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PhD Degree

The following outline of requirements constitutes the basic structure of the PhD program but, depending on the student's background, etc., variations may be worked out with the Program Head. Refer to the PhD Timeline below as your general guideline and refer to the Student Handbook for the complete description of requirements and policies.

A. Overall Requirements for the PhD

  1. Completion of forty-seven (48) units of graduate-level courses, exclusive of dissertation, approved by the Program and including:

    1. Twenty (21) units from the core courses: Math 527 a, b; Math 575 a, b; Math 583 a, b; Math 586 a, b.
    2. Nine (9) or more units of advanced 500-level Mathematics courses
    3. Nine (9) or more units from departments other than Mathematics.
    4. Nine (9) or more units of elective course work either in mathematics and/or interdisciplinary topics.

    Note: If the student has taken courses equivalent to some of the core courses, the core requirements may be revised. The total requirement of 47 units will not, however, be reduced. The Graduate College requires that 50% of the units must be in courses for which ABC grades are given. It is also a Graduate College requirement that all courses must be at the 500 level or above, with the exception that up to six units of 400-level courses may be taken outside the major area (which, for the Program means outside of the Department of Mathematics) with prior approval from the Program. The Program's requirement of a minimum of 9 units of course work outside the Mathematics Department is strictly adhered to.

    Eighteen (18) dissertation units (920) are required for graduation. Students may register for more, but the maximum number of 920 units allowed per semester is nine (9).

    Registration of Units: Every student will be required to register for a minimum of 9 units until all course requirements are met. After all course requirements are met, students are required to register for 6 or more units (unless other restrictions apply). A student should register for dissertation units after passing the Comprehensive Examination and when he/she is ready to start (or have started) their dissertation research. The semester during which preparation for the Comprehensive Examination takes place would be an appropriate time to register for 900 level research units. Students should always discuss registration for 900 and 920 units with their advisor.

  2. Successful completion of the Qualifying Examination at the PhD level.
  3. Successful completion of the Oral Comprehensive Examination
  4. Satisfactory completion of the Research Proposal Writing requirement
  5. There is no foreign language requirement.

B. The Qualifying Examination

This important exam is described in detail in Section VI of the Student Handbook.

C. The PhD Comprehensive Examination

At an appropriate point during the course of the student's second or third year, he/she should be involved in serious research (through, for example, an independent study project) with a faculty member who in all likelihood will become their dissertation advisor. When it becomes clear that the student is ready to take the PhD Comprehensive Examination, a research paper (or an appropriate article, or articles) is identified for the student to study and discuss with his/her advisor. The student, on consultation with the advisor, will then select his/her Comprehensive Examination committee. The committee must be made up of a minimum of four tenured or tenure-track UA faculty affiliated with the Program. The committee chair is usually the student's advisor. In some cases it may be appropriate to have a committee member (with tenure, tenure-track, or equivalent status) who is not affiliated with the Program on the committee. At the request of the committee chair, the Program can obtain a waiver to enable this to happen. This is usually straightforward but should not be assumed to be automatic and must be formally approved by the Dean of the Graduate College.

Approximately six weeks before the intended exam date, the student needs to work with the Applied Mathematics staff to process the required Graduate College paperwork in order to schedule the exam. (The Comprehensive Exam should be scheduled within two years of passing the written Qualifying Exam). The Comprehensive Exam must be conducted on the UA campus.

The committee will administer the exam which consists of the following components:

  1. A written research report on the topics chosen for the Oral Examination (described in more detail below). The report is to be written following the standard journal article format used for the second-semester term paper and the third-semester project. A copy of the report will be given to each member of the student's Comprehensive Exam committee at least two weeks before the Oral exam. If the committee does not find the report satisfactory, the Oral Exam will not take place.
  2. A professional resume and a link to the student's web page will be given to the committee members at the same time as the written report. If the committee does not find these requirements satisfactory, the Oral Exam will not take place.
  3. A "road map" describing (i) the student's plan of study after the comprehensive exam, i.e. a timetable for the proposed research and graduation, future course-work, etc.; and (ii) plans for professional development, e.g. meetings and workshops to be attended, teaching and outreach projects, internships, etc.
  4. An oral exam following the standard rules of the Graduate College.

    5. Note: Copies of the Comprehensive Exam research report, the resume, the road map, and the web page address must also be deposited with the Applied Mathematics office at least two weeks before the date of the exam. If this is not done, the exam will not be allowed to proceed.

    The comprehensive examination is intended to test the student's fundamental knowledge in the fields of their major and minor subjects of study and to determine readiness to undertake a PhD quality research project. This assessment is based upon the student's ability to describe and discuss the chosen research paper and the affiliated areas of science and mathematics. However, at the time of the exam, a specific dissertation topic does not need to have been identified.

    The exam itself usually begins with a short presentation (45 minutes at most) in which the student presents a summary of research paper(s). The student is then questioned about the paper, its scientific setting (background, importance, future work), and other related mathematical and scientific material. Students are encouraged to discuss the research paper (and the research area in general) with members of their committee before the exam.

    Note: If the student has declared a Minor outside of Applied Mathematics, the Comprehensive Exam Committee must include one faculty member representing the Minor department.

    The Comprehensive Exam is conducted as a closed exam and no portion of it is open to the public.

D. Retake of the Comprehensive Examination

Occasionally, a student fails the Comprehensive Examination. This indicates that the committee has serious reservations about the student's ability to perform quality research. In this eventuality, the student may be allowed to retake the exam within a reasonable time frame, but the decision to do so must be made in consultation with the student's advisor and the Program Head. In the rare event of the student failing the exam on the second attempt, he or she will not be allowed to continue in the Program.

E. Third Year Oral Review

If a student has not passed their comprehensive exam by the end of their third year, they will be subject to a constructive review to help them identify a dissertation topic and make any necessary adjustments to their program of study. This review will be undertaken at the end of the sixth semester by an ad hoc committee consisting of the Program Head, one faculty chosen by the student, and a third faculty member at the discretion of the Head. The student will present some of her/his previous research, an extensive study plan (clearly showing how they intend to complete their doctoral requirements), a resume, a web page, and a plan of study that, in addition to plans for future coursework, explains how you intend to find a dissertation topic and advisor.

This review will take place every semester until the Oral Exam has been scheduled.

If the student has already been scheduled (or completed) their comprehensive exam by the end of the third year, the review will take the form of a brief meeting with the Program Head and the student advisor where she/he will present a plan of study (including a description of the Comprehensive Exam), a resume and a web page.

F. PhD Plan of Study and Advancement to Candidacy

Students in the Program in Applied Mathematics are required to file the PhD Plan of Study at the same time the Comprehensive Exam is scheduled. After successful completion of the Comprehensive Exam, the Committee Appointment form (formerly known as the Advancement to Candidacy form) must be filed. Please see the Program Coordinator for procedures.

G. Dissertation

The most important requirement for the PhD degree is the completion of a dissertation that contains original contributions by the candidate to the solution of a mathematical problem in a scientific discipline or to the development of mathematical methods for classes of such problems. The quality and scope of such contributions should be of a sufficiently high standard as to warrant publication in a reputable journal. Dissertations are housed indefinitely in the University Library and in the Program office and are available to the public. Requirements for style and format are set forth by the Graduate College and the exact specifications can be found on the Graduate College website at http://grad.arizona.edu/academics/degree-certification/diss-theses/manuals. A draft of the dissertation must be given to each committee member at least two weeks prior to the Final Oral Defense Examination.

H. Outside Review of Dissertation

The Program in Applied Mathematics requires an evaluation of the dissertation by a reviewer outside of the University of Arizona who is an expert in the field of the research. The choice of external reviewer is made by the student's advisor. A draft of the dissertation must be sent to the external reviewer at least two weeks prior to the Final Oral Defense Examination. The reviewer will submit a report to be made available to the members of the Final Oral Defense Examination committee at the time of the exam.

I. Research Proposal Writing Requirement

To complete their doctoral work, students are required to write a research proposal following the guidelines of a National Science Foundation post-doctoral fellowship, or according to the format of some other agency (such as DOE or NIH) with prior agreement from the Program Head. This proposal will be reviewed by Final Defense committee members. Other types of proposals may also be permitted subject to approval by the Program Head. The proposal must be submitted to the student's committee, and the Applied Mathematics office, at least one week prior to the Final Oral Defense Examination. If the committee does not find the proposal satisfactory, the student will not be allowed to present his or her Final Dissertation Defense.

Although many students leave this requirement to the last minute, the best time to write this proposal is at the time they are seeking post-graduation jobs, i.e., many months before the final defense! Writing the proposal at this stage is a valuable exercise in developing a compelling personal statement and it thus becomes a valuable professional instrument rather than a last-minute chore.

J. PhD Final Dissertation Defense Examination

The Final Dissertation Defense Examination begins with a presentation by the candidate of some significant aspects of the thesis research. The presentation portion of the exam is open to the public. Members of the student's committee then question the candidate in depth in order to satisfy themselves as to its originality and significance.

K. PhD Recommended Timeline

Year 1
  • Complete Core Courses
  • Prepare for Qualifying Exam
  • Identify Research Areas of Interest
Year 2
Year 3
Years 4&5
  • Identify employment opportunities (ongoing)
  • Complete doctoral research
  • Select committee members for Dissertation Defense Examination
  • File Committee Appointment Form (formerly known as Advancement to Candidacy) at least 6 months prior to Final Defense
  • Select outside dissertation reviewer
  • File Announcement of Dissertation Defense Examination Form
  • Pass Dissertation Defense Examination
  • Electronically submit final copy of dissertation to Graduate Degree Certification office according to the specified deadline for the semester in which the defense takes place.

    • Note: All Graduate College forms referred to above must be processed through the Applied Mathematics office for the Program Head's review and signature. The office staff will then forward the forms to the Graduate College Degree Certification Office.