# DEPARTMENT OF MATHEMATICS

##### Introduction

The graduate program in Mathematics is aimed at guiding students toward original research. At the end of the program (normally by the end of the fifth year of study) students are expected to submit and defend a PhD thesis that contributes new results to the body of mathematical knowledge. To successfully deal with problems arising in modern mathematics, graduate students need to acquire a considerable background. This is achieved through course work and independent study. The program provides for two sets of examinations to verify the progress:

• Preliminary Examination (normally passed by the end of the first year)
• Qualifying Examination (often passed by the end of the second year)

It is vitally important that the students begin thinking about a possible dissertation supervisor sufficiently early in the program (normally toward the end of the first year, or the beginning of the second). Students should be constantly browsing different areas of mathematics and discussing math informally with classmates and faculty, as they search for the fields which attract them most.

Students are advised to learn more about the research of the faculty members as early as possible, starting with the current listings of faculty and their research interests. Graduate students may organize seminars in which they invite faculty to explain their areas of research. Students are expected to attend department colloquiums, and may benefit from some of the many research seminars.

Graduate students with learning disabilities are encouraged to contact the Directortor of Graduate Studies and the Office of Services for Students with Disabilities at ssd@northwestern.edu.

##### Courses

Graduate courses are an essential component of a mathematician’s training. Graduates of the Northwestern mathematics program are expected to be well-rounded, in part by having taken courses in a wide variety of subjects. In addition, several courses are research-oriented and presentation-based, offering training for some of the specific skills needed by research mathematicians. Beyond these general expectations, the departmental and University legislation provides for certain minimal requirements related to the course work.

The minimal full-time course load for Mathematics graduate students in the first three years is three courses per quarter. It is possible to register for four  courses with no increase in tuition. Graduate students in each of their first three  years are required to enroll in at least seven regular mathematics courses (i.e., not including 499 independent study courses) each year, and at least one such course per quarter. Students who have passed their qualifying examination may count post- candidacy research (TGS 500) toward this requirement. Substitutions or exceptions require the advanced approval of the Graduate Committee.

Graduate students must fulfill a residency requirement of nine quarters (3 years) of full-time registration in ordinary courses. After fulfilling this residency requirement and passing the Qualifying Examination, a student should register for TGS 500 (Advanced Doctoral Study). This reduces the tuition and maintains full-time status. If a graduate student has passed the qualifying exam but not yet fulfilled the residency requirement, the student may register for Math 590 in lieu of TGS 500.

By the end of each academic year, each student is required to submit a list of the courses taken in order to demonstrate compliance with the course work and registration requirements outlined herein. Each student receiving a university fellowship for the upcoming summer must submit, prior to the end of Spring Quarter, to the Graduate Program Assistant a program of study for the summer months, approved by a supervising faculty member.

##### Independent Study

Independent study courses, Math 499 (also called "reading courses") are an excellent way for students to explore specific topics not covered by the regular course offerings. As such, they play an essential role for graduate students in search of an adviser. A student wishing to enroll in 499 must file a brief syllabus, prepared by the faculty member supervising the Independent Study, with the Graduate Program Assistant by the first day of class using a form provided by the staff. Independent study courses should not duplicate regular courses offered by the department. In particular, if a course was offered in the near past or will be offered in the near future, then a request to enroll in a 499 course in a similar subject would most likely be denied.

##### Preliminary Examination

All students must pass the Preliminary Examination in order to continue with the program. The Preliminary Examination consists of written examinations in the following three subjects: algebra, analysis, and geome- try/topology. Incoming students are invited and encouraged to take the Preliminary Examination upon entrance. There is no penalty for failing to pass a preliminary examination taken upon entrance to the program. Incoming students will be required to take the first year course in each of the prelim subjects they do not pass upon entrance, unless they submit documentation of having passed a comparable course at another institution (or other compelling evidence of familiarity with a subject.

The Preliminary Examination is given during New Student Week in September and at the end of the academic year (typically during Weinberg College’s final exam week in June). Graduate students must take the Preliminary Examination in all three subjects by the end of their first academic year. Students who do not pass the Preliminary Examination by the end of their first year must pass a make-up examination in September at the beginning of their second year in order to continue in the program beyond the first quarter of the second year. In the presence of strong evidence of other mathematical accomplishment, this rule may be modified or waived with the approval of the Graduate Committee.

An award may be offered by the Graduate Committee at the end of each academic year to a student who has achieved an outstanding performance in the Preliminary Examination.

Syllabi for the preliminary examinations and copies of past examinations are available online at the department website. The rules for the Preliminary Examination are posted here.

##### Master’s Degree Requirements

Passing the Preliminary Examination and completing at least nine approved courses over at least three quarters of registration will satisfy the departmental requirements for a master’s degree. The deadline for filing the form with the Graduate School in order to receive the degree at the June degree ceremony is the beginning of Spring Quarter. Students intending to apply for a master’s degree should approach the Graduate Program Assistant at least 3 weeks prior to this deadline.

# The Qualifying Examination is an oral examination. After the student has decided upon a field of interest, he/she asks a faculty member to chair a committee to conduct his/her Qualifying Examination. The chair, in consultation with the student, selects at least two other faculty members to comprise the committee. The committee members, in consultation with each other and the student, assign topics and a reading list to the student. The committee must be comprised of at least three faculty members, at least two of whom are NU graduate faculty

• Students preparing for the Qualifying Examination must submit for approval the qualifying examination form to the Graduate Program Assistant at least three weeks prior to the exam date.
• The Qualifying Examination must be taken by the end of the first quarter of a student’s third year. Failure to pass the exam by the end of the third year may jeopardize financial support in the subsequent year.
• Students must have a prospectus (dissertation proposal) approved no later than the end of the fourth year of study, which falls on the last date of the 16th quarter. A student failing to meet this milestone will be considered not in good academic standing and therefore will be placed on academic probation. Deadlines will be altered in the case of an approved leave of absence or a childbirth accommodation. The prospectus must be approved by a faculty committee comprised of a minimum of three individuals. At least two members of this committee, including the chair, must be members of the Northwestern University Graduate Faculty.
• Students may choose to have their prospectus approved by the faculty committee members via electronic communication or an in-person meeting. The student must then submit the PhD Prospectus form through TGS Forms in CAESAR for program approval. The student’s adviser must email the Graduate Program Assistant to confirm the student’s successful completion of the prospectus. Students are notified via email by TGS of approval of their Prospectus form.
##### Dissertation and Thesis Defense

Each student must complete original research and produce a dissertation acceptable to the student’s adviser and the Thesis Committee. The Thesis committee is comprised of a minimum of three faculty members. At least two members of this committee, including the chair, must be members of the Northwestern University Graduate Faculty. The last milestone of the graduate program is an oral defense of this dissertation.

Submission of a dissertation to TGS is performed electronically. Information on this process can be found in the Degree Requirements section of the Student Services page of the TGS website. Students must consult the TGS website for the latest information on submission requirements and guidelines. They must, however, give electronic or hard copies of the dissertation to each member of your committee at least three weeks prior to the scheduled date of the oral examination.

The student must also provide an electronic copy of the thesis as well as the location and time  of thesis defense to the graduate program assistant three weeks prior to the scheduled date of the thesis defense. The examination on the dissertation will be open to the public, but the examining committee has discretion to exclude the public during part of the examination for additional questioning of the candidate and confidential deliberations.

In order to maintain flexibility in the graduate program, and, in particular, to accommodate exceptional cases, the Graduate Committee will consider written requests from students’ advisers to waive or modify requirements. However, it is expected that such requests will be granted only rarely.

##### Funding

There are several types of financial aid awarded by the department and the University on the basis of merit. Financial aid is typically offered to graduate students for five academic years. The Graduate Committee decides whether to renew a student’s support based upon his/her progress from year to year. For a first year student, progress will be determined from course grades and reports solicited by the Graduate Committee.  For this purpose, a GPA of 3.0 or lower is not considered adequate preparation for further study. Students are expected to pass the Preliminary Examination by September of the second year and the Qualifying Examination by the end of the first quarter of the third year.

Financial aid from Northwestern consists of a mixture of fellowships and graduate (teaching) assistantships.  A student with a non-English speaking background may be required to demonstrate fluency in spoken English before he or she can hold a graduate assistantship. Failure to do so may result in academic probation, loss of funding, and even exclusion from the program. See Graduate Assistantship for the precise requirements.

It is a policy of the Graduate School that teaching assistants and University Fellows may not undertake other employment without the prior permission of The Graduate School. This includes any stipend or fellowship at another institution opportunities. Such exceptions require explicit permission of the student’s adviser and the departmental graduate committee.

##### Residency During Fellowship

The department is required to certify that all graduate students on any fellowship are working full-time on mathematics research.  Therefore, we normally expect fellowship students to be in residence. Exceptions are possible for extraordinary opportunities. Such exceptions require explicit permission of the student’s advisor and the departmental graduate committee.

##### Outside Funding

We encourage students to apply for funding from outside sources during their study at Northwestern. Such outside funding will be banked for the student and can be used as fellowship quarters for a possible 6th Funding through the student’s advisor’s and the department’s grants is not considered as outside funding for this purpose. If any banked quarter accrues to the student as a result of such funding, it will be banked as a fellowship or teaching assistantship quarter, depending on the duties the funding replaced. According to TGS regulations, a student can bank at most four quarters from all funding sources not coming from TGS.

##### Teaching Policy

Teaching experience is considered to be an integral part of the training of all graduate students, and is required each year of all students, with the exception of students in their fifth year of study. First-year students usually fulfill this requirement by assisting with grading. Upper-level students fulfill this requirement by serving as a teaching assistant for at least one course in an academic year.

Potential employers, particularly academic employers, often specifically inquire about the quality of teaching performed by job candidates from the department. Our students have found teaching experience to be a valuable asset.

##### Teaching

Undergraduate teaching is one of the primary responsibilities of the Mathematics Department and graduate students play an important role in this function. As a part of their training, and as a necessary condition for most forms of financial aid, all teaching assistants are involved with departmental teaching activities each quarter they are in residence. Most often they assist faculty members by conducting the recitation sections of calculus and other undergraduate courses. They may on occasion be asked to grade for graduate courses.

##### Course Assignment Responsibilities

Responsibilities to each course assignment begin on the first day of classes of that quarter and end 48 hours after the final examination.The work of teaching assistants varies with the course and the faculty member who has primary responsibility but generally includes answering questions about homework problems in the recitation section (which meets weekly for each course), grading homework,  as well as proctoring and grading quizzes, mid-terms, and final exams. The faculty member teaching the course makes specific arrangements with his/her teaching assistants concerning their duties. Recitation sections meet on Tuesday or Thursday and the lectures are on Mon/Wed/Fri, although there may be occasional departures from this model. Classes begin on the hour and last for 50 minutes.

It is department policy that a teaching assistant’s assigned instructional duties should take no more than four hours per course per week. A teaching assistant who feels he/she is overburdened should consult with the professor in charge of the course. If problems persist, the Director of Graduate Studies should be notified.

Teaching assistants should have at least three formal, announced office hours each week during which students may meet with them. These office hours serve for all of the classes for which the graduate student is a TA. Teaching assistants should choose times when students are likely to be able to come, so as to avoid multiple requests for appointments at other times. Holding some office hours in the early evening is another possibility. One good strategy is to have office hours straddle two class time-slots, for example 1:30-2:30p, to make them available to more students. TAs for courses with graded homework should schedule office hours for the day before the homework is due, not the day after. Check with your professor to see what his/her preference is.

The department and the graduate school organize several different training sessions for its new teaching assistants and all new teaching assistants are expected to attend. For second year graduate students, there is a required teaching seminar during the fall quarter.

Faculty members are required by college legislation to have their undergraduate courses evaluated by the students.  This includes an opportunity for students to evaluate their recitation sections. The information collected is returned to the faculty member and to the department chairperson. This, and other methods of evaluation, are used by the department in assessing how well teaching assistants are carrying out their duties. A graduate student whose teaching is determined to be unsatisfactory may become ineligible for University financial support.

Every year we will have a few openings for graduate students to teach a class in one of our calculus sequences, where they will receive supervision and mentoring from an experienced faculty member. To be eligible for this opportunity, the student must:

• have demonstrated excellence as a TA, as evidenced by CTECs and in-class evaluations
• have evidence of initiative and interest in teaching (such as teaching in SPS or working with the Searle Center in one of their programs for graduate students)
• have the permission of their adviser

Graduate students undergo teacher training during the fall quarter of their second year by participating in Math 580: Seminar in College Teaching. First  year students perform supervised grading of exams in one undergraduate course per quarter.

##### Summer School Teaching Opportunities

In order to provide teaching experience and some summer support, some students are given the opportunity to teach in the Summer School. This is usually offered to advanced students, chosen on the basis of evaluations of performance as a teaching assistant.

##### Minimum In-class Teaching

In-class teaching is an essential responsibility of a mathematician in a research and/or teaching career. A graduate student must have served as teaching assistant (or, in exceptional cases, lecturer) in a role that involves meeting with a class or running a discussion section (grading does not count) in at least three quarter-long courses at Northwestern or other institutions before completing a PhD.

##### Language

Teaching is an essential element of the education and training of PhD students. For this reason, TGS and the Mathematics Department require that all students serve as Teaching Assistants during their career at Northwestern. In order to be able to teach, students must be certified as proficient in the English language. If the student is from an English-speaking country or has attended an undergraduate institution where the instruction is in English, this is automatic. Otherwise, TGS offers four ways to demonstrate proficiency detailed at the following siteWe hope that all students meet the proficiency requirement by the end of the first year. Northwestern has many resources and classes for learning English.

It is up to the student to avail themselves of these resources. Students who have  not met the English proficiency requirement by end of the first year will use fellowship quarters for their support beginning in the Fall quarter of the second year. Any fellowship quarter used in the second year will be deducted from the student’s total of six allotted fellowship quarters.

Students who do not meet the requirement by the end of the second year will be deemed to be not making satisfactory progress as defined in this Handbook and will be placed on probation. As with any probation, the student will be informed in writing the steps needed to meet the requirement and any deadline that must be met. Failure to complete these steps could lead to exclusion from the program.

##### Travel/Book Fund

Each graduate student has access to limited yearly funds available to use for travel or book purchases related to graduate studies.  First year graduate students receive $1,000 total per year ($500 cap on book purchases). All other graduate students receive $750 total year ($250 cap on book purchases). The travel/book fund is available to use from Sept 1st to August 31st each year. Funds may not be carried over to the next academic year and unused funds will be forfeited at the end of the year.

The Mathematics Department has limited funds allocated to support graduate student travel to present work or attend conferences and seminars of relevance to a student’s educational and/or professional development. The student should first approach his or her adviser to inquire about funding through a grant held by the adviser. Please note that funding from federal grants will carry additional limitations beyond the standard University policy, such as requiring US air carriers for foreign travel, and no alcohol.

The student should also seek funding from other sources, such as a hosting institution or conference organizers. After these steps are taken, the student may apply for travel support from the Mathematics Department directly. The student should submit the application(s) as early as possible to secure funding before booking travel, but certainly at least one month prior to the date of travel. The student will receive a written response within 10 days.

To  apply,  the student should submit by email to the Graduate Program Assistant a description of the conference (including titles, dates, web URL’s) and the intended use and total amount of funding requested. The application should also specify other sources and amounts of funding that the student has applied for/secured.

After the travel is complete, the student should create an expense report and submit original receipts to the department office. The student also should submit by email to the Graduate Program Assistant a summary paragraph describing how the completed travel benefits the student’s educational and/or professional development. Once the summary paragraph has been received and approved, funds will be released for reim- reimbursement.

• NoteA student seeking travel support for a conference or seminar at which he or she has an opportunity to contribute a presentation must concurrently apply for a Graduate School Travel Grant. Applications are available at The Graduate School web site. The student may consult with the Business Administrator regarding budgetary questions on the application as necessary.

A graduate student may utilize the travel/book fund to make a book purchase through the office at any time during the year. To make a purchase, the student should email the Graduate Program Assistant the website link to purchase the book. Amazon is the most commonly used vendor, though other vendors can be used. The Graduate Program Assistant will then complete the purchase and inform the student when the book is ready to be picked up from the Math Office (Lunt 201).

Four awards are given to graduate students:

1. An award for the excellent performance on the Preliminary Examination.
2. The Gelfand Award is given at the end of the year to that student who has contributed most to the department.
3. The Best Thesis Award
4. The Graduate Teaching Assistant Award

# One of the best ways to get to know faculty and other graduate students is to attend the departmental teas. These are held in the Common Room, Lunt 218, every afternoon on weekdays at 3:45pm. Students are strongly encouraged to participate actively in departmental seminars and are expected to attend department colloquiums.  The colloquiums are meant for a general mathematical audience.  A complete list of seminars and colloquiums is available on the Math Calendar.

##### Facilities
• The R. P. Boas Mathematics Library is located on the first floor of Lunt Hall.  It has a large collection of books and journals. Some books may be placed on reserve for beginning graduate courses so all the students can have access in the library. Journals must be used in the library, though they may be signed out briefly for copying. The librarian can explain the details of book circulation policies. Normal library hours are weekdays from 8:30a-5p.  Graduate students have  after hours access to this library via Wildcard. Other books and journals of interest, especially those in applied mathematics, can be found in the Science-Engineering Library, east of the Technological Institute.
• Computing Facilities. Many members of the Math Department at Northwestern University use computers in important ways in their research. The department has its own subnetwork with about 50 computers in offices occupied by faculty members and staff. There is also a public computer lab in Lunt B10 with computers used as terminals to access accounts on the Math servers. These computers and three laser printers are housed in Lunt Hall and are used exclusively by the faculty and graduate students in the Math Department. Computers in the Math Department support Mathematica for symbolic and numeric mathematical calculations as well as graphical display of their output, TexT/LaTeX and related typesetting packages.  Consult the computer system administrator, Miguel Lerma, for additional information.

Graduate students can obtain accounts on the department’s internal computer network and are encouraged to become familiar with its use. Each entering student should see Miguel A. Lerma to set up an account. The department discourages excessive personal use of departmental computer resources. The university provides many additional computer services through Information Technology. One of the most useful of these is the campus WiFi network, accessible using your NetID.

##### Where to Find It
• Mailboxes: Mailboxes for faculty and graduate students are located outside Room 201.
• Photocopy machine: Room 202. Obtain a user number from the department office to use the machine.
• Office supplies: Paper, pencils, pens, stationery, etc. are available in reason- able quantities from the office staff in Room 201.
• Texts for TA’s: Desk copies of textbooks are available from the Math Office (Lunt Hall, Rm 201).
• Computers: Several Linux computers and one Microsoft Windows XP are in Room B10. There is one laser printer in the hall of the basement and one laser printer in Room 202.

To remain eligible for all forms of financial aid,  students must meet the criteria for satisfactory academic progress set by The Graduate School. In addition, students must fulfill the Preliminary Exam requirement set by the department. For students who are unable to make satisfactory academic progress, the department follows the TGS policies for academic probation and dismissal. Students wishing to appeal a dismissal decision should follow the appeal process guidelines set forth by TGS.

The graduate program is aimed at guiding students toward original research. At the end of the program (normally by the end of the fifth year of study) students are expected to submit and defend a PhD thesis that contributes new results to the body of mathematical knowledge.

##### Nondiscrimination Statement

Northwestern University does not discriminate or permit discrimination by any member of its community against any individual on the basis of race, color, religion, national origin, sex, pregnancy, sexual orientation, gender identity, gender expres- sion, parental status, marital status, age, disability, citizenship status, veteran status, genetic information, reproductive health decision making, or any other classification protected by law in matters of admissions, employment, housing, or services or in the educational programs or activities it operates. Harassment, whether verbal, physical, or visual, that is based on any of these characteristics is a form of discrimination. Further prohibited by law is discrimination against any employee and/or job applicant who chooses to inquire about, discuss, or disclose their own compensation or the compensation of another employee or applicant.

Northwestern University complies with federal and state laws that prohibit discrim ination based on the protected categories listed above, including Title IX of the Education Amendments of 1972. Title IX requires educational institutions, such as Northwestern, to prohibit discrimination based on sex (including sexual harassment) in the University’s educational programs and activities, including in matters of employment and admissions. In addition, Northwestern provides reasonable accommodations to qualified applicants, students, and employees with disabilities and to individuals who are pregnant.

Any alleged violations of this policy or questions with respect to nondiscrimination or reasonable accommodations should be directed to Northwestern’s Office of Equity, 1800 Sherman Avenue, Suite 4-500, Evanston, Illinois 60208, 847-467-6165, equity@northwestern.edu.

Questions specific to sex discrimination (including sexual misconduct and sexual harassment) should be directed to Northwestern’s Title IX Coordinator in the Office of Equity, 1800 Sherman Avenue, Suite 4-500, Evanston, Illinois 60208, 847-467-6165, TitleIXCoordinator@northwestern.edu.

A person may also file a complaint with the Department of Education’s Office for Civil Rights regarding an alleged violation of Title IX by visiting www2.ed.gov/about/offices/list/ocr/complaintintro.html or calling 800-421-3481. Inquiries about the application of Title IX to Northwestern  may be referred to Northwestern’s Title IX Coordinator, the United States Department  of Education’s Assistant Secretary for Civil Rights, or both.