The PhD program of the Harvard Department of Mathematics is designed to help motivated students to develop their understanding and enjoyment of mathematics. It seems to be generally the case that enjoyment and understanding of the subject, as well as enthusiasm in teaching it, are greater when one is actively thinking about mathematics in one's own way. For this reason, a PhD dissertation involving some original research is a fundamental part of the program. The stages in this program may be described as follows:

The word "help" (in the opening sentence above) is to emphasize that the student is expected to take the initiative in pacing him or herself through the PhD program. In theory, a future research mathematician should be able to go through all three stages with the help of only a good library. In practice, many of the more subtle aspects of mathematics, such as a sense of taste or relative importance and feeling for a particular subject, are primarily communicated by personal contact. In addition, it is not at all trivial to find one's way through the ever-burgeoning literature of mathematics, and one can go through the stages outlined above with much less lost motion if one has some access to a group of older and more experienced mathematicians who can guide one's reading, supplement it with seminars and courses, and evaluate one's first attempts at research. The presence of other students of comparable ability and level of enthusiasm is also very helpful.

The University requires a minimum of two years academic residence (16 half-courses) for the PhD degree. On the other hand, five years in residence is the maximum usually allowed by the Department of Mathematics. Most students complete the PhD in four to five years. Please review the program requirements timeline.

There is no prescribed set of course requirements, but the department runs several introductory graduate courses (e.g. Math 212a, 213a, 230a, 231a, and 232a) to help students acquire the necessary broad basic background in mathematics. Students are required to register and enroll in four courses each term to maintain full time status with the Graduate School of Arts and Sciences, but students may substitute TIME for one course when teaching or engaged in independent study or research.

The department gives a biannual qualifying examination (usually in at the beginning of the fall and spring terms). One's first goal should be to bring one's basic knowledge up to such a point that one can pass the qualifying exam. Some students are able to pass as soon as they enter, and all are urged to take the exam at the beginning of the first term. Those who do not pass simply try again on the following occasion. All students are expected pass the qualifying exam before the end of their second year.

The qualifying examination covers algebra, algebraic geometry, algebraic topology, complex analysis, differential geometry, and real analysis. More details about the qualifying exams can be found here.

After passing the qualifying exam students are expected to find a PhD dissertation adviser. Besides the dissertation, there are three other department requirements for the PhD degree.

The minor thesis: The student chooses a topic outside her or his area of expertise and, working independently, learns it well and produces 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.

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. Accordingly, every student is advised to acquire an ability to read mathematics in French, German, and Russian and is required to demonstrate it by passing a two-hour, written language examination in two of these three languages. Usually students are asked to translate one page of mathematics into English with the help of a dictionary if needed. A student who thinks it is pertinent to his or her field of interest may substitute Italian for one of the languages mentioned above. The first language requirement should be fulfilled by the end of the second year; the second language requirement should be fulfilled by the end of the third year. More information on the graduate program requirements timeline can be found here.

Upon completion of one language exam and having taken eight real courses (not TIME), students can apply for a continuing Master's Degree. This may be useful for higher paying summer jobs (and as another line on your resume). Applying for the continuing Master's degree entitles students to apply for tickets to attend the University Commencement exercises.

Most research mathematicians are also university teachers. In preparation for this role all our students are required to take a teaching apprenticeship and to have two semesters of classroom teaching experience, usually as a teaching fellow. During the teaching apprenticeship the student is paired with a member of our teaching staff. The student will attend some of the adviser's classes and then prepare (with help) and present his or her own class, which will be videotaped. The apprentice will receive feedback from the adviser and from members of the class. Teaching fellows are responsible for teaching calculus to a class of about 25 undergraduates. They meet with their class three hours a week. They have a course assistant (usually an advanced undergraduate) to grade homework and to take a weekly problem session. Usually there are several classes following the same syllabus and with common exams. There is a course head (a member of our teaching staff) who will coordinate the various classes following the same syllabus and who is available to advise teaching fellows. Sometimes graduate students also act as graduate course assistants for advanced courses or run tutorials for small groups of undergraduates studying subjects not taught in our regular courses.

How a student goes through the second and third stages varies considerably among individuals. While preparing for the qualifying examination or immediately after, the student should be taking or auditing more advanced courses and trying to decide upon a field of specialization. Unless prepared to work independently, she or he should choose a field that falls within the interests of some member of the faculty who is willing to serve as thesis/dissertation advisor. Members of the faculty vary a great deal in the way that they go about thesis/dissertation supervision, and the student should take her or his own needs in this direction into account as well as the faculty member's field in making a decision. Some faculty members expect more initiative and independence than others, and they vary in how busy they are with other students. In the event that no member of the department suits a particular student, there is also a possibility of asking an MIT professor for guidance. The student must take the initiative to ask a professor if she or he will act as the dissertation advisor. If one has trouble in deciding under whom to work, it is possible to spend a term reading under the direction of two or more faculty members simultaneously on a tentative basis. The sooner a decision is made the better, but students will need a provisional advisor by the second year.

It is important to keep in mind that no technique has been or ever will be discovered for teaching students to have ideas. All that the faculty can do is to provide an ambiance in which one's nascent abilities and insights can blossom. Moreover, PhD dissertations vary enormously in quality, from hard exercises to highly original advances. Finally, many very good research mathematicians begin very slowly, and their dissertations and first few papers could be of minor interest. On the whole, we feel that the ideal attitude is: (1) a love of the subject for its own sake, accompanied by inquisitiveness about things which aren't known; and (2) a somewhat fatalistic attitude concerning "creative ability" and recognition that hard work is, in the end, much more important.

The qualifying exam is designed to measure the breadth of a student's knowledge in mathematics. Passing the exam early is mainly an indication that a student has attended an undergraduate university with a broad undergraduate program in mathematics. It is not a good predictor of the quality of the eventual PhD dissertation.

A student may take the qualifying examination any number of times, beginning in the first term. The exam may prove a useful diagnostic in helping to identify areas in which a student's knowledge is weak. There is absolutely no stigma attached to taking the exam several times, but students are expected to pass the examination by the second year in residence in order to begin more specialized study leading to research work.

Before passing the qualifying exam, students should enroll three beginning 200 level (or 100 level) math courses each term and may substitute TIME for one course to prepare for the exam.

The exam is given at the beginning of each term. It 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 10 points. In order to pass in each subject, a student must obtain at least 20 of the available 30 points in that subject. Students are considered to have passed the qualifying exam when they have passed in all six subjects (120 of 180 points) in one sitting, or they have passed at least four subjects in one sitting and obtained an A or A- grade in the basic graduate courses corresponding to the subject(s) not passed. Students are expected take the suggested course(s) at the first opportunity.

Once the qualifying exam has been passed, students no longer need to take math courses for a letter grade and may elect to receive the grade (EXC) excused. Students should inform the instructor at the beginning of the term if they are electing to take (EXC) as a grade.

The questions on the qualifying exam aim to test a student's ability to solve concrete problems by identifying and applying important theorems. The questions should not require great ingenuity. In any given year, the exam may not cover every topic on the syllabus, but it should cover a broadly representative set of topics and over time all topics should be examined.

The syllabus is on a seperate page.

Every student, whether or not they have outside support, is required to have two terms of classroom teaching experience as preparation for their likely future role as teachers.

For those students without outside support: In the first year, students automatically receive an offer of the full departmental stipend with no obligation to teach. The department offers second, third and fourth year students half the departmental stipend without an obligation to teach, and students are required to teach to cover the other half of the stipend. However, the department caps the amount of teaching students are required to do: If a student teaches one section of calculus, or the equivalent, the department will supplement the stipend up to the full stipend. Fifth year students are expected to teach to cover the full stipend. Again, the department caps the teaching required: If a student teaches two sections of calculus, or the equivalent, and the pay does not support the annual stipend, the department will supplement the stipend up to that year's level. Students may arrange to teach twice in an earlier year to reduce teaching in the final year.

Equivalence is based on what the department perceives to be the time commitment of a teaching job. We consider the following to be equivalent to one section of calculus:

One tutorial (teaching) Two CA sections of the core Two CA sections of applied math Two jobs at the Math Question Center Two GCA courses in our department

We consider the following to be equivalent to two sections of calculus:

Two tutorials (teaching) Three CA sections of the core Three CA sections of applied math Four GCA courses in our department One section of calculus (teaching) and one CA section of core/applied math One section of calculus (teaching) and two GCA courses in our department

There are other possibilities, which will be judged on an ad hoc basis, but this list gives an idea of what is expected.

Students with outside support available in their first year are expected to take it in their first and second year. Students who choose to defer outside fellowship support to a later year will need to teach twice in an earlier year to cover their full stipend in the deferral year. For instance, a student with one remaining year of outside fellowship support who chooses not to take the funding in their fourth year, will need to teach two sections of calculus or the equivalent in their fourth year in order to receive the full departmental stipend. Similarly, a student with two remaining years of outside fellowship support who chooses not to take the funding in their third year, will need to teach two sections of calculus or the equivalent in their third year in order to get the full departmental stipend.

The department's teaching staff helps students find appropriate teaching jobs. While we would like to accommodate student's teaching preferences, the teaching staff works under many constraints. It is necessary to balance student preferences with those of other graduate students and the needs of the department. Students who have done a good and conscientious job on previous teaching assignments are more likely to get their preference in subsequent years. Students need to make teaching plans well in advance and in consultation with the teaching staff. Students should not change their teaching commitments at short notice.

Students making satisfactory academic progress may teach beyond the minimal requirement outlined above. Students who are not required to teach as part of their funding package may teach if jobs are available. Pay for additional teaching will supplement other sources of funding. Students must take the initiative to find additional jobs. In assigning teaching positions, first preference will be given to those required to teach for funding. After that, positions will be allocated to those with the strongest teaching credentials.

Below are some examples of 10-month stipends for 2013/2014 academic year:

from Ravi Vakil.

This part of the page was put together by Stephanie Yang.

Writing papers and submitting them

• Kleiman's notes on mathematical writing

Applying for jobs

• Joe's guide to job searching and job talks
• AMS guide to job searching
• Tex package

Writing a CV

• Example 1 and template (courtesy of Hank Zee)
• Example 2 and template with style file (courtesy of Hank Zee)

Writing a Cover Sheet

• The AMS guide to cover sheets, including templates in all formats

Noam D. Elkies	Professor of Mathematics	Number theory, computation, classical algebraic geometry, music.
Dennis Gaitsgory	Professor of Mathematics	Geometric aspects of representation theory.
Robin Gottlieb	Professor in the Teaching of Mathematics
Benedict H. Gross	George Vasmer Leverett Professor of Mathematics	Algebraic number theory, Diophantine geometry, modular forms.
Joseph Harris	Higgins Professor of Mathematics	Algebraic geometry.
Michael J. Hopkins	Professor of Mathematics	Algebraic topology.
Arthur Jaffe	Landon T Clay Professor of Mathematics and Theoretical Science	Analysis, probability, symmetry, and geometry related to quantum and statistical physics
Mark Kisin	Professor of Mathematics	Number theory and arithmetic geometry.
Peter Kronheimer	William Caspar Graustein Professor of Mathematics	Topology, differential and algebraic geometry, and their applications.
Jacob Lurie	Professor of Mathematics	Algebraic geometry, algebraic topology, and higher category theory.
Barry Mazur	Gerhard Gade University Professor	Number theory, automorphic forms and related issues in algebraic geometry.
Curtis T. McMullen	Maria Moors Cabot Professor of the Natural Sciences	Riemann surfaces, complex dynamics, hyperbolic geometry.
Martin Nowak	Professor of Mathematics and Biology	Mathematical biology, evolutionary dynamics, infectious diseases, cancer genetics, game theory, language.
Wilfried Schmid	Dwight Parker Robinson Professor of Mathematics	Lie groups, representation theory, complex differential geometry.
Yum-Tong Siu	William Elwood Byerly Professor of Mathematics	Several complex variables.
Shlomo Sternberg	George Putnam Professor of Pure and Applied Mathematics	Differential geometry, differential equations, Lie groups and algebras, mathematical physics.
Clifford Taubes	William Petschek Professor of Mathematics	Nonlinear partial differential equations and applications to topology, geometry, and mathematical physics.
Horng-Tzer Yau	Professor of Mathematics	Probability theory, quantum dynamics, differential equations, and nonequilibrium physics.
Shing-Tung Yau	William Caspar Graustein Professor of Mathematics	Differential geometry, partial differential equations, topology, and mathematical physics,

Junior and visiting faculty interests comprise a diverse and important addition to the department. As these appointments vary in length from one term, on the part of visitors, to three-year appointments as a Benjamin Pierce Lecturer on Mathematics, Assistant Professor of Mathematics, they will be listed annually in the courses of instruction.

The request must be sent in writing. They may reply with e-mail if you include your address, and will state the cost. Upon receipt of the requested dollar amount, they will send you a copy.

Titles and names of dissertations written since 2001 are listed on this page.