News
 The online application for 2016 admissions to the Department of Mathematics will be available on the Graduate School of Arts and Sciences Web site in late August 2015. GSAS Apply.
The deadline for fall 2016 applications is December 15, 2015. All applications and supporting materials must be submitted online before 5:00 p.m. Eastern Standard Time. Late applications are not accepted. Please check the Harvard Graduate School of Arts and Sciences (GSAS) website for Prospective Students for more information on admissions and financial aid.
Admissions
To apply for admissions and financial aid, or for additional information on admissions requirements for the PhD program in pure 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 Larissa Kennedy (617) 4955377 Office 333i EMail: lkennedy@math 
Preparing the Application
The statement of purpose for graduate applications is carefully weighted by the admissions committee. The applicant's statement should convince the committee that he or she is able to communicate effectively and with a deep understanding of mathematics. It is not intended to be a biographical sketch or a reflection on one's decision to enter the field.
Three letters of recommendation are required. Letter writers should be faculty or others qualified to evaluate the applicant's potential for graduate study in mathematics. The letters must be submitted online and by the application deadline.
Applicants should include any research papers, publications, and other original works they would like to have evaluated by the admissions committee.
The department requires all applicants to submit GRE Mathematics Subject Test scores. Applicants should check on the ETS website for test dates in their area to insure the scores will be submitted before the application deadline. While the admissions committee reviews all applications submitted before the deadline, missing math subject test scores provide 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 applicants to upload an electronic copy of undergraduate transcripts. Hard copies of official transcripts are not required at the time of application.
PhD Program in Pure Mathematics
The department does not grant a terminal Master's degree, but the Master's 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 Master's study may make is to better prepare students for the Qualifying Exam.
All graduate students are admitted to begin their studies in the fall term. The department plans on an entering class of about twelve students. Since the admissions committee receives in excess of two hundred applications, the competition is keen.
Funding Graduate Study
Applicants are urged to apply for all funding available to them. If no outside funding is available to the applicant, financial aid in the form of scholarships, research assistantships, and teaching fellowships is available. In general, students who do not have 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 in the fifth year.
The statement of purpose for graduate applications is carefully weighted by the admissions committee. The applicant's statement should convince the committee that he or she is able to communicate effectively and with a deep understanding of mathematics. It is not intended to be a biographical sketch or a reflection on one's decision to enter the field.
Three letters of recommendation are required. Letter writers should be faculty or others qualified to evaluate the applicant's potential for graduate study in mathematics. The letters must be submitted online and by the application deadline.
Applicants should include any research papers, publications, and other original works they would like to have evaluated by the admissions committee.
The department requires all applicants to submit GRE Mathematics Subject Test scores. Applicants should check on the ETS website for test dates in their area to insure the scores will be submitted before the application deadline. While the admissions committee reviews all applications submitted before the deadline, missing math subject test scores provide 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 applicants to upload an electronic copy of undergraduate transcripts. Hard copies of official transcripts are not required at the time of application.
PhD Program in Pure Mathematics
The department does not grant a terminal Master's degree, but the Master's 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 Master's study may make is to better prepare students for the Qualifying Exam.
All graduate students are admitted to begin their studies in the fall term. The department plans on an entering class of about twelve students. Since the admissions committee receives in excess of two hundred applications, the competition is keen.
Funding Graduate Study
Applicants are urged to apply for all funding available to them. If no outside funding is available to the applicant, financial aid in the form of scholarships, research assistantships, and teaching fellowships is available. In general, students who do not have 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 in the fifth year.
The department strongly recommends applicants to seek out and apply for all sources of financing available to them for graduate study. Recommended sources for funding US graduate students are 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. 
Harvard's School of Engineering and Applied Sciences 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 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 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 graduate 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.
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 twohour, 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.
Nonnative 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.
 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 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, 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 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 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.
Past 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
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 fifth 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 halfyear stipend, the department will supplement the teaching salary.
Work restrictions on student visas may limit international student teaching.
Please check with the International Office or the Graduate Program Administrator
before finalizing your teaching plans.
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 fifth 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 halfyear stipend, the department will supplement the teaching salary.
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:
The following are considered to be equivalent to two sections of calculus:

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

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.
Junior and visiting faculty areas of interest provide an
additional resources for graduate students. As these
appointments vary in length from oneterm to threeyear
appointments, the names of junior faculty are listed in
the courses of instruction.
In addition the department hosts a number of weekly seminars, and students have access to resources and seminars at the Harvard Center of Mathematical Sciences and Application (CMAS).
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
Requesting Letters of Recommendation
Guidelines for requestions recommendations by Ravi Vakil.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
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 18005213042. 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 openaccess 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. 
Birkhoff Library

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