The PhD program of the Harvard Department of Mathematics is designed to help motivated students develop their understanding and enjoyment of mathematics. 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:

Students are expected to take the initiative in pacing themselves 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 graduate 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. Most students complete the PhD in four or five years. Please review the program requirements timeline.

There is no prescribed set of course requirements, but 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.

The department gives the qualifying examination at the beginning of the fall and spring terms. The qualifying examination covers algebra, algebraic geometry, algebraic topology, complex analysis, differential geometry, and real analysis. Students are required to take the exam at the beginning of the first term. More details about the qualifying exams can be found here.

Students are expected pass the qualifying exam before the end of their second year. After passing the qualifying exam students are expected to find a PhD dissertation advisor.

The minor thesis is complementary to the qualifying exam. In the course of mathematical research, students will inevitably encounter areas in which they have gaps of knowledge. The minor thesis is an exercise in confronting those gaps to learn what is necessary to understand a specific area of math. Students choose a topic outside their area of expertise and, working independently, learns it well and produces a written exposition of the subject.

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

The minor thesis must be completed before the start of the third year in residence.

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, students are required to demonstrate the ability to read mathematics in French, German, or Russian by passing a two-hour, written language examination. Students are asked to translate one page of mathematics into English with the help of a dictionary. Students may request to substitute the Italian language exam if it is relevent to their area of mathematics. The language requirement should be fulfilled by the end of the second year. For more information on the graduate program requirements, a timeline can be viewed at here.

Non-native English speakers who have received a Bachelor's degree in mathematics from an institution where classes are taught in a language other than English may request to waive the language requirement.

Upon completion of the language exam and eight upper level math courses, students can apply for a continuing Master's Degree.

Most research mathematicians are also university teachers. In preparation for this role, all students are required to participate in the department's teaching apprenticeship program and to complete two semesters of classroom teaching experience, usually as a teaching fellow. During the teaching apprenticeship students are paired with a member of the department's teaching staff. Students attend some of the advisor's classes and then prepare (with help) and present their own class, which will be videotaped. Apprentices will receive feedback both from the advisor 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 (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. A course head (a member of the department teaching staff) coordinates the various classes following the same syllabus and is available to advise teaching fellows. Other teaching options are available: graduate course assistantships for advanced math courses and tutorials for advanced undergraduate math condentrators.

How students proceed through the second and third stages of the program varies considerably among individuals. While preparing for the qualifying examination or immediately after, students should begin taking more advanced courses to help with choosing a field of specialization. Unless prepared to work independently, students should choose a field that falls within the interests of a member of the faculty who is willing to serve as dissertation advisor. Members of the faculty vary in the way that they go about dissertation supervision; some faculty members expect more initiative and independence than others, and some vary in how busy they are with current advisees. Students should consider their own advising needs as well as the faculty member's field when choosing an advisor. Students must take the initiative to ask a professor if she or he will act as a dissertation advisor. Students having difficulty deciding under whom to work, may want to spend a term reading under the direction of two or more faculty members simultaneously. The sooner students choose an advisor, the sooner they can begin reaearch. Students should have a provisional advisor by the second year.

It is important to keep in mind that there is no technique for teaching students to have ideas. All that faculty can do is to provide an ambiance in which one's nascent abilities and insights can blossom. PhD dissertations vary enormously in quality, from hard exercises to highly original advances. Many good research mathematicians begin very slowly, and their dissertations and first few papers could be of minor interest. 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 students' knowledge in mathematics. While some students are able to pass the qualifying exam in one try, 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.

Students are required to take the qualifying examination at the 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 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.

The department runs tutorials and offers several introductory graduate courses (e.g. Math 212a, 213a, 230a, 231a, and 232a) to help students acquire the necessary broad basic background in mathematics to pass the exam.

The 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 10 points. In order to pass each subject, students must obtain at least 20 of the 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 recommended course(s) at the first opportunity.

Once students have passed the qualifying exam, they 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 elect 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.

All students are required to have two terms of classroom teaching experience as preparation for their likely future role as teachers.

Students with no outside support: In the first year, students automatically receive an offer of the full departmental fellowship with no obligation to teach. The department offers second, third and fourth year students half the departmental stipend without an obligation to teach, but students are required to teach to cover the other half of the stipend. The department caps the amount of teaching students are required to do, so if a student teaches one section of calculus, or the equivalent, and the teaching pay does not support the half-year stipend, the department will supplement the teaching salary. Fifth year students are expected to teach to cover the full stipend. 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 stipend level. Students may arrange to teach twice in an earlier year to reduce teaching in the final year.

Teaching equivalence is based on the time committement the department perceives to be involved in teaching jobs. The following are considered be equivalent to one section of calculus:

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

The following are considered to be equivalent to two sections of calculus:

Two tutorials (teaching) Three CA sections for Gen Ed Three CA sections of applied math Four GCA courses in math 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 teaching possibilities, which will be allowed on an ad hoc basis, but this list gives an idea of what is expected.

Students offered outside support: If students are offered outside support, they are expected to take the funding. For students allowed to defer outside fellowship support, the department expects students to use the funding in the first and second years. Students who choose to defer funding in the third or fourth year will need to teach twice one of the deferral years to cover the full stipend. For instance, a student who chooses to defer funding in their third year, will need to teach two sections of calculus or the equivalent in one of the deferal years in order to receive the full departmental stipend.

The department's teaching staff helps students find appropriate teaching jobs. While student's teaching preferences are considered, the teaching staff works under many constraints, and it is necessary to balance student preferences with the needs of the department. Students who have done a 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 on short notice or plan to travel during a term when they are teaching.

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 is given to those required to teach for funding. After that, positions will be allocated to those with the strongest teaching credentials.

The department provides continuing students with summer support. In addition, a number of teaching opportunities are available during the summer.

Professional Development

Mathematical Job Search Sites

AMS Employment site	Mathjobs AMS Employment Services
EIMS Employment Information in the Mathematical Sciences	NSF Mathematical Scienses Postdoctoral Research Fellowships

Senior Faculty Research Interests

How to obtain copies of past PhD theses

PhD dissertations are listed in descending order by year, author, and title on this page. All scholars can order copies of most Harvard dissertations from 1982 to the present by contacting UMI/ProQuest at 1-800-521-3042. Permission of the author is usually required to copy theses within the last five years. Most PhD dissertations submitted from March 2012 forward are available online in DASH, Harvard's central open-access repository. Access to Harvard theses/dissertations that are not available through ProQuest depends on the school of origin and its associated library. Check the Hollis catalogue to see where a thesis is housed. Harvard affiliates with IDs and PINs can access the full text of most Harvard PhD theses since 1990 from the Proquest database.