## MATHEMATICS

### The First Two Years

### Advising

When students arrive, each is assigned an advisor. However, a specific professor can be requested. Students are required to take the qualifying examinations (quals) in the fall term of the first year. While taking the quals will not answer every question as to which way to proceed, they are a valuable source of information about where a student stands mathematically.

The first year is a time to get to know Harvard, the faculty, and fellow students. This is a time to get a sense of what sort of mathematics is done here, at what level, in what style, and by whom. By the end of the first year, it should be possible for the student to have some idea of the area that is most interesting to her or him and with whom she or he might work.

While preparing for the qualifying examination or immediately after taking it, the student should enroll in more advanced courses with the idea of choosing a field of specialization. Unless prepared to work independently, the field chosen should fall within the interest of some member of the faculty who is willing to serve as dissertation advisor. One method of choosing a professor with whom to work is to spend a term reading under the direction of two or more faculty members simultaneously, on a tentative basis. Another method might be to talk to professors about course matters. Faculty members vary a great deal in the way that they go about dissertation supervision; one’s needs in that direction should be taken into account. It is up to the student to ask a professor whether she or he will act as dissertation advisor. Most students choose an advisor during their second year. It is not usually a good idea to wait longer than two years before doing so.

The director of graduate studies and the chair are always available for consultation if problems arise in choosing an advisor or in resolving other issues that might arise. In the event that no member of the department suits a particular student, there is also a possibility of asking a Massachusetts Institute of Technology (MIT) professor for guidance.

During the dissertation stage, regular meetings with the professor chosen should be arranged. Early on, the student should consult her or his advisor regarding the selection of the required second and third readers to form the dissertation committee.

### The Qualifying Examination

The examination is given twice each year at the beginning of the fall and spring terms. The qualifying exam consists of three, three-hour papers held on consecutive afternoons. Each paper has six questions, one each on the subjects: algebra, algebraic geometry, algebraic topology, differential geometry, real analysis and complex analysis. Each question carries ten points. In order to pass the examination, a student must obtain at least twenty of the available thirty points in that subject. A student is considered to have passed the qualifying exam when they have passed in all six subjects, or they have passed in four subjects and obtained an A or A- grade in two basic graduate courses corresponding to those subjects not passed. Once the qualifying exam has been passed, students no longer need to take math courses for a letter grade and may receive the grade “excused.”

A student may take the qualifying examination any number of times, beginning in the first term. A student is not penalized in any way for failing to pass the examination once or several times, but students are expected to pass the examination by the end of the second year in residence in order to begin real mathematical research.

The sole use of the qualifying examination is to measure the breadth of a student’s mathematical knowledge. The department offers a basic sequence of mathematics courses for the first four terms in residence and the successful completion of this sequence plus minimum memory skills should amply prepare the student for the qualifying examination. The basic courses are: Math 212a (real analysis) Math 231a (algebraic topology) Math 213a (complex analysis) Math 232a (algebraic geometry) Math 230a (differential geometry). There is no graduate course covering the qualifying exam syllabus in algebra. A student who wishes to replace the algebra section of the qualifying exam with a basic graduate course should take 221 (commutative algebra). These courses cover substantially more mathematics than the qualifying examination requires; a student who passes the examination upon entrance will also find these courses interesting. A qualifying examination syllabus and samples of prior exams and solutions are available on the department’s website.

It is extremely rare for a student not to pass the qualifying examination by the third year. However, if that were to happen, there would be a consultation between the student and the advisor. Any solution that might be suggested would depend on the student’s individual situation and research progress.

### Courses

The department does not have a prescribed set of course requirements, but the University requires a minimum of two years of academic residence for the PhD degree. (See the *GSAS Guide to Admission and Financial Aid* for financial residence requirements.)

In addition to courses, students may register for three types of TIME. Two may be used for credit toward the PhD degree—TIME-C as credit for specific studying, such as for the qualifying examination, and TIME-R as credit for research. TIME-T may be used for preparing for lectures when a teaching fellow but may not be used for credit toward the degree. TIME can be used as part of the four required courses per term. Without the permission of the director of graduate studies, a student should not register for more than one half-course of TIME in a term in which she or he is not required to teach, or more than two half-courses of TIME in a term in which she or he is required to teach. (For a more complete explanation, see the references to TIME in the index.)

### The Minor Thesis

For the minor thesis, students choose a topic outside their area of expertise and, working independently, learn it well and produce a written exposition of the subject. The exposition is due within three weeks, or four if the student is teaching. The minor thesis must be completed before the start of the third year in residence.

The topic is selected in consultation with a faculty member, other than the student’s PhD dissertation adviser, chosen by the student. The topic should not be in the area of the student’s PhD dissertation. (For example, a student working in number theory might do a minor thesis in analysis or geometry). At the end of the allowed time, the student will submit to the faculty member a written account of the subject, and be prepared to answer questions on the topic.

The minor thesis is complementary to the qualifying exam. In the course of mathematical research, the student will inevitably encounter areas in which s/he is ignorant. The minor thesis is an exercise in confronting gaps of knowledge and learning what is necessary efficiently.

### Language Requirement

Mathematics is an international subject in which the principal languages are English, French, German, and Russian. Almost all important work is published in one of these four languages, although much Russian work is translated into English. For the PhD, every student is required to acquire an ability to read mathematics in two of these three foreign languages. The student’s competence is demonstrated by passing a two-hour written examination. Usually the student is asked to translate into English a page of text from a mathematics book or journal. Students may, if they wish, use a dictionary. If another language is specifically appropriate to the student’s PhD program, the student may request approval from the director of graduate studies to substitute that language. If the student’s native language is one of those required, the requirement may be waived.

The first language requirement should be fulfilled by the end of the second year; the second language examination should be passed by the end of the third year.

### Teaching

All graduate students are required to gain at least two terms of classroom experience in teaching. Teaching may be a source of support for some students. Students without outside support are usually required to teach once in each of years two through four and twice in year five. If teaching positions are available, students may arrange to teach twice in an earlier year to avoid teaching twice in the fifth year.

Teaching fellows ordinarily prepare and teach their own sections of undergraduate calculus. Participation in course-wide meetings, examination writing, grading, and holding office hours also are part of the duties, but routine homework grading is done by a course assistant. There are a few upper-class tutorial seminars taught by experienced teaching fellows.

All students must complete a teaching course and apprentice program run by the department in the term before they start teaching. Students without outside support usually complete the course in their first term and the apprenticeship during their first year.

From time to time there may be additional teaching fellow positions or graduate course assistant positions (to aid professors by running review sessions and grading homework and examinations) available for those students who wish to supplement their funding. These positions are only available to those who are making good progress on their academic work. Preference will be given to successful teachers.

### AM Degree

The Master of Arts (AM) degree is not a prerequisite for the PhD but may be obtained by students on their way to a PhD. The formal requirements for the continuing AM degree are a minimum academic residence of one year and eight half-courses in mathematics at the 100 or 200 level, with at least four at the 200 level, and candidates must pass one of the two language examinations required for the PhD.

Applicants are not accepted in the program for the terminal AM in mathematics.

### PhD Degree

The degree of doctor of philosophy is awarded to students who have demonstrated their mastery of the basic techniques of mathematics and their ability to do independent research. The former is tested in the qualifying examination, the latter in the dissertation. The dissertation, however, is the more important of the two.

The University requires a minimum of two years’ academic residence (16 half-courses). On the other hand, the PhD usually takes four to five years.

### The Dissertation

The PhD dissertation is an original treatment of a suitable subject leading to new results, usually written under the guidance of a faculty member. Many of the more advanced courses and seminars are designed to lead the student to areas of current research.

Traditionally, dissertation defenses are held in March and April for a May degree. Degrees are conferred three times during the year in November, March, and May, but most students finish for the May degree. The University Commencement is held once in May.

Dissertations presentations are scheduled some weeks prior to the University dissertation submission deadline date. A final draft of the dissertation must be placed in the Birkhoff Library two weeks prior to the advisor’s presentation to the faculty. The student’s advisor presents the dissertation to the faculty during the departmental meetings that coincide with the timetable of the University.

Once the faculty members agree the dissertation can move forward, the student may proceed with his or her oral defense. The oral defense will be in the style of a seminar with a public presentation of about fifty minutes with ten minutes for questions. The student’s dissertation committee members will attend the defense and formally approve the dissertation.

When the dissertation is accepted, the student can submit the dissertation to the registrar. The final manuscript must conform to the requirements described online in *The Form of the PhD Dissertation*.