Ph. D. Requirements

In addition to the information on this page, the Mathematics Graduate Program Handbook provides comprehensive policies regarding satisfactory progress towards satisfying the Ph.D. requirements.  Students should refer to the graduate catalog, available on the Graduate College website, for more details on graduate college requirements for PhD candidates.

Graduate Students are expected to follow the policies and procedures for both the UA Graduate College and for the Department of Mathematics. Policies are updated frequently and it is the student’s responsibility to comply with current policies. Graduate College policies can be viewed on-line at https://grad.arizona.edu/new-and-current-students; university policies can be found at https://catalog.arizona.edu/.

Course Requirements

Students are required to complete 36 units of graduate credit in the major and 12 units in a supporting minor, which may be declared in Mathematics, although outside minors are encouraged.  Units may not be counted towards both the major and minor. In addition, 18 units of dissertation (Math 920) must be completed.  Students cannot register for 920 until they have passed their oral comprehensive exam.  This rule can be waived by the Associate Head for the Graduate Program in exceptional circumstances.

Traditional Core Courses

For each of the traditional core courses, Algebra, Real Analysis, and Geometry–Topology, students must either take the course and receive a grade of B or better in both semesters or earn a high pass on the corresponding written qualifying exam.  The material in these courses is essential knowledge for all mathematicians, and it is assumed in all further advanced course work in the department.

Further Mathematics Coursework

Two year-long Mathematics course sequences that are not dual-numbered and are not part of the required core of algebra, real analysis, and geometry-topology are required.  Students should seek advice on appropriate courses from their advisor (if they have one already), faculty in the area in which they plan to do research or the director of the graduate program.  For many students Complex Analysis is a good choice for one of these sequences.

Outside of Department Courses or Internship

Students who entered prior to Fall 2022, unless they elect to adopt the new Community and Professional Development requirement (see below), must take two courses (6 units) outside the mathematics department. These may be applied toward the minor, if appropriate. The spirit of this requirement is that students should learn to communicate with and appreciate the perspectives of users and producers of mathematics in other disciplines.  The requirement may either be satisfied by either taking two courses (6 units) outside the Mathematics Department or by doing an internship.

Courses which fulfill this requirement should (a) have significant content in mathematics or mathematics education; and (b) not be substantially equivalent to courses in the mathematics department. We maintain a list of a priori acceptable courses. For courses not on this list students should ask the Associate Head for the Graduate Program if they would fulfill the requirement. A priori unacceptable courses include those cross listed in mathematics or taught by a mathematics faculty member. An exception is that courses offered by the math department in mathematics education may be used to satisfy the outside course requirement by students whose dissertation is not in mathematics education.

An internship with a company or government lab may satisfy this requirement if it involves mathematics in a significant way. Students should consult with the Associate Head for the Graduate Program before such an internship to see if it would satisfy the requirement. The student's internship supervisor may be asked to provide documentation of the amount and mathematical nature of the work involved in the internship.

Students who entered the program Fall 2022 or later are encouraged but not required to take out-of-department courses or internships. If taken, these may be applied towards their course units and/or Professional Development requirements (see G2. Professional Development Requirements).

Minor

The University requires that PhD students declare a minor. PhD students in mathematics may declare their minor in mathematics or in a supporting discipline. Requirements for the minor are determined by the minor department. Up to 12 units of course work may be in the minor. Students contemplating a minor other than Mathematics should consult with the Director of Graduate Studies and their thesis advisor regarding the suitability of their plans. Typical minors include Statistics, Computer Science, Education, Physics, and Applied Mathematics, though others may be considered.

Program of study

The Plan of Study should be completed after the student has passed their qualifying exams. Each student must present a coherent collection of courses in which the work outside of Mathematics is related to part of the studies in Mathematics. There are many such possibilities, including: algebra, and computer science or discrete methods in operations research; probability, and statistics or reliability/quality control; numerical mathematics, and computer science or computational science; mathematical foundations and history, and education; analysis, and physics or optics; etc.

Graduate Faculty Advisor

All students in the Mathematics Ph.D. program are required to have a Graduate Faculty Advisor (also called Major Professor) in order to maintain Satisfactory Progress. First year students should select a faculty advisor and have it approved by the Director of Graduate Studies (DGS). If the student cannot find an advisor, the DGS will work with the student to appoint one. Students may change their advisor in consultation with the DGS at any time. Once an advisor is chosen, the student will inform the Graduate Coordinator. By the time a student starts to prepare for their Comprehensive Exam, it is expected that the chair of Comprehensive Committee will take over the role of Graduate Faculty Advisor will  It is typical for the Dissertation Advisor to serve as Graduate Faculty Advisor once the research area is selected. The primary responsibilities of a Graduate Faculty Advisor include:

1. Be a source of academic information for their graduate students

2. Provide assistance with details in determining the Plan of Study

3. Be proficient in inputting, managing, and approving forms in GradPath as needed to assure smooth progression to final degree

4. Meet periodically with their students and provide regular, timely input to determine academic progress. This may include review of Plan of Study and Prospectus as prepared by the student.

 When selecting the Graduate Faculty Advisor, the student should contact the faculty member to discuss expectations for both the faculty member and the student. The two shall meet throughout the academic year and at the end of each semester the student will complete the end of year conversation form to discuss with the advisor. A survey will be filled out at least twice yearly to report progress to the Graduate Office.

Research Tutorial Groups

Students must enroll in MATH 596G and complete a research tutorial group (RTG) project starting in the spring semester of their first or second year of enrollment. In the spring, MATH 596G is a one-unit course in which faculty members present short lecture series on research topics of current interest. In the following fall, students choose one of the proposed topics and work with the corresponding faculty member on a research project. This project and a presentation of it at the end of the fall semester is the basis for three more units of credit in MATH 596G. The RTG project is meant to be an early introduction to research in mathematics and forms part of the evaluation of the qualifying exam.

Qualifying Examination

The qualifying examination is based on the following assessment options:

• A written exam in algebra
• A written exam in analysis
• A written exam in geometry and topology
• An MS thesis

Students must attempt at least three assessment options. Two of the assessments must be chosen from the traditional core exams (the first three options). Each written exam is offered in August and January. There is no limit to the number of attempts for the written exams. Students may attempt more than three assessment options. Students with prior preparation may attempt the examinations upon entrance to the program, or after one semester.

Each of the options has three possible grades: fail, pass, and high pass. In general, a grade of high pass indicates the student is ready to go on to advanced course work and to prepare the comprehensive exam. For the MS thesis option the meaning of pass is that the thesis is acceptable for the MS degree. The meaning of high pass is that the quality of the thesis indicates the student is capable of PhD level work. The thesis need not contain original work, but the quality should indicate that the student has the potential for such work. The grade for the MS thesis is determined by the thesis committee. Prior to scheduling your thesis defense, you will need to get your MS committee approved by submitting the Committee Approval form to the Graduate Office. Once your committee is approved, you will need to print out the Results of the MS thesis form and take this to your thesis defense for a final grade.  Both forms can be found on the forms page on our website.

The written exams in algebra, analysis and geometry/topology  cover material from the traditional core courses, Algebra (MATH 511A-B), Real Analysis (MATH 523A-B), and Topology–Geometry (MATH 534A-B). They also include a small amount of undergraduate level material. For the algebra exam this undergraduate material is from linear algebra. For the analysis exam it is from rigorous advanced calculus. For the geometry/topology exam it is from undergraduate complex analysis. Short lists of topics on the exams and copies of recent examinations are available on the web.

To successfully complete the Ph.D. qualifying examination, a student is expected to obtain a result of high pass in two of the assessment options and a result of pass or high pass in a third. The Graduate Committee will be responsible for making the final determination as to whether the student has successfully completed the Ph.D. qualifying examination and may take into account all factors relating to the student's work.  

Students must successfully complete the qualifying exams before the end of their sixth semester to continue in the PhD program.

Comprehensive Examination

The purpose of the comprehensive examination is to determine whether the student has mastered the necessary general and specialized knowledge required to carry out dissertation research. The comprehensive exam has written and oral parts. To complete the written part, students write a short paper which may give an account of a research problem of interest, a significant example, or significant computations. The written part must be approved by the examining committee, which consists of a minimum of 4 tenured or tenure-track faculty, at least two weeks before the oral examination. The oral examination consists of a representation by the student, typically lasting one hour, followed by questions from the examining committee.

As part of the comprehensive examination, students are encouraged to prepare a detailed plan for the last years of their program. This plan should include a discussion of courses to take, seminars to participate in, faculty beyond the dissertation advisor to interact with, and possibly conferences to attend and professional development activities to undertake.

The Oral Comprehensive Examination is primarily, but not exclusively, on material in the area of concentration. The examination covers background material for the general area together with advanced references in a more specific sub-specialty.

Prospectus

After completing the comprehensive exam, students are expected to prepare a prospectus in consultation with their advisor. The prospectus is a detailed plan for the last years of their program. This plan should include a discussion of the research being undertaken as well as courses to take, seminars to participate in, faculty beyond the dissertation advisor to interact with, and possibly conferences to attend and professional development activities to undertake. Students must complete a departmental approval form.

G1. Professional Development Requirements (for students entering prior to Fall 2022)

PhD students must complete two professional development requirements chosen from this list:

• a foreign language requirement,
• a computing requirement, and
• a communication skills requirement.

Details of each requirement are given below. The requirements have been designed so that to a great extent they should be satisfiable by activities that would normally be undertaken by any good PhD student. The need for foreign language and computing skills varies considerably among fields of mathematics and so students should consult with their advisors on which requirements would be the best choice. Advisors may also suggest that students complete more than the minimum of two of these requirements. Students are urged to complete the professional development requirements as early in their programs as possible. In all cases, they must be completed before advancement to candidacy.

Foreign Language Requirement

A substantial portion of the mathematical literature is written in languages other than English, principally French, German, and Russian. Knowledge of Spanish is important for some fields of Mathematics Education. Being able to read and accurately translate these texts is a valuable skill in Mathematics and Mathematics Education research.

In order to fulfill the foreign language requirement, students will demonstrate their abilities to read and accurately translate mathematical texts in French, German, or Russian, (or, for students in Mathematics Eduction, texts relevant to that field in Spanish) by passing an examination given by a faculty member approved by the graduate committee. The student will prepare a careful, written translation of a text chosen by the examining faculty member (typically 5–10 pages) in a limited amount of time (typically 48–72 hours), with the aid of a dictionary and language reference works, but without the assistance of computers or other people. As a minimum standard, the translation must be mathematically accurate. We maintain a list of approved examiners.

Grading of language examinations is a significant burden on faculty and students should not make frivolous attempts to pass the examination without sufficient preparation. Faculty members may administer an oral “pre-test” to gauge whether the student appears to be prepared for the examination.

Results of foreign language examinations should be communicated to the graduate office by the examining faculty member using the language examination form.

Computing Requirement

Machine computation is an increasingly important component of mathematical research. Students for whom such computation will be relevant should master the needed programming skills and software packages during their graduate careers.

To fulfill the computing requirement, students should demonstrate their mastery of the relevant skills by carrying out a significant computing project under the supervision of a mathematics faculty member. This project might be tied to course work, the student's MS thesis, or his or her dissertation research. The precise nature of the project will be determined by the student and the sponsoring faculty member, but projects must have substantial mathematical content and should typically involve the following aspects of computing:

• formatted input and output
• appropriate use of data structures and algorithms
• use of structured programming techniques, possibly including calls to externally provided subroutines or functions.

Projects may be written in a standard programming language such as C or Fortran, or may use software packages such as Matlab, Maple, GAP or Pari. 

At the conclusion of the project, working code and documentation must be made available on the department's web site. The completion of the requirement should be communicated to the graduate office by the sponsoring faculty member using the computing examination form.

Communication Skills Requirement

The ability to communicate effectively, both verbally and in writing and to audiences of varying levels of sophistication, is essential to a successful career in industry, research, or teaching. The communication skills requirement gives students an opportunity to develop their capabilities in a variety of directions. To complete the requirement students must:

• prepare a basic web page containing information on the student's research, teaching, and other professional activities and make this page available on the department's web site
• prepare a professional CV and post it on the web site
• write articles or proposals and give lectures or presentations for audiences of various levels of sophistication so that at least one activity occurs in each row of the following table of examples. At least one of these activities must be verbal, and at least one must be written.

The entries in the table are meant to be illustrative and do not exhaust the possibilities. Written components should use TeX or other scientific text processing software. Verbal components may involve the use of such technologies as overheard transparencies or presentation software. Each component must be sponsored by a faculty member who will review the text or presentation and provide constructive feedback. When the sponsoring faculty member is satisfied with a student's performance on a component of the requirement, this fact should be communicated to the graduate office using the communication skills progress form.

G2. Professional Development Requirements (for students entering Fall 2022 or later or who opt in)

Students entering Fall 2022 or later will be subject to the following professional development requirements. Students who entered prior to Fall 2022 have the option of following these requirements or the requirements in the handbook listed under “Professional Development Requirements prior to Fall 2022.” Students who wish to use the new requirements must formally “opt in.” Note that those who opt in to the new requirements will be exempt from both the prior Professional Development requirement (G1) and the out-of-department course requirement.   These requirements are to be reviewed again during the 2024-2025 academic year.

Context: Students graduating from the Mathematics PhD programs move on to many different careers, including those in the academic, industrial, and government fields. In each of these fields, it is essential that applicants have more skills than those developed singularly through PhD research and teaching. Successful applicants to academic positions must be able to contribute to multiple axes of the academic endeavor, including mentoring, service, and leadership. Successful applicants to industrial positions must be able to work in teams with both their supervisors and colleagues and be able to communicate effectively with stakeholders. It is notable that most recently there has been a significant trend among graduating PhD students away from academic positions and toward industry regardless of the field of mathematical study.

Purpose:  The overarching goal of these requirements is to help students develop into well-rounded individuals who will gain important skills and competencies that will help them be competitive toward multiple career paths.  

The goal of professional development and community development requirements are to develop important “soft skills” necessary for future employment and job success. These include, but are not limited to communication, teamwork, critical thinking, organization, networking, problem solving, and leadership. Many of these are interpersonal skills which require interaction with people outside the mathematics community. Professional and community development are done throughout one’s career. The goal of the professional development requirement is to set a precedent for students to become lifelong learners toward bettering themselves as professionals. The goal of the community development requirement is to set a precedent for students to engage in their communities and work to improve them.

Requirements: In order to help balance and guide students through this process and create an individualized experience that allows students to handle changing circumstances throughout their time in the program, students will work with advisors each year to plan and assess the amount of work to be devoted to professional development and community development each year. For instance, if a student serves as an officer in an organization that takes significant time one year, it may be prudent to do community development with a lower time commitment the following year. Some years require more time to devote to studies (such as during qualifying exams or when finishing a dissertation), and this should be taken into consideration when planning, and also revisited during the year if changes need to be made. The student-advisor relationship will be relied upon to ensure that the student is able to complete these activities taking into account any mental or physical limitations such as anxiety, and to ensure students find a program that promotes their development without getting in the way of other necessary pieces of the graduate program such as exams and defenses.

Students will discuss a plan for community development and professional development at the beginning of each year and touch base with their advisor through the year with updates. It is expected that students will explore a few areas of work with the community (each with a low time commitment) in their initial years and then take a leadership role in later years in one or more areas of their choice. This will provide a wealth of different experiences and help the students develop a narrative for use in later job applications and interviews. Professional and community development activities may be paid; being paid does not preclude them from being considered as professional development or community development requirements for the year. The advisor and the student will need to agree on use of such requirements. It is generally expected that the time commitment for all of these requirements averages no more than 1-2 hours per week, and any commitments that will be longer (for instance, taking an out-of-department course) should be specifically discussed in this context as to whether it is a good plan for the student.

The Graduate Committee will review all plans each Fall semester in order to ensure that the plans are not too onerous on the student and satisfy the spirit of the requirements as stated above.

The following constitute the requirements:

Professional development requirement

Professional development is required yearly after year 1. In jobs, this is sometimes called training and sometimes people get certification. Possible examples are:

Community development requirement:

It is expected that the student will start to take leadership roles in the later years of their program. Here are several communities that may be considered:

Broader Impact requirement:

All National Science Foundation grants require the applicants to describe BOTH the proposal’s “Intellectual Merit” and “Broader Impact.” Intellectual Merit is the scientific content, and the research work would satisfy this piece. Broader Impacts are areas that benefit society. All students are expected to participate in at least one Broader Impact activity prior to completion of the PhD, which will likely come from the community and/or professional development activities. Some possible areas of broader impacts could be:

Evaluation:

At the end of each year, students will be evaluated in terms of the amount and quality of their community development and professional development and create recommendations for the following year. Part of this procedure will require keeping a current CV and writing a short narrative (not a list) on their community development and professional development activities as well as broader impact efforts. The evaluation will take place in the yearly Career Conversations between the Student and their Advisor, to be submitted to the Graduate Office. Professional and community development for the year will be evaluated at Satisfactory, Needs Improvement, or Unsatisfactory. Students with fewer than four years of Satisfactory community and professional development will need to petition for completion of the requirements and may be asked to remediate. The broader impact requirement can be satisfied at any time and completion will be submitted via form to the Director of Graduate Studies.

Advancement to Candidacy

A student may advance to candidacy once they have completed all program requirements other than the dissertation.  Students must advance to candidacy at least one semester prior to their dissertation defense. Failure to do so will delay the date of the dissertation defense.

Dissertation

The dissertation is a polished written account of a substantial new contribution to the mathematical sciences, publishable in a reputable journal. It is evaluated by an internal committee of at least 4 members who must be tenured or tenure-track faculty members or approved as equivalent by the Graduate College. One member may come from the minor department. Otherwise the members must be from the Mathematics Department. (Exceptions to this last rule may be granted by the Graduate Committee.) The dissertation committee approves the dissertation after a final oral defense. Students must give a copy of the dissertation to each member of the committee at least four weeks prior to the oral defense. Students have the option of also including an external reviewer who is not on the faculty of the University of Arizona. The inclusion of such an outside reviewer can provide the student with valuable feedback as well as help make the student's research known outside the local community. Students should register for Math 920 while working on their dissertation. The Graduate College requires 18 units of Math 920.

Students are encouraged to form their dissertation committee as soon as possible after the comprehensive exam. Requirements for how often this committee must meet may be found in the handbook.

The dissertation is by far the most important component of the PhD program and its quality and originality will have a major impact on the beginning of the student's professional career. Writing a quality dissertation should be the student's top priority.

Final Oral Examination

The final oral examination is a presentation and defense of the student's dissertation; the first part of the exam is open to the public.

Sample Plans

The following gives an example of how students might schedule their coursework, community & professional development, teaching, and other milestones. It is meant to be a guideline and not prescriptive.

Note that underlined items are possible Broader Impact activities.

Typical electives by research area (important core courses are in parentheses):

Ph.D. Degree Requirements: MATHEMATICS EDUCATION OPTION

Course Requirements

The course requirements are 36 units of graduate credit in the major and 12 graduate units in a minor in Education (or related field) and 18 units of dissertation (Math 920).

Courses in Mathematics

Students will normally either take the first year graduate core courses in Algebra (MATH 511A-B), Real Analysis (MATH 523A-B), and Topology–Geometry (MATH 534A-B), or otherwise learn this material by the end of their first year of Ph.D. studies for the Qualifying Examinations. The remaining 18 units will be chosen in consultation with an advisor. These remaining units will include one year-long Mathematics course sequence that is not dual-numbered and is not part of the required core of algebra, real analysis, and topology-geometry. Some of the units will include relevant courses in Mathematics Education research (to be discussed with an advisor).

Courses in Education (or related field)

The 12 units in Education (or related) will be chosen in consultation with an advisor to ensure a coherent program of study. The courses will primarily be in Education. Courses in psychology, anthropology, sociology, women's studies, etc., may also be appropriate, depending on the student's research focus. Some suggested Education courses are listed below. EDUC 500, 501, 600, 601, 602; TTE 521, 524, 532, 545, 621, 640. Two courses in research design and methods (e.g., EDUC 600, 601, 602, or appropriate research methods courses in other fields such as sociology, anthropology, agriculture, ...) are required.

Teaching Experience or Practicum

Two or more years of pre-college teaching experience are required. Students can fulfill this requirement through 9 units of practicum in local schools. Such students will take 3 units per semester to complete one practicum at the elementary school level, one at the middle school level, and one at the high school level. (Note: these 9 units do not apply toward the required 36 units of mathematics nor the 12 minor units.)

Program Of Study

The same stipulations as given for the Ph.D. program in Mathematics.

Qualifying Examination

Same as for the Ph.D. program in Mathematics.

Comprehensive Examination

Similar guidelines to those for the Ph.D. program in Mathematics, but the area of concentration will be in Mathematics Education.

Prospectus

Same as for the Ph.D. program in Mathematics.

Professional Development Requirements

Same as for the Ph.D. program in Mathematics except that the foreign language requirement may be satisfied in Spanish or American Sign Language as well as French, German, or Russian.

Advancement to Candidacy

Same as for the Ph.D. program in Mathematics.

Dissertation

Same guidelines as for the Ph.D. program in Mathematics. The dissertation will be in Mathematics Education.

Final Oral Examination

Same as for the Ph.D. program in Mathematics.