Who is this advice for?
It is for Polytechnic undergraduate math majors who want to pursue a Ph.D. in mathematics or computer science and a career in mathematical research.
What is it like going through a Ph.D. program in mathematics?
A good Ph.D. program in mathematics is an extremely rigorous and challenging experience. You will have to take two years of difficult graduate courses and at the same time pass one or more special examinations (written or oral, depending on the school) that test your mathematical knowledge and skills. After that, you will work for 2-4 years under the supervision of a Ph.D. advisor on a thesis (also known as a dissertation), in which you are expected to develop new mathematical ideas and prove new and previously unknown mathematical theorems.
Should I get a Ph.D.?
If you need to ask, the answer is no. The only people who should pursue a Ph.D. in mathematics are those who are so passionate about mathematics that they cannot imagine doing anything else. While you are a Ph.D. student, you should be prepared to live, breathe, sleep, eat mathematics and do nothing else. Afterward, there is a lot of competition for the more attractive jobs that require a Ph.D. in mathematics. If you want to become a successful research mathematician, you must be willing to live almost anywhere in the U.S. or even abroad. If your love of mathematics falls short of this, then we recommend that you apply your talents in mathematics in other ways.
How can I afford the cost of getting a Ph.D.?
In most Ph.D. programs, the students are fully supported by a full tuition scholarship plus a stipend that should provide enough money to live very modestly. You may be expected to do a certain amount of teaching for the stipend. No one should ever pay any tuition or take out a loan to get a Ph.D. in mathematics.
I'd like to stay at Polytechnic for my Ph.D. program.
There is a widely accepted principle in academia that students should not stay at their undergraduate institution for their Ph.D. studies. We strongly agree with this. In our experience very few students succeed when they stay at the same school for their undergraduate and graduate programs. It is difficult to explain exactly why this is so; it appears to have something to do with students feeling too comfortable in a familiar environment and not struggling hard enough to prove themselves.
So what is your advice about choosing a Ph.D. program and advisor?
After you get your Ph.D., when you meet others in your field, the first three questions you'll be asked are "Where did you get your Ph.D.?" "Who was your advisor?", and "What is your thesis about?". People's initial view of you will be based on your answers to these questions. For this reason, the most important factors in the value of the Ph.D. that you will ultimately receive are:
How do I figure out the rank and reputation of a Ph.D. program?
Do not rely on word-of-mouth. No matter how much you trust your professors and classmates, treat their recommendations and judgments with skepticism. Our views are highly prejudiced, skewed towards our friends, and often badly out of date. Use these recommendations as a starting point, but investigate everything carefully using objective evidence as described below.
Look at rankings of departments produced by surveys of graduate departments. Rankings are useful but usually out of date, so you have to be careful. You can, however, check the rankings at www.phds.org. The key measures to look at are "Program is effective in educating researchers", "Scholarly quality of program faculty is high", and "A large percentage of faculty have research grants", because these are all fairly direct measures of a program's reputation. It is, however, possible for a program to have a good reputation but be on the way down. To avoid this, you want make sure that the current faculty is still productive in its research. In addition to the advice given below on choosing a Ph.D. advisor, you can look at the measures "The average number of recent publications per faculty member is high", "The average number of recent citations of faculty's work is high".
Look at how many faculty are currently supported by NSF research grants. The NSF is by far the dominant source of research grants for mathematical research. Looking at who is or is not supported by an NSF research grant is a good way to assess both individual mathematicians and mathematics departments.
To search for NSF research grants in mathematics that are currently active for a professor or institution, do the following:
How do I figure out a professor's rank and reputation in the research mathematical community?
Treat word of mouth with skepticism. Listen to what professors and students tell you, but make sure you can corroborate what you've heard by objective evidence, as described below. If you can't find objective evidence backing up what you hear, then it is all smoke and mirrors.
Look up the professor's NSF research grants using the instructions above.
Look up the professor on the Mathematical Genealogy Project. This will tell you how many Ph.D. students the professor has successfully advised and who they are. You can then google the students to see where they are working now. You can also do everything described below for the students as well as the professor.
Look up citation records. The availability of specialized search engines and databases has made it easy for anyone to figure out the impact of a professor's research.
It is not enough just to look up a professor's publications. You are not in a position to judge the quality of the work. The number of articles and books written by the professor is also not particularly meaningful.
What really matters is the impact of the professor's research on other people's research. You want to know whether anybody else has actually read and used the papers. This is easy to measure, because any time people use any of the ideas or results from an academic article, they have to cite the article in their paper. The number of citations an article has received is a good measure of how much impact the paper has had in mathematics. Although it is possible to distort the numbers through self-citations (the author of an article citing one of his other papers) or citations by weak articles, a professor's citation record is usually a very accurate measure of the significance of that professor's research. Even if a professor has written 500 papers, if no paper has received more than a few citations, it is pretty safe to say that this professor's research has had virtually no impact on the field.
When assessing a professor's citation record, it is important to look at citations of research articles only. Survey papers and books always have inflated citation numbers that should be discounted. Also, what constitutes a healthy number of citations varies wildly depending on the field. Papers in areas close to computer science, applied mathematics, or other areas of science (physics, biology) will get significantly more citations than papers in the pure mathematics. You can, however, effectively compare different professors within the same field.
You can look up a mathematician's citation record using:
Shouldn't I choose the graduate school based on what specialty I am interested in?
Although your own interests and tastes should have some influence on your decision, we think it should be less important than the factors detailed above. If you are serious about pursuing a career in academic research, you will need to land yourself either a tenured faculty position or a permanent position in a research laboratory somewhere. This is not easy to do, and the factors above are quite crucial. If you are able to attend a highly ranked school and work with a Ph.D. advisor who is one of the top researchers in an area you are passionately interested in, then the decision is easy. Otherwise, it is a difficult compromise. Our advice is to sacrifice the factors discussed above as little as possible and be as flexible as you can in what you do your thesis on. Try to remember that once you have tenure, you will be able to do any kind of research you want.