The core courses in mathematics are MATH 511AB (Algebra), MATH 523AB (Real Analysis), and MATH 534AB (Topology-Geometry). The material in these courses is essential knowledge for all mathematicians, it forms the basis for the qualifying examination, and it is assumed in all further advanced course work in the department. The first step in the PhD program is to master the material in these courses.
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 large-scale writing. Term papers may be presented at a mini-conference at the end of the spring semester and will be used in the overall evaluation of the qualifying exam.
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.
The qualifying exam serves as a primary indicator of a student's readiness to pursue research in mathematics. Qualifying exam outcomes are “high pass”, “pass”, and “fail”. An exam result of “high pass” indicates that a student is ready to go on to advanced course work and to prepare the comprehensive exam and advance to candidacy for the PhD. A result of “pass” indicates that the student may eventually be capable of a PhD, but that a preliminary Master's degree is required. A result of “fail” indicates that a student will not continue in the PhD program, but may try to obtain a Master's degree.
The exam has three parts, covering material in Algebra, Analysis, and Topology-Geometry. (A more precise description of the material covered is discussed in another document.) All three parts are offered as separate exams each August and January. Each part may be attempted at most twice and (for students starting in the fall) all parts must be completed by January of the third year in residence. The parts are evaluated on the scale “high pass”, “pass”, “fail”. When all three parts of the exam have been attempted, an evaluation of the qualifying exam as a whole may be made, again on a scale of “high pass”, “pass”, “fail”. Information used in the global evaluation includes the result of the three parts, and any course work, term papers, and RTG projects completed. As a general guideline, an overall evaluation of “high pass” requires an evaluation of “high pass” on at least two of the three parts, and an evaluation of “high pass” or “pass” on the third. An overall evaluation of “pass” always requires an evaluation of “pass” or “high pass” on all three parts of the exam. In particular, an evaluation of “fail” on two attempts of any part of the exam implies an evaluation of “fail” on the exam as a whole.
A master's thesis will be a valuable experience for many students, especially those with weaker backgrounds. Writing such a thesis allows the student to develop skills and overcome problems associated with working on a large-scale project and doing significant writing. It also allows students to demonstrate their capabilities beyond the standard measures of course work and exams. Finally, writing and defending a Master's thesis puts the student in excellent position for the comprehensive exam.
Students are required to complete a total of 48 units of graduate credit, 12 of which may come from a supporting minor. 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. For many students, one of these sequences will be Complex Analysis, MATH 520AB.
Each student must present a coherent collection of courses in which the work outside of Mathematics (see below) is related to the work in Mathematics. There are many such possibilities, including: algebra with computer science or discrete methods in operations research; probability with statistics or reliability/quality control; numerical mathematics with computer science or computational science; mathematical foundations and history with education; or analysis with physics or optics; etc.
Students must take two courses (6 units) outside the mathematics department. The spirit of the outside course requirement is that students should learn to communicate with and appreciate the perspectives of users and producers of mathematics in other disciplines. 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. 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 by students in mathematics proper (i.e., not students pursuing the math education options) to satisfy the outside course requirement.
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 thesis advisor regarding the suitability of their plans.
PhD students must complete two professional development requirements chosen from this list:
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.
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, sample texts, and suggested preparation courses.
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.
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:
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. We maintain a page with examples of suitable projects.
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.
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:
| Verbal | Written | |
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| non-mathematical audience |
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| general mathematical audience |
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| specialist audience |
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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. When the student has completed all components of the communication skills requirement, he or she should petition the graduate commitee to approve passage of the requirement using the communication skills petition.
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 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 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 tenured or tenure-track faculty members, one of which may come from the minor department, and by a specialist not the the faculty of the University of Arizona. The dissertation committee approves the dissertation after a final oral defense.
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 thesis should be the student's top priority.
Students should refer to the graduate catalog, available on the Graduate College web site, for more details on graduate college requirements for PhD candidates.