Ph.D. Degree Requirements & Operating Guidelines

INTRODUCTION

These guidelines are intended to help familiarize graduate students with the policies governing the graduate program leading to the degrees of Doctor of Philosophy (Ph.D.) in Applied Mathematics. This material supplements the Graduate School regulations and degree requirements. Students should be familiar with these regulations and requirements.

THE DOCTOR OF PHILOSOPHY PROGRAM

The Doctor of Philosophy (Ph.D.) Degree in Applied Mathematics is primarily a research degree, and is not conferred as a result of course work. The granting of the degree is based on proficiency in Applied Mathematics, and the ability to carry out an independent investigation as demonstrated by the completion of a doctoral dissertation. This dissertation must exhibit original mathematical contributions in a significant area of application. In particular, a thesis on a mathematical topic with no demonstrable application is not acceptable, nor is it appropriate to study some application using mathematical techniques which are only routinely used in that application.

REQUIREMENTS FOR THE PH.D. PROGRAM

1. COURSE REQUIREMENTS FOR THE PH.D. PROGRAM

The courses required for the Ph.D. are as follows:

• AMATH 567, 568, 569
• AMATH 584, 585, 586
• Three of AMATH 570, 571, 572, 573, 574, 575
• AMATH 507 or 515
• AMATH 506
• Two Applications Courses
• AMATH seminars (2 credits)
• AMATH 500 (journal club) or AMATH 502 (clinic)(2 credits)

It is strongly recommended that students take AMATH 567, 568, 569, and AMATH 584, 585, 586 during their first year of academic study. Students with exceptional preparation who are going on into the Ph.D. program may waive some core courses with the agreement of the Graduate Program Coordinator.

The application course requirement is satisfied with a coherent sequence of two (2) graduate (500 level) graded courses in a given field. Other alternatives may be appropriate. For example, at most three (3) credits may be in a 400 level course if the student receives a minimum grade of 3.0 in this course. The following courses offered by the Applied Mathematics Department are also designated as Applications Courses: AMATH 503, 504, 505, 521; AMATH 422, 423, 441. In any event, the courses selected in the field of application must be approved by the student's advisor and the Graduate Committee. Traditional fields of application include all branches of engineering, the physical sciences, biological sciences, computer science, economics and management science, and certain areas of medical science. Other fields may be approved by the Graduate Committee where appropriate. Mathematics and statistics are not considered to be fields of application for this purpose.

2. EXAMINATION REQUIREMENTS FOR THE PH.D. PROGRAM

Students in the Ph.D. program need to pass the following exams:

• The preliminary exam
• The general exam
• The final exam (thesis defense)

3. AREAS OF SPECIALIZATION

Every candidate for the Ph.D. Degree is expected to demonstrate proficiency in at least two distinct areas of mathematical specialization which are relevant to his/her doctoral research. This requirement is partially satisfied by passing twelve (12) graded credits in 500 level Applied Mathematics or Mathematics courses (not including courses used to satisfy M.Sc. Degree requirements). S/NS grades will not be acceptable for this twelve (12) credit requirement. Principal specialization areas include:

• Advanced numerical analysis (beyond AMATH 581, 584, 585, 586)
• Advanced ordinary differential equations / Dynamical systems (beyond AMATH 568)
• Advanced partial differential equations / Nonlinear waves (beyond AMATH 569)
• Asymptotics / Perturbation methods
• Integral equations
• Control / Optimization (beyond AMATH 507, 515)
• Advanced probability and statistics / Stochastic Processes (beyond AMATH 506)

4. CANDIDACY

Students in the Ph.D. program attain the status of Candidate for the Doctor of Philosophy degree upon

• Forming a Supervisory Committee, selecting two specialization areas, satisfying course requirements, and designing a doctoral research plan.
• Passing the General Examination.

5. RESIDENCE REQUIREMENTS

The candidate for the Ph.D. degree is required to spend at least two years in residence as a graduate student at the University of Washington, one of which is full-time residence.

6. SATISFACTORY PERFORMANCE AND PROGRESS

At all times, students need to make satisfactory progress towards finishing their degree. Satisfactory progress in course work is based on grades, which are expected to be 3.4/4.0 or better. Satisfactory progress on the examination requirements consists of passing the different exams in a timely manner. For specifics on what constitutes a timely manner for the different exams, see below.

In order to continue in the Ph.D. program, a student is expected to demonstrate satisfactory progress towards the advancement of their degree program. Departmental funding is contingent on satisfactory progress.

Note that regardless of whether a student is enrolled for full- or part-time study, the Graduate School requirement is that the Ph.D. degree be completed in less than ten (10) calendar years of graduate study. In the event that this time limit is exceeded, the student must petition the Graduate School for an extension. This petition must be supported by the Department which is required to review all the courses that the student has taken for credit to ensure that the current equivalents of these courses are not significantly different. In the event that a needed course has been substantially updated, the student is required to retake this course.

The Graduate School rules regarding satisfactory progress are outlined in the Graduate School Memorandum No. 16

EXAMINATIONS

1. THE PRELIMINARY EXAMINATION

The Preliminary Examination is offered by the Department during the first week of Winter quarter, and again the first week of Spring quarter. Entering Ph.D. students are expected to take the Preliminary Exam following the first quarter of full-time study. Full-time students should pass the Preliminary Examination in their first year of study. This general rule does not hold for students on a part-time program of six credits or less per quarter. Exception to the general rule may be granted upon request to the Applied Mathematics Faculty.

The Preliminary Examination consists of three written, two-hour exams, covering a core of undergraduate material necessary for successful completion of the Ph.D. program. These core areas are:

• Differential Equations:
  For a review of this material, see Boyce & DiPrima, Elementary Differential Equations, Seventh Edition, Wiley, 2001.

• Linear Algebra:
  For a review of this material, see Strang, Introduction to Linear Algebra, Third Edition, Wellesley-Cambridge Press, 2003.

• Advanced Calculus:
  For a review of this material, see Taylor and Mann, Advanced Calculus, Third Edition, Wiley, 1983. Many of these topics are covered at a more elementary level in any standard calculus text, such as Salas, Hille and Etgen, Calculus, 9th edition, Wiley, 2003.

Old preliminary exams are collected here.

The outcome of a student taking the preliminary exams for the first time is (i) or (ii):

(i) Pass:
A student passes the preliminary examination by passing all three area examinations at the Ph.D. level in one sitting.

(ii) Retake
If a student did not pass the preliminary exams he/she may choose to retake the preliminary exams once. If a student does not obtain a pass on the examination the first time, but receives an outstanding result on one of the three parts of the examination he/she may be excused from retaking this part. The possible outcomes of a retake are (i) or (iii), see below.

(iii) Fail:
The student is judged as not demonstrating satisfactory understanding of the examined materials on a retake of the preliminary exams. The student will not be allowed to continue in the Ph.D. degree program.

In every case, a thorough assessment is made by the Applied Mathematics Faculty. Factors other than the examinations can be taken into consideration to the benefit of the student. If there are questions regarding the written performance, an oral examination can be administered by the Graduate Program Coordinator and the examiners during the first two weeks following the written examination.

2. THE GENERAL EXAMINATION

The General Examination is administered after the student has passed the Preliminary Examination, typically before the end of the student's second year. The purpose of this examination is threefold:

1. To determine whether the student has acquired the necessary mathematical background (including two areas of specialization) as well as an applications background appropriate for his/her proposed research,

2. To determine whether the student is able to draw on this background to make progress in the proposed research,

3. To determine whether the proposed research topic and approach have the potential of leading to an acceptable dissertation.

The General Examination is an oral exam administered by the supervisory committee (see below). Its duration is approximately one and a half hours and it has the following two components:

1. A thesis proposal (open to the public) consisting of a thirty minute (maximum) presentation on the topic of the dissertation. The student will formulate a research topic which has the potential of leading to original mathematical contributions in a significant area of application. It is suggested that a brief thesis proposal outlining this work be prepared and given to all members of the Supervisory Committee prior to the General Examination.

2. The thesis proposal is followed by an oral examination (not open to the public) testing on the thesis topic area and two areas of specialization taken from the list above.

The student will receive a pass of fail depending upon their performance. The General Examination can be taken up to a maximum of two times.

It is expected that the General Examination will be attempted no later than the end of the second academic year of study and passed no later than the first quarter of the third year of full-time study. If the candidate has not attempted the General Examination within this period, the Supervisory Committee shall determine if satisfactory progress is being made. The Committee must determine if justifiable reasons for postponement exist and advise the Graduate Program Coordinator of these reasons.

3. THE FINAL EXAMINATION

The preparation of a dissertation brings the student to the Final Examination. This consists of an oral presentation of the completed research in a seminar open to the public. After the seminar, the student is expected to defend his/her contributions and results by responding to questions from the audience in a public session, and by members of the Supervisory Committee in a closed session.

If the student's Final Examination has not yet been scheduled after five years of full-time study, then the Supervisory Committee will formally meet and reconsider the candidate's ability to complete a Ph.D. dissertation. The considerations of the committee will focus on the candidate's progress, understanding, and contributions to the dissertation. The two possible conclusions of this examination are: (1) a reasonable conclusion of the dissertation can be foreseen and a specific time limit will be established, or (2) a reasonable conclusion of the dissertation is not expected and the student will be informed that satisfactory progress toward the degree is not being made.

SUPERVISORY COMMITTEE

As soon as possible, and no later than the end of the Spring quarter of the first year, a student should form a supervisory committee. Initially, the supervisory committee may consist of only a advisor. At this stage, the Supervisory Committee plays an advisory role in designing a course of study appropriate for the student's research interests, and in formulating a dissertation topic. A full Supervisory Committee should be formed no later than four months prior to the student's General Exam. All names of committee members must be submitted to Graduate School via its on-line form.

The Supervisory Committee should have a minimum of three (3) regular members plus the Graduate School Representative (see below), and will be constituted of at least two (2) faculty members from Applied Mathematics, one of whom is to be the Chairman of the Committee. If the proposed dissertation advisor is a member of the Applied Mathematics faculty, then the advisor will be the Chairman. In special cases, the dissertation advisor may be from another department, and is then also a member of the Supervisory Committee.

One additional faculty member from an appropriate area will be appointed by the Graduate School to be Graduate School Representative (GSR); faculty suitable for this position may be suggested by the student. The GSR should have some familiarity with the anticipated area of research and may not be from the Applied Mathematics faculty. Thus, the minimum number per committee is four. In the event that not all three members, exclusive of the GSR, are familiar with the proposed area of research, it is advisable to select additional faculty so that a Dissertation Reading Committee, consisting of three members, may be formed from the Supervisory Committee. Two members of the Dissertation Reading Committee must be from the Applied Mathematics faculty.

While the principal source of guidance during the process of choosing specialization areas and a research topic is the thesis advisor, it is strongly advised that the student maintain contact with all members of his/her Supervisory Committee. It is expected that the student meet with the Supervisory Committee at least once a year to discuss his/her progress until the doctoral thesis is complete.

SUGGESTED ACADEMIC WORKLOAD

A full-time student is expected to continue to register for at least fifteen (15) quarter credits. These credits may initially include course work related to the specialization areas, or other areas suggested by the Supervisory Committee. If the dissertation advisor is not on the Applied Mathematics faculty, the Doctoral Dissertation credits will be equitably distributed between AMATH 800 and the advisor's departmental 800 registration.

During the dissertation research, it is expected that the candidate will show personal dedication and incentive in accomplishing his/her stated research goals.

ANNUAL PROGRESS REPORT

Students are required to submit an Annual Progress Report to his/her Supervisory Committee by the end of the Winter quarter each year. The Annual Progress Report Web template or LaTeX template may be helpful when creating this report.

FINANCIAL ASSISTANCE

Financial support for Doctoral studies is limited to five (5) years after admission to the Ph.D. program in the Department of Applied Mathematics. In unusual circumstances support for an additional period can be granted upon approval of a petition, endorsed by the student's thesis supervisor, to the Graduate Program Coordinator.

INITIAL ADVISING APPOINTMENT

All new students should make an appointment to see a designated faculty advisor before registering for the first quarter of study. During the departmental orientation and the initial advising appointment, the student will be assisted in the selection of courses for the first quarter of study; the advising procedures for subsequent quarters will also be explained at this time. Since an integral part of the Applied Mathematics program of study is an applications field, it is suggested that each new student give some thought to identifying this field. Of course, the selected applications field may change as the student's interest evolves after exposure to different topics.

MASTER OF SCIENCE PROGRAM

Students in the Ph.D. program may obtain an M.Sc. Degree while working towards their Ph.D. degree by satisfying the requirements for the M.Sc. degree.

Time Table to Ph.D. Degree