News
 September 8, 2013: Quals of Fall 2013 and solutions are posted
Admissions
To apply for admissions and financial aid, or for additional information on requirements for the PhD program in mathematics, please go to the appropriate Graduate School of Arts and Sciences web site listed below. All other inquires may be directed to the Graduate Program Administrator of the Mathematics Department.
Here are the addresses and links:
Admissions
Graduate School of Arts and Sciences Harvard University Phone: (617) 4955396 EMail: GSAS Admissions 
Mathematics Department
Graduate Program Administrator Susan Gilbert (617) 4965211 EMail: sgilbert@math 
The application deadline for fall 2015 admission will be available in the
late fall of 2014. Please check the
Harvard Graduate School of Arts and Sciences (GSAS) website
for Prospective Students for more information on the application process.
Preparing for the Application
The statement of purpose for graduate applications is very different from writing a personal essay. It is neither a biographical sketch nor a reflection on your decisions to enter the field. It should be written for a particular audience: The admissions committee, which consists of faculty members in the department. Your statement should convince the committee that you are able to communicate effectively and with a deep understanding of mathematics.
Request three letters of recommendation from faculty or others qualified to evaluate your potential for graduate study. The letters must be submitted online and by the application deadline.
The Department requires all applicants to submit GRE Mathematics Subject Test scores. Applicants should check ETS test dates to insure the scores will be submitted before the application deadline. While the admissions committee reviews all applications submitted by the deadline, missing math subject test scores are one less data point available to evaluate the application. Depending on the applicant pool and the strength of the application materials, the missing subject test scores may put the application at a disadvantage.
The Graduate School requires scores to be submitted for the General Test of the Graduate Record Examinations (GRE).
Applicants whose native language is other than English and who do not hold the equivalent of a US Bachelor degree from an institution at which English is the language of instruction must submit scores from the Internet based test (IBT) of the Test of English as a Foreign Language (TOEFL).
The Graduate School requires that applicants upload a copy of their undergraduate transcripts. Additionally, applicants must submit a hard copy of the official university/college transcript bearing the seal of the institution and signature of the registrar or appropriate official.
PhD Program in Pure Mathematics
The department does not grant a terminal Master's, but a Master's degree can be obtained "on the way" to the PhD by fulfilling certain course and language exam requirements.
In general, there is no transfer status application to the Graduate School of Arts and Sciences or to the Department of Mathematics. No formal credit is given for an MSc or MA earned elsewhere. All applicants are considered to be applying as first year graduate students. The only difference that your previous degree will make is that you may be in a better position to prepare for our Qualifying Exam, which is required of all students.
All graduate students are admitted to begin their studies in the fall term. We plan on an entering class of about ten to twelve students. Since we normally receive over two hundred applications, the competition is keen.
Funding Graduate Study
Applicants are encouraged to apply for outside funding, but financial aid in the form of scholarships and/or teaching fellowships is available. In general, students without outside support will get scholarship support in their first year, but students are required to act as a teaching fellow for one half course (i.e. for a one term course) in their second through fourth years and for two half courses if they stay for a fifth year.
The department provides continuing students with summer support in the amount of $4,080. In addition, a number of teaching opportunities are available during the summer.
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 threeyear appointments as a Benjamin Pierce
Lecturer on Mathematics, Assistant Professor
of Mathematics, they will be listed annually in
the courses of instruction.
The statement of purpose for graduate applications is very different from writing a personal essay. It is neither a biographical sketch nor a reflection on your decisions to enter the field. It should be written for a particular audience: The admissions committee, which consists of faculty members in the department. Your statement should convince the committee that you are able to communicate effectively and with a deep understanding of mathematics.
Request three letters of recommendation from faculty or others qualified to evaluate your potential for graduate study. The letters must be submitted online and by the application deadline.
The Department requires all applicants to submit GRE Mathematics Subject Test scores. Applicants should check ETS test dates to insure the scores will be submitted before the application deadline. While the admissions committee reviews all applications submitted by the deadline, missing math subject test scores are one less data point available to evaluate the application. Depending on the applicant pool and the strength of the application materials, the missing subject test scores may put the application at a disadvantage.
The Graduate School requires scores to be submitted for the General Test of the Graduate Record Examinations (GRE).
Applicants whose native language is other than English and who do not hold the equivalent of a US Bachelor degree from an institution at which English is the language of instruction must submit scores from the Internet based test (IBT) of the Test of English as a Foreign Language (TOEFL).
The Graduate School requires that applicants upload a copy of their undergraduate transcripts. Additionally, applicants must submit a hard copy of the official university/college transcript bearing the seal of the institution and signature of the registrar or appropriate official.
PhD Program in Pure Mathematics
The department does not grant a terminal Master's, but a Master's degree can be obtained "on the way" to the PhD by fulfilling certain course and language exam requirements.
In general, there is no transfer status application to the Graduate School of Arts and Sciences or to the Department of Mathematics. No formal credit is given for an MSc or MA earned elsewhere. All applicants are considered to be applying as first year graduate students. The only difference that your previous degree will make is that you may be in a better position to prepare for our Qualifying Exam, which is required of all students.
All graduate students are admitted to begin their studies in the fall term. We plan on an entering class of about ten to twelve students. Since we normally receive over two hundred applications, the competition is keen.
Funding Graduate Study
Applicants are encouraged to apply for outside funding, but financial aid in the form of scholarships and/or teaching fellowships is available. In general, students without outside support will get scholarship support in their first year, but students are required to act as a teaching fellow for one half course (i.e. for a one term course) in their second through fourth years and for two half courses if they stay for a fifth year.
The department strongly recommends applicants to seek out and apply for all sources of financing for graduate study such as NSF Graduate Fellowships and NDSEG Fellowships. Applicants from the UK are urged to also apply for the Kennedy fellowships and applicants from UK, New Zealand, Canada and Australia, for Knox fellowships. International students may apply for the Fullbright IIE or any home country fellowships available for study abroad. 
A list of courses offered by the Mathematics department can be found here. 
Masters and PhD Degrees in Applied Mathematics The Harvard University School of Engineering and Applied Sciences (SEAS) offers programs for both the Master's degree and the PhD degree in Applied Mathematics. Please visit the SEAS Web site for more information on degrees in applied mathematics. 
School of Engineering and Applied Sciences
http://www.seas.harvard.edu 
Guide To Graduate Study
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 everburgeoning
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 halfcourses) 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 twohour,
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.
 Acquiring a broad basic knowledge of mathematics on which to build a future mathematical culture and more detailed knowledge of a field of specialization.
 Choosing a field of specialization within mathematics and obtaining enough knowledge of this specialized field to arrive at the point of current thinking.
 Making a first original contribution to mathematics within this chosen special area.
The Qualifying Exam
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,
threehour 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 Qualifying Exam Syllabus
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.
Some Old Qualifying Exams
Some old departmental qualifying exams are available here (all links are PDF's)

Some PDF files of questions arranged by topics. 
Collected by Danny Calegari and Tom Coates
source. 
Teaching Requirements
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.
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:
We consider the following to be equivalent to two sections of calculus:

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

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 10month stipends for 2013/2014 academic year:
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 10month stipends for 2013/2014 academic year:
Student A in year 1 students are not required to teach and receive
 
Student B in year 2 teaches one section of calculus and receives:
 
Student C in year 3 CAs two sections of the core and receives:
 
Student D in year 3 GCAs two math courses and receives:
 
Student E in year 4 without outside support decides not to teach and
receives:
 
Student F in year 5 teaches two sections of calculus
and receives:

The department provides continuing students with summer support in the amount of $4,080. In addition, a number of teaching opportunities are available during the summer.
Professional Development
This part of the page was put together by Stephanie Yang.
Writing papers and submitting them
Applying for jobs
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
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
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. 
YumTong 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. 
Hugh Woodin  Professor of Philosophy and of Mathematics  Set theory, determinacy, and strong axioms of infinity. 
HorngTzer Yau  Professor of Mathematics  Probability theory, quantum dynamics, differential equations, and nonequilibrium physics. 
ShingTung Yau  William Caspar Graustein Professor of Mathematics  Differential geometry, partial differential equations, topology, and mathematical physics, 
How to obtain copies of past PhD theses
In general, past PhD theses from any university can be obtained from:  Past Harvard PhD theses may be obtained directly from:  


The request must be sent in writing. They may reply with email 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.
Titles and names of dissertations written since 2001 are listed on this page.
Birkhoff Library

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