Ph. D. Requirements
In addition to the information on this page, the web page Satisfactory Progress gives some policies regarding satisfactory progress towards satisfying the Ph.D. requirements. Students should refer to the graduate catalog, available on the Graduate College web site, for more details on graduate college requirements for PhD candidates.
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 MATH 511A, Real Analysis MATH 523A, and Geometry–Topology MATH 534A, 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.
Students are encouraged to write a term paper in lieu of a midterm exam in one of their core courses in the spring semester. Such a paper gives the student an opportunity to demonstrate his or her skills beyond the standard course and exam measures and is valuable training for future largescale writing. Term papers may be presented at a miniconference at the end of the spring semester and will be used in the overall evaluation of the qualifying exam.
Further Mathematics Coursework
Two yearlong Mathematics course sequences that are not dualnumbered and are not part of the required core of algebra, real analysis, and geometrytopology 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 MATH 520A is a good choice for one of these sequences.
Outside of Department Courses or Internship
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 satisifed 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.
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 should consult with the Associate Head for the Graduate Program and their dissertation advisor regarding the suitability of their plans.
Program of study
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.
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 oneunit 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 511AB), Real Analysis (MATH 523AB), and Topology–Geometry (MATH 534AB). 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 start 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 tenuretrack faculty, at least two weeks before the oral examination. The oral examination consists of a repesentation 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 subspecialty.
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.
Professional Development Requirements
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 “pretest” 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.
Verbal  Written  

nonmathematical audience 


general mathematical audience 


specialist audience 


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.
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 commitee of at least 4 members who must be tenured or tenuretrack 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 on the web page Satisfactory Progress.
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.
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 511AB), Real Analysis (MATH 523AB), and Topology–Geometry (MATH 534AB), 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 yearlong Mathematics course sequence that is not dualnumbered and is not part of the required core of algebra, real analysis, and topologygeometry. 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 precollege 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.