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 as your general guideline.

A. Overall Requirements for the PhD

  1. Completion of forty-seven (47) units of graduate-level courses, exclusive of dissertation, approved by the Program and including:

    1. Twenty (20) 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. Foreign language requirement: There is no foreign language requirement.

B. The Qualifying Examination

This important exam is described in detail in Section VII 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. Approximately eight 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).

A five-member committee, usually chaired by the student's prospective advisor, 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. The report will be given to 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, a web page, and a plan of study will be given to the committee members together with the written report. If the committee does not find these requirements satisfactory, the oral exam will not take place.

  3. An oral exam following the standard rules of the Graduate College.

    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 two faculty members representing the Minor department.

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 and a web page.

This review will take place every semester until the comprehensive 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 after successful completion of the Comprehensive Exam. (Due to the interdisciplinary nature of the Program, special permission was granted by the Graduate College for this time extension). It is recommended that the Advancement to Candidacy form be submitted at the same time as the Plan of Study. 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/PDF/ETD_Diss_Manual.pdf. A draft of the dissertation must be given to each committee member prior to scheduling the Final Dissertation 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 reviewer will complete a report to be made available to the members of the Final Dissertation Defense Examination committee at the time of the exam.

I. 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.

J. PhD Recommended Timeline

Year 1
  • Complete Core Courses
  • Prepare for Qualifying Exam
  • Identify Research Areas of Interest
Year 2
  • Pass Qualifying Exam
  • File Masters Specialist Plan of Study and Masters Completion of Degree Requirments forms
  • Receive MS Degree (if desired)
  • Investigate research opportunities in one or more areas
  • Identify possible research advisors
Year 3
  • Select an advisor
  • File Application for Comprehensive Exam
  • Pass Comprehensive Exam
  • File Doctoral Plan of Study
  • Begin doctoral research
Years 4&5
  • Identify employment opportunities (ongoing)
  • File Advancement to Candidacy form
  • Complete doctoral research
  • Select committee members for Dissertation Defense Examination
  • File Announcement of Final Dissertation Defense Examination
  • Select outside disseration reviewer
  • Pass Dissertation Defense Examination
  • Electronically submit Dissertation to Graduate Degree Certification office