# Degree Requirements for PhD in Mathematics

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## Degree Requirements for PhD in Mathematics

### Mission

The doctoral program trains mathematicians for further research, high-level teaching, and industrial employment. Students gain a solid grounding in mathematics while being exposed to contemporary problems and applications.

### Summary of requirements

The degree program requires at least 72 hours beyond an undergraduate degree, including at least 48 hours of course credits and at most 24 hours of research credits (MATH 998 or MATH 999), and culminates in a doctoral dissertation. There are two sets of exams, in addition to the dissertation proposal, the dissertation itself, and the dissertation defense. Students who already possess a M.S. degree or other graduate work may receive up to 30 credit hours towards the degree, however they are not exempt from the preliminary exam requirement. Additional requirements include 4 of the 5 specified core courses (one must be MATH 675), two areas (major and minor) of emphasis, participation in advanced seminars, preliminary and qualifying exams, and formal advancement to candidacy, all as described below.

Students should periodically check Patriot Web to see if they have met degree requirements. Under "Student Services", go to "Student Records" and finally "Degree Evaluation".

### Beginning study, required core courses and preliminary exams

PhD students are required to take 4 of 5 core courses, one of which must be MATH 675. The core courses are

- MATH 675 (Linear Analysis)
- MATH 621 (Algebra)
- MATH 631 (Topology)
- MATH 677 (Ordinary Differential Equations)
- MATH 685 (Numerical Analysis)

Students should also take the Graduate Seminar (1 credit) for at least six semesters (maximum nine). Students who enter GMU with an MS degree should take the Graduate seminar until they advance to candidacy or until they have six seminar credits.

Core courses, seminar, and all other courses are included in a required Program of Study Form that students must complete as part of advancement to candidacy. Students may find it helpful to fill out a Proposed Coursework Form to guide them in their choice of classes (the Proposed Coursework is not mandatory).

The preliminary exams are intended to test breadth of a student's basic mathematical knowledge. Students are required to pass written preliminary exams on three of the core subjects.

For the first year, a typical full-time student takes coursework including four of the five core courses, then studies and passes three preliminary exams.

Each preliminary exam is based on material presented in the corresponding core course. More information on preliminary exams, including syllabi and previous exams can be found here.

A student may generally not attempt an exam more than three times. After an exam is passed, students should receive a copy of the required Results of the Preliminary Exam, which will be needed for advancement to candidacy.

Passing three preliminary exams is sufficient to satisfy the creative component of the Master's Degree in Mathematics.

### The Qualifying Exam

PhD students are required to take a qualifying exam on one major and one minor topic . As soon as possible after the preliminary exams, a student will choose a dissertation advisor based on his/her mathematical interest and compatibility. The advisor will shape the program of study for the qualifying exam.

In preparation for this exam, a student will study two bodies of material in a deeper way, with an eye toward an eventual research topic. The goal is to get students to the point at which they can do research in the major field of research, and have some depth knowledge in a second content area of mathematics.

Students are expected to master one topic (the *major area*) at a sufficiently deep level to be judged ready to do research in this topic. They are additionally required to understand another topic (the *minor area*) at an advanced level comparable with several semesters of graduate work. The expectation is that students' knowledge in the major area is equivalent to approximately four courses (including a 600-level introductory course if one exists), and in the minor area is approximately three courses of material (including a 600-level introductory course if one exists). The specific topics constituting a major and minor area are left quite flexible, although they must be distinct from each other. These topics vary widely depending on the individual student, advisor, committee, and research area. The preparation work in the major and minor areas are typically done as part of independent studies or reading courses.

The major and minor areas must be in two *different *areas of mathematics. Though some fields may overlap, the two exams should cover substantially different material, even if there are connections between fields.

For example, a student may study convex geometry as a major area and commutative algebra as a minor area, or a major in dynamical systems and a minor in analysis. When appropriate (such as in applied major fields), the student may choose a minor field outside the department, but it must be highly mathematical in nature and approved by the qualifying exam committee as well as the graduate director. The specific body of material on the exam must be listed on the Qualifying Examination form. Advisors should be directing students to appropriate material for the qualifying exam, with the goal of preparing students for research.

The grade on the qualifying exam will be “Pass,” “Conditional,” or “Unsatisfactory.” If a student receives an “Unsatisfactory,” he or she is allowed to retake it once approximately six months later; a subsequent "Unsatisfactory" could result in termination of funding. A “Conditional” may either require retaking the entire test, or retaking a portion of the exam with restricted topics, as determined by the committee. These conditions should be given to the student when the committee determines the “Conditional” score. The results of the Qualifying Exam should be recorded on this form with the signatures of the examination committee and the graduate director.

Advisors determine the content and subject areas of the qualifying exam, typically in consultation with the student. Advisors may suggest specific faculty to conduct part of the exam in the minor area. The qualifying examination committee will grade the student's qualifying exam, and often has an informal advisory role in the student's education. The composition of the committee and the topics for the qualifying exam must be approved by the advisor and the graduate director, using the Qualifying Examination form.

The qualifying exam must be a written exam, though it may have an oral component as well, possibly following the evaluation of the written component. The exam format will be determined by the examining committee, but it typically includes a total of two to four hours of examination time. Some examinations are done for both topics together, and others are done separately for the major and minor fields.

### Dissertation Committee

Students choose a dissertation committee in consultation with his or her advisor. This occurs around the same period in which they take the qualifying exam. The dissertation committee consists of three department members (who often coincides with those on the qualifying exam committee) and one member external to the department. The doctoral dissertation committee members are expected to take an interest in the student’s progress and act in informal advisory roles throughout the rest of the student’s time at George Mason. Students are encouraged to talk frequently with dissertation members so that the thesis topic and results are familiar to the members far before the thesis is defended.

Students should fill out the e Doctoral Dissertation Committee form before or as soon as possible following passing the qualifying exam. It should be filed with the graduate director no later than one month following successful completion of the qualifying exams.

Should the student choose to change the committee, a new Doctoral Disseratation Committee form is needed. It should be filed more than a month prior to the defense of the thesis, subject to the approval of the graduate director.

### Thesis Proposal and Advancing to Candidacy

Approximately one semester after passing the qualifying exam, each doctoral student prepares a written dissertation proposal while taking Math 998 (Doctoral Dissertation Proposal). The proposal must be approved by the thesis committee. Students should fill out this form, obtain signatures from the committee members, and submit the form to the graduate director, maintaining his or her own copy for purposes of advancing to candidacy. The thesis proposal does not require an oral defense, though it is strongly encouraged. The committee members may insist students present their proposal orally before signing the approval of the dissertation proposal. Students are advised to give the proposal to the committee members at least two weeks before any oral defense or a request for signature.

The thesis committee typically consists of the three qualifying exam committee members and a fourth member who must come from outside the department (as required by the university guidelines for doctoral students). After the dissertation proposal is approved, students must file the Advancement to Candidacy Form. This form is due with all relevant signatures in the COS Graduate Admissions Office within 6 years from when a student first enrolls as a degree-seeking student. Part-time students should speak with the Graduate Chair for accommodations if needed, however even part-time students are subject to this rule.

Keep in mind that the Advancement to Candidacy Form may take a couple weeks to process, and students may not take MATH 999 before it has been approved. The form should be submitted at least two semesters in advance of graduating, so as not to run into problems with the requirement for MATH 999.

### Thesis and Defense

After advancing to candidacy, a student will work on a doctoral dissertation while enrolled in MATH 999. Students must enroll in at least 3 credits of MATH 999 in the term in which a student advances to candidacy. A student must continue to enroll in 3 credits of MATH 999 until all credit requirements are met, including a total of 72 graduate credits and a minimum of 12 credits between MATH 998 and MATH 999. Students may subsequently enroll in fewer credits of MATH 999 until they graduate.

The dissertation is a written piece of original thinking which demonstrates a doctoral candidate’s mastery of the subject matter. A student is expected to produce new and original research worthy of publication in a peer-reviewed journal. After the thesis is completed, the committee will review the thesis and examine the student in a public oral thesis defense. Student must give copies to their dissertation committee members at least two weeks before the thesis defense. After the defense, the committee should complete the Defense of Doctoral Thesis form. Students are typically expected to finish the thesis within two years of advancing to candidacy.

There is a formal process for submitting a thesis including a format review and a submission consultation. The process requires several steps and may take a few days. You can find this information as well as the deadlines for the final submission of all materials can be found on University Dissertation and Thesis Services (UDTS) website.

Don’t forget that you must have a public defense at least two weeks prior to the defense. You would need to submit a Defense Announcement Submission Form with the following information for your defense:

- Date, time and place
- Dissertation title
- Committee chair
- Committee members
- Abstract

The College of Science will then send out the required announcement.

Close to completion of all the PhD requirements, students must apply to graduate as this is not an automatic process!

### Additional requirements and restrictions

#### Grades in the core courses

Ph.D. students must earn a B or better in each of the four core courses taken to satisfy the core course requirement. If a student has already completed equivalent coursework elsewhere, he/she may meet this requirement by receiving appropriate transfer credit or taking an (approved) advanced course in the field (and earning a B or better). Students are still required, however, to pass the preliminary exams unless they have been explicitly waived. Similarly, passing preliminary exams does not substitute for the core course requirement.

#### Seminar

Students must attend a 1-credit weekly student seminar (MATH 795). Students are required to have 6 seminar credits as part of their course requirements, and may not count more than 9 seminar credits toward their degree. This seminar is graded on presentations and attendance. It consists of both faculty presenting potential thesis topics and presentations by students (on topics that they choose with faculty guidance, but not necessarily on research). Each student will give a presentation each semester and will receive comments by an assigned faculty discussant. All faculty interested in having PhD students will periodically be discussants and will present topics to the students. Students must take these credits before the semester in which they advance to candidacy. MATH 795 may not be taken concurrently with MATH 999.