Handbook for Graduate Studies in Mathematics
2008
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, 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). 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 (to be added).
2. The
Master's Program
(2.0) Common Requirements
(2.0.1) Admission Requirement
For regular admission, a baccalaureate degree in mathematics or its equivalent is required. 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 real analysis at the level of Math 551, and of linear algebra at the level of Math 543. 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 unless this requirement is similarly waived in writing as a result of the Basic Exam. Entering graduate students who have not had any advanced calculus 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.
(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 696
(Graduate Seminar).
(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 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.
(2.1.4) 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.
Math 441 must be taken if equivalent has not been taken as an undergraduate.
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, Optimization.
II. Probability theory/mathematical statistics: Stat 561-562; (other approved Stat courses provided 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 level. It includes several courses specifically designated for this option. A number of the required courses are offered over the summer or in the evening, for the convenience of working teachers.
(2.4.2) Core Curriculum
Students must complete Math 455 (or 451), Math 530, Math 533, Math 535, Stat 505 and either Math 534 or Math 536 (both are recommended). Courses equivalent to these may, on occasion, be substituted with the approval of the student's Advisory Committee.
(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 consisting of four sections: algebra, geometry, analysis, 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.
(2.4.4) Other Requirements
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.
(2.5) 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.
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 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 Secondary 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 each subject area, they will be considered to have passed the Ph.D. Entrance Examination if they subsequently enroll in the Ph.D. program within 5 years.
As a necessary condition in considering the awarding or continuation of graduate assistantships in the Ph.D. program, M.S. students who proceed directly into the Ph.D. program are required to pass the M.S. Advanced Exam/Ph.D. Entrance Examination at the Ph.D. level by no later than their second attempt overall, and by no later than the second time the exam is offered after they enroll in the Ph.D. 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
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 with probationary status. Deficiencies must be completed within the first year. The entrance examination must still be passed by the end of the third semester of full time graduate work.
(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 graduate faculty outside the Department of Mathematics.
(3.4) Ph.D. Entrance Examinations
A common Ph.D. Entrance Exam/M.S. Advanced Exam structure is adopted. Students choose two areas in which to be examined from Algebra, Real Analysis, Differential Equations, and Topology. The Examination procedures are fully described in Attachment 2.
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 the examinations, or other exceptions to examination procedures, must be approved in advance by the Graduate Programs Committee.
(3.5) Course Requirements
Four courses in a major area must be taken at the700-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 at least one subject from each of the two groups is selected.
A. Algebra/Number Theory/Discrete Mathematics/Set Theory.
B. Analysis/Differential Equations/Numerical Analysis/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 requirement must be approved by the Graduate Programs Committee.
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.
Any exceptions to the above program requirements must be approved in advance by the Graduate Programs Committee.
(3.6) Ph.D. Qualifying Examinations
With the approval of the Ph.D. committee or the Ph.D. Advisory Committee of the student, the student must specify two minor areas and complete the corresponding courses (2 courses in each of the two minor areas) with a GPA at least 3.5 in the course work for each of the two minor areas. 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: One three-hour written examination over the major area, followed by an oral exam.
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 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. 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.
(3.11) Duration of Financial Support
A Ph.D. student will be supported by the department of mathematics for up to five (5) years, assuming that the student remains in good standing and is making satisfactory progress. Any extension must be approved by the graduate 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
1. Oversee the advisee's plan of study to make sure that the advisee is making progress towards the fulfillment of the degree requirement.
2. 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.
3. Remind the advisee of the academic deadlines.
4. 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.
5. 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).
6. 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.
7. 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.
8. 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.
9. A Ph.D. student is expected to passed 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. Algebra/Number Theory/Discrete Mathematics/Set Theory.
B. Analysis/Differential Equations/Numerical Analysis/Topology.
10. 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.
11. Using an independent study to fulfill any of the major or minor course requirement must be approved by the Graduate Programs Committee.
12. 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.
13. 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.
14. Consult the graduate director or the associate graduate director should you have questions concerning any of the above.
15. The funding of a GTA is subject to termination if the student fails to
fulfill these requirements.
Information on the Ph.D. qualifying exam can be found in Sections (3.6) and (3.7).
1. 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.
(d) The one month advanced notification stated above may be waived by the graduate director.
(e) A student who is admitted to the Ph.D. program must take the entrance
exam examination by the end of the third semester after first enrolling. Any
delay in taking the examinations must be approved by the Graduate Studies
Committee.
2. Examining Committees
(a) Each exam has an examining committee. The graduate director is an ex officio member unless he is an examiner.
(b) There will be two members other than the graduate director for each committee, who 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 appropriate faculty and/or the graduate studies committee.
(d) Each examining committee acts independently from each other examining committee in preparing its exam.
(e) Instructors who regularly teach the related subject should be expected
to serve on an examining committee in the field.
3. Content of Exams
(a) Lists of topics for each exam are to be made up by appropriate faculty (committees). The lists are effective for a 5-year period, and will be reviewed and revised (if necessary) every 5 years. Specific reference texts covering the topics are to be listed and placed on reserve in the math library. The lists of topics are also placed in the math library. Topics are to be standard.
(b) Faculty teaching the related courses are expected to cover a reasonable amount of the material listed in (a). (The exams are topic oriented, not course oriented.)
(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.
4. Exam Format
(a) Each exam consists of a written exam and a possible and separate oral exam.
(b) The written exam is to be taken in a 2 hour period.
(c) If the outcome of the written exam is not conclusive, the oral exam may be given. The oral exam will be given at the discretion of the committee with the approval of the graduate director.
(d) When oral exams are needed, they should be given within five working days upon the completion of all the written exams.
(e) Each oral exam must be attended by both members of the appropriate committee, and may be attended by the graduate director.
Information on the Basic Exam
(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 each Semester (Fall and Spring)
starts whenever there are MS students enrolled in the program. The take-home
part should be turned in by noon of the Tuesday of the first week of the
semester. 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
6. The M.S. Advanced Exam/Ph.D. Entrance Exam
(a) Each student must pass two two-hour exams from two subjects of the following: Algebra, Real Analysis, Differential Equations, and Topology.
(b) There are three grades for each subject: Ph.D. Pass, M.S. Pass, and Fail.
(c) The student must receive at least two M.S. Passes from the 3 person examination committee of one subject to warrant an M.S. Pass for that subject.
(d) The student must receive at least two Ph.D. Passes and one M.S. Pass from the 3 person examination committee of one subject to warrant a Ph.D. Pass for that subject.
(e) To be awarded an M.S. degree in Option A and Option C, the student must obtain an M.S. pass in each of the two subjects.
(g) To be admitted into the Ph.D. program, the applicant must obtain a Ph.D. pass in each of the two subjects.
7. Retake of The Exams
(a) Each student can only take a given exam two times during their graduate study in mathematics.
(b) A student who fails the exam on a subject must retake the exam on the same two subjects when the exam is administered the next time.
8. 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 (including the oral exam).
(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.
(d) A memo from the graduate director is to be sent to all members of the faculty and to all advisors listing all students who have passed the exams.
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:
1) 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)
2) Develop the ability in doctoral students to read mathematical literature in journals and books.
3) Provide to doctoral students supervised experience in assembling research background on a topic or subject and developing and delivering a presentation.
4) 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:
1) 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)
2) 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.
3) 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.
4) 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.)
5) 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.