Graduate Schools and Fellowships in Mathematics
Choosing the Right Grad School
Your choice of a graduate school is a major step in your career as a mathematician.
Below are some criteria you may consider for choosing the right school for you. Not
all the criteria apply to everybody. We strongly urge you to seek advice from
several faculty members familiar with you and/or your field of interest early in
the fall semester of your senior year to try to sort out what may be the best
``fit'' between you and a graduate school.
A thesis (dissertation) advisor plays a very important role in the student's graduate education. After all, a dissertation generally consists of making a new advance, solving an unsolved problem. And since the problem is not yet solved, the advisor, presumably, has no solution for it either. Therefore it takes a good mathematician, with sound intuition and deep insights to help his/her advisee choose a problem that will, on the one hand, not wind up being so hard that a student can't solve it or parts of it, nor, on the other hand, so easy that a student wouldn't get to learn a lot of beautiful mathematics on his/her way to solving the problem.
Most math departments do not have the resources to excel in all branches
of mathematics. You should ascertain that a university you are planning to apply to
has top-quality tenured faculty members pursuing research in your potential field
of specialization. But you don't necessarily have to go to a leading grad school to
get a good advisor. There are a number of mathematics departments in this country
which may not be at the top of the pack overall, but which have one or more
top-quality mathematicians who can be excellent thesis advisors. Your academic
advisor at Harvard can help you sort out which departments are strong in which
Breadth and Depth
As important as it is to choose a school with strong reputation in your field of
interest, it is also important to balance this criterion with considerations about
the overall breadth of the department. It is common for one's interests
to change as learning progresses, and you want to leave yourself the freedom to
switch fields or advisors without leaving your chosen graduate department.
The Strength of Your Peers
The quality of other graduate students in the program is also very important. During the first few years of graduate study you will learn much from other graduate students, so it is very helpful to have talented peers.
On the other hand, going to the most competitive school may not be the best
decision for everybody. You can sometimes get more time and attention from faculty
if you are at the top of a weaker pack, rather than near the bottom of a stronger
one. Again, your academic advisor can help you decide what would be best balance in your case.
The Student-to-Faculty Ratio
An equally important factor is the student-to-faculty ratio. In some
universities, it is not uncommon to have 10 to 15 grad students working
under one advisor. Obviously under these conditions, some students won't
get the attention they require.
Visiting the Schools
When making a final decision on graduate schools, visit all your serious choices. Talk to faculty members about their research interests and the number of dissertations they are advising. Ask where their former students are employed.
A good source of information are the graduate students currently enrolled in the
program. Talk to them about accessibility and reputation of potential advisors in
your field of interest. Find out what is the average number of years students take
to complete their dissertation, and (very important) what is the dropout ratio.
Look into the housing options (very rarely do grad students live in the dormitories
for more than a year, if at all) and the teaching load of teaching assistants. Try
to decide if you will be comfortable spending some number of years at that
university or in that town.
Finally, you should talk to your academic advisor, the Director of Undergraduate Studies, and
other faculty you know well. They can evaluate your choices, give you the
latest information about the departments you are considering, and help you
decide what is best for you.
Applying to Harvard
The Harvard Math Department encourages its own undergraduates to go elsewhere for
graduate study because it is a good idea for a student to get to know other
mathematicians, to be exposed to alternative tastes and styles of doing
The Admissions Process
An important criterion used by many admissions committees is the content of faculty recommendations. Graduate schools look closely at evidence not only of mathematical ability, but of motivation and tenacity. In contrast to college admissions, extracurricular activities and non-academic character traits are not given much weight. Usually two or three recommendations are required, and you should be making the effort throughout your undergraduate career to let a few faculty members know you well (e.g., by dropping by their office hours, inviting them to the semi-annual faculty dinners held at your House).
If you are writing a thesis, you might ask your thesis advisor for a letter of recommendation. Working on your thesis during the summer and fall of your senior year with your thesis advisor provides an opportunity for him or her to get to know you better.
If you took a course from a professor and did relatively well, you may want to ask
him or her to write a letter of recommendation right away. He or she may not
remember all the details of your performance a year or two later. The recommendation
can later be fine tuned to address specific graduate schools or fellowships. If you are
thinking of taking some time off after college and applying later, you may still
want to get your recommendations written while you are in school. Remember that
faculty members often take sabbaticals, change universities, etc. Ask your
recommenders to send copies of your recommendations to your undergraduate house to be included in your personal folder; they may be useful later on. To place a copy of your
recommendation in your undergraduate house file you need to obtain a form from your House's
Senior Tutor. Harvard will keep these files indefinitely, and will mail the
letters you want to graduate schools or employers at your request.
You will be required to write one or two application essays. Typically you
will need to describe your academic background, your achievements to date,
what experiences led you to want to get a Ph.D. in math, and what areas of
research interest you most. Those essays give you an opportunity to
explain away some bad grades you may have had, to demonstrate your
new-found love of math and resolve to succeed, and convince the admissions
committee that you not only have the requisite intellect, but
that you are going to persevere through some tough moments in your career
and finish your dissertation. They want to make sure that you aren't
going to grad school just because you could think of nothing better to do,
or because you missed the LSAT deadline. You don't have to feel that you should
know your dissertation topic by your senior year in college. You should
merely demonstrate that you really are interested in mathematics and are
relatively focused and determined.
Grades and Test Scores
Besides recommendations and essays, other criteria for admissions
include grades and scores on the Graduate Record Examination (GRE).
Most selective math departments do not put too much weight on the
differences between good and great scores, but having poor grades
in math courses or poor GRE scores can hurt your chances of admissions.
Many schools look at your transcript to see evidence of substantial
exposure to serious mathematics (e.g. some graduate level courses)
and are quite understanding about some poor grades.
Taking the GRE
Most universities require applicants to take two parts of the GRE -- the general and the subject tests. The general part is similar to the SAT. The questions on the subject test in Mathematics may be quite different from the math most students learn at Harvard, and you are well advised to look over the material tested ahead of time. Even if many of the problems may seem easy, you have less than a minute per question, and if you have to derive everything, you won't finish the test.
GRE, at least in the usual ``paper and pen'' format, is offered only 3 times a year: in October, December and April. Although you can take both the general and the subject part on the same day, most students prefer not to. Therefore many take at least the general part of the GRE during their junior year. In any case, it's a good idea to get the GRE out of the way soon. When you start working on your thesis and filling out applications for grad schools and fellowships, worrying about the GRE will be the last thing you will want to do.
Also keep in mind that you have to register to take the GRE more than a month in advance; if you want to take the test at a place somewhere near Cambridge, you should get moving several months before the test date. For example, if you plan to take GRE in October, and would prefer to take it in Boston rather than, say, in Swampscott (remember, you have to be there at 8am) you should register as early as July. GRE information pamphlets are available from OCS, from the GSAS admissions office at Byerly Hall. You can also obtain them from ETS web page, http://www.ets.org/index.html or from http://www.gre.org.
Note that if you apply for National Science Foundartion (NSF)
Graduate Fellowships, then the NSF will actually pay for your
GRE test - provided you take it in December. See the NSF application for more
The deadlines for graduate school applications range from early
December to early January. Most schools usually require you to complete
your application folder by January 1st or 15th.
The deadlines for fellowship applications start as early as October.
Fellowships and Financial Aid
Unlike what many people think, you don't need to pay to go to graduate
school in mathematics. Grad students are usually paid (albeit not much) to
study. There are several ways post-graduate education in mathematics is financed.
A few students are able to win national fellowships. The national fellowships are awarded by various government and private foundations, and some pay rather large stipends plus tuition for the first three to five years of graduate school. Currently, we are aware of support being offered by the following organizations:
Flyers put out by some of these fellowships are posted on the undergraduate bulletin boards-one is opposite room 320 and another is opposite room 503.
The Office of Career Services (OCS) Web site offers useful information for applying to graduate school and for finding sources of funding. Please review the Fellowships section on the OCS Web site at http://www.ocs.fas.harvard.edu/students/fellowships.htm. Also, OCS has a library with information on fellowships, and its staff can help you enormously in your pursuit.
The Harvard Graduate School of Arts and Sciences (GSAS) offers fellowship information online at http://gsas.harvard.edu. The online publication ``Financing Graduate Studies'' is available to download, and the interactive database ``Graduate Guide to Grants'' will allow you to search for specific grants and provides contact information, application guidelines and restrictions, and deadlines.
Finally, check the Web sites of the universities where you plan to apply. Like GSAS, many graduate schools offer graduate funding options specific to their institutions.
Generally, undergraduates are urged to try their luck in all
nationwide fellowships. Harvard students have been very successful in
winning these awards. Some fellowships have early deadlines (as early
as October) and some pay more attention to the GRE scores and grades
than university admissions offices.
A university itself may also offer a number of fellowships for students who will accept its offers of admission. These can be as lucrative as the nation-wide fellowships, and you generally don't need any separate application - your application for admission automatically enters you into the competition.
Students who do not get fellowships usually receive teaching assistantships. Those generally carry a tuition waiver and a stipend that is sufficient for living expenses in exchange for teaching, grading, or assisting in low-level math courses. (Many departments won't let first-year students teach. These often pay incoming students a stipend, and have them begin teaching in their second year). Some students can get research assistantships which let
them stop teaching and concentrate exclusively on research.
Few students enter graduate school in mathematics for an M.A. However, if you think that this is something you may want to do, you should discuss your situation with your academic advisor or the Director of Undergraduate Studies. In many schools, masters degrees are awarded only on the route to a Ph.D.; also, students who drop out from a Ph.D. program after a year or two are often able to get the masters degree on their way out (you should check the policies of individual departments). If you are applying for a masters program only, you may have less (in some universities, significantly less) chance of getting financial aid. Some universities do not admit students who want to get a terminal M.A. degree.
Harvard's A.B.-A.M. Degree Program
Harvard students with Advanced Standing may wish to apply for the
A.B.-A.M. degree. These students must meet both the academic and
course requirements for each of these two degrees. A given course can
be counted for only one of the two degrees, i.e., one course cannot
meet the requirements for the A.B. degree and then be counted again
for the A.M. degree. In addition to the course requirements, any
candidate for the A.M. degree in Mathematics at Harvard must take
a special language exam to demonstrate the ability to read
mathematics in either French, German or Russian. These tests are
administered by the department only once a semester. For more
information on the language exams please contact the Graduate Program
Coordinator. Any undergraduate
who wishes to apply for the A.B.-A.M. degree must file an
application form for the graduate program in mathematics just as any
other student files for graduate work at Harvard. For information on
the degree, contact the office of Advanced Standing.
There are many fellowships and scholarships for study abroad. Information about them can be obtained on the Office of Career Services (OCS) website, under ``Global Opportunities'', or from your House Fellowships Tutor. Particularly useful are the Guide to Grants and the Guide to Study Abroad, published annually by OCS. Some of the math/science oriented fellowships are the:
In the past, math majors from Harvard have also been particularly successful in competing for the Marshall Scholarships, and for the Fulbright Grants, especially for study in Israel (it seems that at least one grant a year almost always goes for a math student to study Logic at the Hebrew University of Jerusalem, and Israel is one of the very few countries where one does not have to know the native language in order to receive a Fulbright grant to study there). Application deadlines for some of the these fellowships start as early as September.
Further Information and Advice
For further information and advise about graduate programs and fellowships in mathematics, please talk to the Director of Undergraduate Studies, Professor Peter Kronheimer, your faculty advisor or the House Fellowship Advisor. They can greatly assist you in determining what recommendations and information outlined here applies in your case, and may help you get more information. And watch for advertisements from fellowships and graduate programs on the undergraduate bulletin boards.