Graduate Degree Programs & Requirements

Admission to the Office of Graduate Studies in Arts & Sciences depends on a candidate's ability and promise of achievement. To demonstrate both, applicants must submit undergraduate records showing above-average performance at a recognized school, and letters of recommendation from people thoroughly familiar with their abilities. Graduate Record Examination general scores and the subject test are required. For information about admission schedules and procedures, please consult the Office of Graduate Studies in Arts & Sciences.

An entering graduate student should display competence in analysis and algebra, equivalent to the content of Washington University's Math 411-412 and 429-430 courses. PhD candidates will also find a knowledge of topology (Math 417-418) to be very helpful. A student lacking in one or more of these prerequisites may have difficulty conforming to the expected timetable for the qualifying exams; time may be allowed to complete this preparation when appropriate.

International students must be proficient in English. Entering students whose native language is not English are expected to perfect their speaking and writing skills before beginning studies at Washington University. A student arriving at the University with an inadequate command of English may be required to take remedial English instruction at his or her own expense. It is best for all foreign students to arrive at Washington University in the early summer, several months before the fall semester begins at the end of August, to ensure that these language requirements are met. 

Doctoral students at Washington University are required to demonstrate proficiency in algebra, real analysis, complex analysis, and geometry. Most students satisfy these requirements by taking a yearlong (two-semester) course in each subject, capped by a final exam that serves as a "qualifier." Students with strong backgrounds may be excused from some of these courses.

Typically, it takes two years for a student to complete the written qualifying exam phase of the program. By the end of the second year, the student usually has some idea of which area of mathematics to choose. By that time the student is also acquainted with several faculty and feels comfortable asking one to direct his/her research.

Once the qualifying exams are passed and a thesis advisor engaged, the next step in the program is for the student to prepare a "minor oral presentation" and a "major oral presentation." Topics for these orals are chosen in consultation with their thesis advisor, and culminate in two public lectures. These should be completed by the end of the year following completion of the written qualifying exams as well as satisfying their language requirement.

After these preliminaries, the essential part of a student's graduate work – the thesis – begins. This thesis may range from fifty to a few hundred pages and is the formal record of the student's achievement in the graduate program. The final official step is for the student to defend the thesis in a public lecture followed by questions from a panel of appointed faculty members of the University in a closed session.

Students typically complete the PhD program in five years. A student who comes here with advanced preparation may finish in less time. In special cases, the time schedule may be lengthened accordingly. Students should plan to develop a close relationship with their thesis advisors so that they may have a realistic idea of their progress.

There are 36 units of graduate-level course work required, with or without a thesis; 6 units may be for thesis research. The minimum residence requirement is one full academic year of graduate study. If the department consents, a student may transfer up to 6 units from other universities. A grade point average of B (3.0) or better must be maintained in graduate course work.

For more information, please visit the Bulletin.

There are four basic graduate course sequences in pure mathematics: 

Code	Title	Units
Math 5021 & Math 5022	Complex Analysis I and Complex Analysis II	6
Math 5031 & Math 5032	Algebra I and Algebra II	6
Math 5045 & Math 5046	Geometry/Topology I: Algebraic Topology and Geometry/Topology II: Differential Topology	3-6
or Math 5047	Geometry/Topology III: Differential Geometry
Math 5051 & Math 5052	Measure Theory and Functional Analysis I and Measure Theory and Functional Analysis II	6

A candidate for the MA in Mathematics must include two of these sequences (12 units) in the required 36 units. The student, in consultation with his or her advisor, will select the remaining 24 units according to the student's interests and needs.

For more information, please visit the Bulletin.

It usually takes 3-4 additional semesters to complete a master's degree (MA) after earning the bachelor's degree (BA). The Office of Graduate Studies in Arts & Sciences has an accelerated BA-MA program in which highly qualified undergraduate majors can earn both the BA and MA degrees with two additional semesters work (usually a total of 5 years). The department offers the MA degree in Mathematics. 

Note that this program is only for WU undergraduates who, if they participate, must do so in the academic year immediately following graduation (no deferments).  According to the Office of Graduate Studies in Arts & Sciences, eligibility requires having completed the BA degree from Arts & Sciences (which would rule out, for example, a second major whose degree is from the Engineering School). 

Some students might find a master's degree earned through this program useful. However, students planning to enter a PhD program in mathematics usually do so directly after completing the BA degree rather earning an MA degree first.  Majors should talk with their advisors early about whether this program would have any advantage for them. 

Majors who are interested in the program should review the Accelerated Bachelor's/Master's Program information in the Bulletin.

The application deadline (subject to change by the Office of Graduate Studies in Arts & Sciences) is the March 15 prior to beginning the MA program in the fall semester following completion of the WU BA degree. Note that the application is made through the Department, which forwards it to the Office of Graduate Studies in Arts & Sciences. Since relevant people in the Department might not be available much during the summer months, it's best to complete the application by the end of the semester's classes.

The program allows participants to count up to 15 units of 400-500 level coursework earned during the 4 years of undergraduate study (with grades of B or better) toward the MA course requirements.  (The point of the 15 unit limit is to avoid having majors push so much specialized work into the BA years that they limit their opportunities to do the exploration outside of the major that an undergraduate should do.) Counting these 15 units makes it possible to finish the master's requirements in one additional year, but the program is still fast-paced and requires a lot of intense work and some careful planning.

Careful course selection throughout the undergraduate program is required. Usually, the Department will not recommend that the Office of Graduate Studies in Arts & Sciences admit a major to this program unless the 15 applicable units of 400-500 level work will be completed by the time of graduation. In addition, the Department expects applicants to have backgrounds comparable to students admitted to our regular master's programs.

For more information about graduate program requirements, contact the Department's Director of Graduate Studies, Professor Gregory Knese.

The Department of Mathematics asks that interested majors apply to the program by the end of the fall semester in their senior year.  If accepted, they must begin the program in the fall semester immediately after graduation: no deferred admissions are allowed by the Office of Graduate Studies in Arts & Sciences.

Students should fill out the Application Form and make an appointment to meet with the Administrative Assistant for the Department's graduate programs, Ms. Mary Ann Stenner, in Cupples I, Room 100.  She will clarify the rest of the procedure for you: submitting the application for Department consideration and getting the required signatures from the Department Chair and Chair of the Graduate Committee.  If the Department approves the application, then Ms. Stenner will submit it to the Office of Graduate Studies in Arts & Sciences which will make the final decision.

International Students: 

For students in the US on a visa as undergraduates: staying on to complete an MA creates a change in visa status, which will involve reapplication paperwork. Be sure to check on the details with an international student advisor at the Office for International Students and Scholars.

Completion of the PhD requires four full years of graduate study (72 hours), with at least 48 hours spent in residence at Washington University. The student must spend at least one academic year as a full-time student; this requirement cannot be met wholly by summer sessions or part-time study. The student may, with Departmental permission, transfer a part of the 72 hours from other universities (24 graduate credits for a Ph.D. and only 6 credits for a Master's). The typical load is nine credit hours per semester. A grade point average of "B" or better is required in graduate course work.

Graduate students in mathematics may ordinarily expect up to five years of support. Continuation of support each year is dependent upon normal progress toward the degree and satisfactory performance of duties.

For the well-prepared student, "normal progress" usually means: At the end of the second year, the student should have successfully completed the specific course requirements and passed six qualifying exams; at the end of the third year, the student should have completed the candidacy requirement; by the end of the fourth year, the student should have completed the 72-hour course requirement, and should be making substantial progress on a thesis.

Students must also complete the Teaching Seminar course (L24 597), which prepares them for both Assistant to the Instructor (AI) work and academic teaching duties, which is an integral part of all scholarly activities. For a typical PhD student, the course is taken twice: once in the spring of the first year and again in the fall of the second year. Students will have Departmental duties (AI, grading, proctoring) of no more than fifteen hours per week as Assistants to the Instructor (AI). Students must also complete a Mathematical Professional Development course (L24 598).

Please note, however, that the sequence outlined above is for "well-prepared" students. The exact point at which any student enters the sequence depends upon their ability and background. When warranted, deviation from the normal sequence is permissible and a tailored program that fits the student's ability and background will be followed.

For more information, please visit the Bulletin.

The 72 hours of coursework must include eight of the following nine courses: 

Code	Title	Units
Math 5021	Complex Analysis I	3
Math 5022	Complex Analysis II	3
Math 5031	Algebra I	3
Math 5032	Algebra II	3
Math 5045	Geometry/Topology I: Algebraic Topology	3
Math 5047	Geometry/Topology III: Differential Geometry	3
Math 5051	Measure Theory and Functional Analysis I	3
Math 5052	Measure Theory and Functional Analysis II	3

Students may omit one of the following classes when satisfying the course requirement: Math 5022, 5047, or 5052.  To satisfy the breadth requirement one must pass the required courses with a B (3.0) or better.  The classes are typically offered in the following time frame:

• Fall:                 Algebra 1, Real Analysis, Complex Analysis I, Algebraic Topology, Differential Geometry
• Spring:             Algebra II, Functional Analysis, Complex Analysis II, Differential Topology

In exceptional circumstances, departmental permission may be requested to replace required courses with a suitable alternative. The student may also petition the department to waive one or more of these courses because of work done previously.

It is in each student's best interest to take the courses that contain the material covered in the qualifying exams as soon as their individual program allows. Sequels to these courses, at the 500 level, are frequently offered; the qualifying exam courses are generally prerequisites to these 500 level courses.

For more information, please visit the Bulletin.

All students must demonstrate proficiency in English.

If English is not the student's native language, they must pass an oral English proficiency exam with a grade of 3 or better. If the student does not score a 3 the first time they take the exam, the Director of the English Language Program through Continuing and Professional Studies (CAPS) will recommend taking one or more classes to improve reading, writing, pronunciation, listening or speaking skills. After the recommended classes have been completed, the student is required to retake the English proficiency exam. Once the student has demonstrated the ability to handle teaching a class (by scoring a 3 or better on the exam), they will qualify for Assistant to the Instructor or Course Instructor duties.

The qualifying exam is a series of six written tests covering a range of topics and the candidacy requirement is an oral presentation and thesis proposal.

The written tests cover the material in one semester of the courses:

Code	Title	Units
Math 5021 & Math 5022	Complex Analysis I and Complex Analysis II	6
Math 5031 & Math 5032	Algebra I and Algebra II	6
Math 5045 & Math 5046	Geometry/Topology I: Algebraic Topology and Geometry/Topology II: Differential Topology	3-6
or Math 5047	Geometry/Topology III: Differential Geometry
Math 5051 & Math 5052	Measure Theory and Functional Analysis I and Measure Theory and Functional Analysis II	6

To satisfy the written exam requirement, the student must pass six out of the nine possible qualification exams, with the requirement that two be

from each of these subgroups:

•  Math 5021 Complex Analysis I, Math 5022 Complex Analysis II, Math 5051 Measure Theory and Functional Analysis I, or Math 5052 Measure Theory and Functional Analysis II
•  Math 5045 Geometry/Topology I: Algebraic Topology, Math 5046 Geometry/Topology II: Differential Topology, or Math 5047 Geometry/Topology III: Differential Geometry
•  Math 5031 Algebra I, Math 5032 Algebra II

To satisfy the qualification examination requirement, the student must pass the final exam for the course with an A- or better.

Because each course varies somewhat in content from year to year, it is recommended that the student take the exams at the conclusion of the course in which they are enrolled. No advantage is gained by delaying the exam. It is required to finish all six qualification exams by the end of the second year of study.

Some students will enter the PhD program with previously acquired expertise in one or more of the required courses. This sometimes happens with students who transfer from other PhD programs, or who come from certain foreign countries. Such students may formally petition the Chair of the Graduate Committee to be exempted from the appropriate course and from its qualifying exam. The petition must be accompanied by hard evidence (e.g., published research, written testimony from experts, records of equivalent courses, or examinations and the grades achieved on them). The Graduate Committee will make the final judgment on all exemption requests.

Once the written phase of the qualifying process is complete, the student is ready to begin specialized study. By the third year of study, the student must complete the candidacy requirement. The student must form a preliminary thesis committee called a Research Advisory Committee that includes their advisor and at least two other faculty members. In discussion with the advisor and preliminary thesis committee, the student will select a topic and body of literature related to this topic. The student will prepare a one-hour oral presentation related to the topic and a two-page thesis proposal that demonstrates mastery of the selected topic. The oral presentation is designed to expedite specialized study and provide guidance toward the thesis. The preparatory work for the thesis proposal often becomes the foundation on which the thesis is constructed.

After the student completes the candidacy requirement, work on the thesis begins.

For more information, please visit the Bulletin.

The student's dissertation is the single most important requirement for the PhD degree: it must be an original contribution to mathematical knowledge and is the student's opportunity to conduct significant independent research.

It is the student's responsibility to find a thesis advisor who is willing to guide their research. Since the advisor should be part of the candidacy requirement, the student should have engaged an advisor by the beginning of the third year of study.

Once the Department has accepted the dissertation (on the recommendation of the thesis advisor), the student is required to defend their thesis through a presentation accompanied by a question/answer period.

For information about preparing the thesis and its abstract as well as the deadlines involved, including the creation of the Research Advisory Committee and the Dissertation Defense Committee, please consult the relevant resources from the Office of Graduate Studies in Arts & Sciences: the Doctoral Dissertation Guide, the Forms page, and the Policies and Procedures page.

Through the PhD in Mathematics program, students have the option to specialize and conduct doctoral research in several areas, including Applied Mathematics. However, we do not have a separate PhD program in Applied Mathematics, and students specializing in this field must satisfy the same requirements described above for the PhD in Mathematics.

Applied Mathematics research in our department tends to focus on theory, analysis, and methods. Prospective students with previous degrees in Mathematics or Applied Mathematics who are interested in these aspects of the field are encouraged to apply. The Mathematics faculty contact for this option is Prof. Ari Stern.

On the other hand, prospective students who are more interested in the scientific and engineering applications of methods in Applied Mathematics may be better suited to doctoral programs in the McKelvey School of Engineering, such as the PhD in System Science & Mathematics. The Engineering faculty contact for this option is Prof. Arye Nehorai.