the Graduate Programs Committee, a Ph.D. student may demonstrate the proficiency of a computer language to replace this foreign language requirement.

### (3.10) Minimum Number of Courses

As a condition of satisfactory progress, all Ph.D. students receiving full time financial support from the Department of Mathematics are expected to take three courses each semester (exclusive of seminar courses) that are either part of program requirements or represent approved or mandated prerequisite courses. More advanced Ph.D. students may, with approval, use research hours (Math 797) in place of one or more of these courses, when appropriate. In addition, they are expected to satisfactorily complete with a grade of B or better at least two graduate mathematics courses in the Department of Mathematics each semester toward fulfillment of their program requirements. Independent Study cannot be used to fulfill this requirement during the first year the student has been admitted into the Ph.D. program. Courses taken for audit do not count toward maintenance of satisfactory progress or these minimum enrollments.

### (3.11) Duration of Financial Support

Students making satisfactory progress in the Ph.D. program are expected to complete their degree within five years of full-time study. A Ph.D. student will be supported by the department of mathematics for a maximum of five (5) years, assuming that the student remains in good standing and is making satisfactory progress. Any extension must be approved by the Graduate Programs Committee.

### (3.12) Grade Point Average

To be awarded the Ph.D. degree, students must have a GPA of at least 3.0 in course work presented for the degree, and separately have a GPA of at least 3.0 in mathematics course work presented for the degree. This does not include credit hours for research (Math 797) or Graduate Seminar (Math 696).

## Attachment 1: Academic Advising Guidelines

- Oversee the advisee's plan of study to make sure that the advisee is making progress towards the fulfillment of the degree requirement.
- Approve and sign the course registration for the advisee, and inform the graduate director or the associate graduate director of the registration information by the end of the second week of the semester.
- Remind the advisee of the academic deadlines.
- Advise the advisee to formulate the plan of study within the first year in the program in consultation with the advisor and the graduate directors. The plan of study must be approved by the advisor and the graduate director.
- A student receiving full time financial support from the department of mathematics must take at least two regular graduate courses from the department of mathematics, unless the student has fulfilled the mathematical course requirement in the program. These two regular graduate courses should exclude the teaching seminar, graduate seminar, and teaching practicum. See sections (2.5.1) and (3.9).
- As a general requirement, students in the M.S. program should include no more than 12 hours of graduate course work outside the department of mathematics in their plan of study.
- In order to be applied to fulfill graduate degree requirements in mathematics, courses taken in another department must have reasonable mathematical content and, at a minimum, be applicable to the corresponding graduate degree program in that department. For the M.S. degree, no more than half of such courses taken may be at the 400-level, and they must be approved by the advisor.
- A student who is admitted to the Ph.D. program must pass the entrance examination by the end of the third semester after first enrolling. Any delay in taking this examination requires a prior approval of the Graduate Programs Committee.
- A Ph.D. student is expected to pass the qualifying examination within three (3) years after first enrolling in the Ph.D. program. Any delay requires a prior approval of the Graduate Programs Committee. Therefore, a Ph.D. student should be advised to plan ahead for the qualifying exams. The major and minor areas for the qualifying exam must be chosen from the two major groups A and B below so that at least one subject from each of the two groups is selected.
- A Ph.D. student must take at least two courses (6 hours of credit) at the 700 level in each of the two minor areas, and at least four courses (12 hours of credit) at the 700 level in the major area. With the prior approval of the Graduate Programs Committee, a total of at most 3 hours of course work in each of the two minor areas may be at 500-600 level.
- Using an independent study to fulfill any of the major or minor course requirement must be approved by the Graduate Programs Committee.
- As a general rule on the duration of financial support, a Ph.D. student will receive at most five (5) years of financial support while enrolling in the Ph.D. program. An M.S. student will receive at most two (2) years of financial support. Any extension of financial support requires approval of the Graduate Programs Committee.
- Surveys in Mathematics may be taken once as an elective in the M.S. program. In order to enroll in this course the student must have satisfied any deficiencies identified by the Basic Exam.
- Consult the graduate director or the associate graduate director should you have questions concerning any of the above.
- The funding of a GTA is subject to termination if the student fails to fulfill these requirements.

A. Algebra/Number Theory/Discrete Mathematics/Set Theory.

B. Analysis/Differential Equations/Numerical Analysis/Topology.

## Attachment 2: Examinations

Information on the Ph.D. qualifying exam can be found in Sections (3.6) and (3.7).

**1. The M.S. Basic Exam**

Information on the Basic Exam: The Basic Exam is given to M.S. students prior to beginning their program in order to assess their background in introductory real analysis and linear algebra.

(a) MS Basic Exam Format: Each exam consists of a 2-hour written exam, a take-home exam, and an oral discussion on the take-home part.

(b) MS Basic Exam Schedule: The exam will be given on the Friday before the start of the Fall semester. The take-home part should be turned in by noon of the Tuesday of the first week of the semester. The oral discussion is optional and at the discretion of the graders in case more information is needed. The oral discussion will be scheduled on the Thursday of the first week of the semester.

(c) Content of the MS Basic Exam

(i) Topics to be covered in Advanced Calculus: Elementary properties of Open/closed/compact/connected sets in R^n. Numerical sequences and series. Limits, Cauchy sequences, convergence. Continuity. Continuity and compactness/connectedness. Uniform continuity. Sequences and series of functions; uniform convergence. Calculus of real-valued functions: Differentiation, mean value theorems, Taylor's theorem. Definition and existence of the Riemann integral. Fundamental Theorem of Calculus. Integration and differentiation of series/sequences of functions.

(ii) Topics to be covered in Linear Algebra: Vector spaces, linear independence, basis, dimension, linear transformation, and matrix representations, rank, range space, null space, eigenvalues and eigenvectors, diagonalizations, canonical forms, inner product spaces, othogonal basis, symmetric and hermitian matrices and properties.

(d) Grading of the MS Basic Exam: The examination committees will send the graduate program committee their course recommendations within a 7 day period after the written exam is conducted. These may include advanced calculus Math 451, or real analysis Math 551, and/or linear algebra Math 343, Math 441, Math 543. The recommendation will be based on the student’s background, and performance on the exam.

Textbooks:

__Advanced Calculus__

Elementary Analysis: The Theory of Calculus, by Kenneth Ross (used for Math 451)

Principles of Mathematical Analysis, Rudin (a standard advanced calculus text)

__Linear Algebra__

Elementary Linear Algebra, Kolman (used for undergraduate linear algebra)

Introduction to Linear Algebra, Strang (used for applied linear algebra)

Linear Algebra, Hoffmann & Kunze (used)

**2. The M.S. Advanced Exam/Ph.D. Entrance Exam**

Two times each year, in the late spring and early fall, the Department administers four subject area exams over a one-week period. The four subjects are real analysis, algebra, topology and differential equations. Each exam is a three hour written exam. These subject area exams together constitute the M.S. Advanced Exam and the Ph.D. Entrance Exam. Each subject area exam taken by the student is graded, in terms of decreasing performance, as either Ph.D. level pass, M.S. level pass, or fail. For a student in the M.S. program, the M.S. Advanced Exam is passed by achieving at least an M.S. level pass in two subject areas. For a student in the Ph.D. program, the Ph.D. Entrance Exam is passed by achieving a Ph.D. level pass in two subject areas.

**3. Policies and time limits for the M.S. Advanced Exam/Ph.D. Entrance Exam**

(a) A student may attempt any set of subject area exams each time they are administered (i.e. one subject area exam or up to four).

(b) A student may attempt a given subject area exam no more than three times. This is a cumulative overall limit that applies while the student is in either degree program. Thus, for instance, if a student attempts a subject area exam twice while in the M.S. program, they may attempt that subject area exam at most once more if they subsequently enter the Ph.D. program.

(c) Passing the M.S. Advanced exam is a graduation requirement in Option A and Option C of the M.S. program. Students have up to three academic years in which to pass from when they first enroll in the M.S. program. Specifically, a student first enrolling in the M.S. program in Year N must pass the M.S. Advanced exam by the end of the Spring semester in Year N+3.

(d) Students entering the Ph.D. program must pass the Ph.D. Entrance Exam within two full academic years from when they first enroll. Specifically, a student first enrolling in the Ph.D. program in Year N must pass the Ph.D. Entrance Exam by the end of the Spring semester in Year N+2.

(e) Ph.D. students that do not pass the Ph.D. Entrance Exam within the allowed period will be dropped from the Ph.D. program. In this event they may be eligible for an M.S. degree based on their course work and exam outcomes, or continue work toward an M.S. degree.

(f) The awarding of a teaching assistantship carries with it the expectation of higher standards. In this regard, a GTA in the Ph.D. program is expected to require no more than two attempts to receive a Ph.D. pass on any given subject area exam attempted. If they fail to achieve this standard, GTA support may be withdrawn. Also note the expectations in (2.5.6) that apply to students continuing with GTA support from the M.S. to the Ph.D. program.

(h) Part-time students are expected to comply with the deadlines above. Extensions of the time in which to pass can be considered by the Graduate Committee based on the particular circumstances involved, including the overall number of credits taken by the student.

**4. Exam Schedule of M.S. Advanced Exam/Ph.D. Entrance Exam**

(a) Second week of classes in Fall.

(b) Two weeks before the final exam week of the Spring term.

(c) Students must notify the graduate director of their intention in writing to take the exams a month before the exams are given, specifying the subject area(s) to be taken.

(d) The one month advanced notification stated above may be waived by the graduate director.

**5. Examining Committees**

(a) Each subject area exam has an examining committee consisting of two designated faculty members and the graduate director as a member ex officio.

(b) Members will serve on the committee for a two year period, with renewal possible.

(c) Members (including replacements for absent members) of each committee will be appointed by the Graduate Director in consultation with the Graduate Committee.

(d) Each examining committee acts independently from each other examining committee in preparing and grading its exam.

(e) Grades are assigned to exam papers based on the judgment of the examiners and not on particular percentages.

(f) Instructors who regularly teach the related subject should be expected to serve on an examining committee in the field if asked.

**6. Content of Exams**

(a) Lists of topics for each subject area exam are to be made up by appropriate faculty (committees) and posted online. The lists are effective for a 5-year period, and will be reviewed and revised (if necessary) every 5 years. Topics are to be standard.

(b) Faculty teaching the related courses are expected to cover a reasonable amount of the material listed in (6a). However, the exams are topic oriented, not course oriented and students should not assume that a given course sequence will cover exam content in its entirety.

(c) Samples of previous exams will be made available to students.

(d) Students are allowed to take the exams without taking the related courses at WVU.

**7. Notification**

(a) A memo from the graduate director announcing the time, place, and the examiners is to be circulated to all faculty and graduate students at the beginning of the Fall semester and at the end of March in the Spring semester.

(b) Each examining committee is to report to the graduate director the outcome of each exam for each student within 5 (five) working days upon the completion of the exams.

(c) Written notification from the graduate director concerning the decision of the examination committee is to be sent to the student within 10 (Ten) working days upon the completion of all the exams.

**8. Appeals and Exceptions**

Students may speak with examiners, as available and agreeable, concerning their performance on a given subject area exam, and may review their answers. The exams themselves remain in the custody of the department and are not returned to the students. Since the grade on an exam is a collective decision of the examiners, students will normally not see the scores assigned by individual examiners. A student may appeal their grade to the Graduate Committee. However if the examiners concur in the assigned grade, the Graduate Committee will usually defer to the examination committee’s judgment unless there is strong evidence that the grade given is inconsistent with usual practice or is based on a mistake or a misreading of the solutions.

Exceptions to the policies in (3) above may be considered by the Graduate Committee on rare occasions when warranted. Personal circumstances, including medical problems, accidents, hardships, etc. may play a role. However the general outlook of the Graduate Committee is that the allowed three attempts and the time period provided in which to pass the respective exams already build in substantial opportunity to overcome most individual difficulties. In considering exceptions, the Graduate Committee will review not just the reasons for the exception, but also the student’s academic and exam performance and their potential for success. Normally, to consider an extension of time to pass the Ph.D. Entrance Exam, the student will be expected to have obtained at least a Ph.D. pass in one subject area and an M.S. pass in another; to have taken advantage of prior opportunities to attempt the exams; to have a strong academic record; and to have made reasonable progress in identifying a research area and thesis supervisor.

## Attachment 3: Qualifying Examination Procedures

This document discusses qualifying examination procedures which apply to the major area written and oral examinations, the prospectus presentation, and voting. Note that in some cases a minor area exam may also be required, as discussed previously in the Graduate Handbook.

**1)Scope and preparation of the major area written examination.**

The purpose of the qualifying examination is to demonstrate that the student has acquired the background necessary to proceed on to research for the dissertation. The major area written examination includes the course work in the major area but need not be restricted to this material, and may also include general mathematics background knowledge and other areas specific to the dissertation research as identified to the student by the thesis supervisor.

The examination should be primarily prepared by the thesis supervisor, preferably in close collaboration with at least one other member of the committee. The examination should be distributed to the committee members for review and comment no later than one week before the examination.

**2)The major area oral examination**

The oral exam, covering the same material as the major area written examination, will be scheduled no later than one week after the written examination. It may be held contemporaneously with the prospectus presentation. Each member of the committee will be invited to question the student. Questions may concern general topics from the major area or involve examination problems. Members of the graduate faculty, but not students, may attend the oral examination in an observing capacity.

**3)Presentation of the written prospectus**

The student will provide the committee, no later than one week before the presentation, with a copy of the written prospectus. The prospectus should describe the proposed subject of the thesis with background information, along with preliminary results and proposed approaches, as available. The presentation should last approximately 30 minutes, but may be longer at the discretion of the committee. The presentation will be open to the department, and the audience may ask questions. Subsequent to the presentation, the committee will question the student, with only members of the graduate faculty allowed to be present in an observing capacity. Acceptance of the prospectus by the committee as part of the qualifying examination represents the judgment of the committee that the student is prepared to begin dissertation research and has identified potential avenues of inquiry. The acceptance of the final dissertation is to be considered an independent judgment of the committee members.

**4)Voting**

After any part of the qualifying examination (including the written and oral examinations on the major area) the committee may meet and decide not to proceed further if the student’s performance has not been satisfactory to that point. In such a case, the vote will be recorded on the signature page of the qualifying examination form and the decision outlined in the space provided, along with the committee’s decision on the time and content of an appropriate reexamination, if applicable.

Given that the qualifying examination proceeds to the prospectus presentation, the committee will meet alone after the questioning following the prospectus presentation to determine whether the student has passed the qualifying exam. Each member will vote pass or fail and the vote will be recorded on the signature page of the qualifying examination form. The student will be deemed to have passed the qualifying examination if all members, or all members but one, vote pass. If the overall vote is to pass the student, the committee may add minor conditions to be fulfilled by the student, as written on the signature page. In this case, the passing vote is not considered final, and the student is not considered to have passed the qualifying examination, until the conditions have been met and accepted via a signed memo from the committee

## Attachment 4: Graduate Seminar

The graduate seminar has the following goals:

- Provide all graduate students with a working knowledge of professional tools for research, technical typesetting, presentation, and computation.

(For example, using Math Reviews & Science Citation Index, internet resources, TeX-based typesetting/word processing, Powerpoint (or equivalent) presentation software, Web publishing, MATLAB for computation and graphics, guidelines for mathematical/technical writing) - Develop the ability in doctoral students to read mathematical literature in journals and books.
- Provide to doctoral students supervised experience in assembling research background on a topic or subject and developing and delivering a presentation.
- Expose graduate students to a broad range of subject areas within mathematics as reflected by the interests of their colleagues and faculty members.

To realize these goals, the following requirements will apply:

- Doctoral students will register for one credit of graduate seminar each semester they are in residence. Masters level students will register for one credit of graduate seminar when taking the professional tools seminar (see 2 below)
- All graduate students must take the one semester graduate seminar in professional tools, offered once yearly. (This will be for one credit hour, meeting once weekly.) Students who have already developed expertise in using one or more of the tools involved may, at the option of the instructor, arrange for an appropriate demonstration of that expertise in lieu of attendance at the corresponding lectures.
- Under the supervision of their dissertation supervisor (or another faculty member if the dissertation advisor has not yet been appointed) each doctoral student will be responsible for delivering three one-period talks during their program of study. One talk will be an expository talk in an area outside their main research interest (which could, for instance be based on material in a chapter of a book, or an article in a general interest journal such as the Monthly or the Intelligencer). One talk will be a general expository talk on basic results within their research area. (Survey papers may provide a good base for assembling the information for this talk.) One talk will be an advanced research talk in the area of their dissertation, which may contain their own work. (This talk may serve as the research prospectus presentation portion of the Ph.D. qualifying examination.) If any talk is not considered satisfactory by the supervisor, an additional presentation must be made. These talks may, if appropriate, be part of an ongoing seminar within the department. A student may substitute a presentation (of at least 20 minutes) at a conference for one of the required talks.
- Doctoral students will be expected to attend 10 mathematical presentations during the academic year. These may include visiting colloquia, faculty presentations, and seminar talks, but at least half should be presentations by other doctoral students. A record of attendance will be maintained for this purpose. (The number of talks required is reduced to 5 during the year that a student takes the professional tools seminar.)
- Special cases that require exceptions to the above, or questions of interpretation or procedure, will be resolved by the Graduate Director, consulting with the Graduate Committee as suitable. Any such decisions may be appealed to the Graduate Committee for final adjudication.

## Attachment 5: Special Admission to the Ph.D. Program

There are three routes of entry into the Ph.D. program.

Students who have already obtained a Masters degree in mathematics or its equivalent will normally be admitted as regular Ph.D. students and will need to meet the Ph.D. basic coursework requirements described in Section 3.5 of the Handbook. Students with deficiencies in their background can be admitted provisionally, with deficiencies made up in the first year of study.

Students applying to the graduate program with only a Bachelor’s degree will normally be placed in the M.S. program. However, some students have an unusually strong undergraduate background that includes many of the traditional first-year graduate courses in mathematics. Typically, but not exclusively, these are students from international degree programs in mathematics in which major area course work is much more extensive than U.S. programs. Such students can be admitted provisionally into the Ph.D. program, with the provision that they pass the Basic Exam and their program of study include 12-18 credit hours of approved coursework, at the M.S. level or above, beyond the basic Ph.D. coursework requirements of Section 3.5.

Finally, there is a special provision whereby well-prepared students enrolled into the M.S. program can, with the permission of the Graduate Programs Committee, transfer into the Ph.D. program after their first year of study. Such students must pass the Ph.D. entrance exam and complete 18 hours of regular M.S. coursework during their first year of study. The GPC will consider the student’s background and performance in deciding whether to grant permission to transfer to the Ph.D. program. After this transfer, the student will be admitted as a Provisional Ph.D. student, with any additional program requirements determined by the GPC. Students potentially interested in this provision should ideally consult with their advisor or the Graduate Director prior to their initial enrollment.