This is the time of year when decisions about which graduate program (of accepting institutions) in the mathematical sciences is best for you. If you are a "typical" student, only a few of the programs you applied for will grant admission with support. Nevertheless, I hope you receive some offers that meet your intellectual needs.
After the application period ends, many students will have questions concerning the process. Two of the most common are: Why was I not admitted to this particular graduate program and Why didn't I get that fellowship? Certainly, I cannot answer these questions in general, but I can give my views on the decision process. The purpose of this article is to help students understand the factors involved in fellowship and admission decisions for graduate study in mathematics.
An Insider's View
There are so many scenarios that it would be absurd to think that my views are "typical" or "representative," but they are still helpful. My perspective comes from my experiences in programs in which I have played some role in the decision-making process. During my 15 years as a faculty member at Cornell University, I participated in admissions in fields such as applied mathematics, biometry, ecology and evolutionary biology, statistics, and theoretical and applied mechanics.
I served on NSF's graduate fellowship committee  for 3 years and have selected postdoctoral fellows, new faculty members, and undergraduates for admission to summer programs. I have also been a member of special admissions program committees at the Institute for Mathematics and Its Applications  and the Center for Discrete Mathematics and Theoretical Computer Science .
Who Gets Admitted to Graduate School and Why?
The criteria are diverse and often highly susceptible to local conditions. Admission procedures are strictly confidential, so I cannot reveal the details of the decisions that I participated in, but I can give you a series of fictitious vignettes. These examples are based on real cases and are constructed from my experiences and observations as a participant in various admission committees.
"John" applies to School A. He has excellent GREs, great letters of recommendation, and an outstanding academic record. Furthermore, his adviser tells him, "There is no chance that you will not get admitted into Institution A." John's adviser mentally recalls his track record with this particular university by saying, "I have had three very good students admitted to this program, and John is the strongest candidate. My previous three students are doing extremely well. John is a 'shoo-in' for acceptance."
Surprisingly, John gets rejected! What happened? Many factors played a role in the matter. First, the committee composition changed since the time the adviser's first group of students was accepted. These new faces had no institutional memory. Second, School A hired a big-shot professor in the field, and she was promised support for her transferring Ph.D. students. Finally, to make matters worse, the job market was not good that year and, consequently, a few students decided to retain their "positions" as graduate students for an additional year. To sum it up, the adviser's friends on the committee were gone, and there wasn't enough room in the department for John. The result: He gets an impersonal (form) rejection letter.
"Barbara" applies to School B and has the credentials to be admitted with graduate student support (teaching assistantship), but "Francisco" from Spain also applies. Barbara and Francisco are both interested in working with the same faculty member, "Professor Jones," who takes at most one new student per year.
Because European academic training can be more comprehensive compared with that provided at American universities, Francisco has a slight advantage over Barbara. Francisco's 4-year B.S. degree required three mathematics courses and a physics course per semester along with writing an undergraduate thesis. His master's degree required two additional years of graduate-level courses in mathematics and thesis. In addition, he has carried out full-time research at a Spanish institute for the past 3 years and now has three publications and two submitted papers.
Unfortunately, Barbara and Francisco are not aware that Professor Jones has not been as productive as before and recently received a grant-rejection letter. In fact, his publication record over the past 3 years, while solid, has not been impressive. Professor Jones weighs his options and, not surprisingly, Francisco becomes the strongest candidate for admission. Barbara was indeed qualified, but circumstances determined her rejection.
"Linda" applies to School C and also has the credentials to be admitted with graduate student support. She has taken three advanced mathematics courses with young assistant professors and done extremely well. These three young faculty members think her potential for graduate school is simply extraordinary. Linda asks for letters of recommendation from two of her young mathematics assistant professors. She requests a third letter from her linear algebra professor (a course she took as a first-year student taught by a very famous mathematician). The fact that she got a B+ in linear algebra (her only B) did not enter into her calculations. The famous mathematician writes a letter that says she is a solid student and "may" have some potential to complete her Ph.D. He went on to say he was not totally excited about her mathematical talent because she ranked in the top 25% of his linear algebra class (not a particularly noteworthy class). Can you guess what happened? That's right, Linda gets rejected.
This is a very important point, so be advised! You must ask people who you are "sure" will write positive letters of recommendation for you. Because students aren't usually allowed to read them, it becomes a matter of trust. The best way to handle it is after you've decided who to ask, be blunt and say, "Will you be willing to write a very strong, positive letter for me?" If they say yes, they are duty-bound to do so. If they say, "No, I can't because, ..." that's OK too. Thank them for their time and move on to the next person.
"Barry" applies for a graduate school fellowship at Foundation X. He has a GPA of 4.0, three terrific letters of recommendation, and very good GREs. Unfortunately, he attends a "nonselective" college in the southwest. One admissions committee member argues that a 3.5 at an Ivy League university is far superior to a 4.0 at a nonselective college. Furthermore, during the interview, Barry says, "In today's world 'anybody' who is very good can be admitted to a selective college." Barry uses his recent admission as an example.
However, this is about who deserves funding the most. After deliberations, the committee is about to recommend the Ivy League student for the last spot on their fellowship list, but one of the committee members recalls reading an interesting statement on Barry's application. He said he "had become interested in mathematical biology while talking to his mother's doctors." Barry's mother has cancer and he transports her to chemotherapy sessions. The committee member suggests to the rest that perhaps Barry chose to attend the "nonselective" college in the southwest so he could be close to his mother. You can write your own ending to this case.
What Have We Learned?
Admission decisions are made under sets of circumstances that are well beyond the imagination of those who apply to graduate or fellowship programs. Hence, the advice that we must apply to a variety of programs from a relatively wide spectrum of institutions is very solid. I followed this advice. In fact, I applied to the mathematics program at the University of Wisconsin (UW), Madison, and was turned down. Fortunately, I had an offer from UW Milwaukee. I got a master's degree from UW Milwaukee and proceeded to reapply for admission to the mathematics program at UW Madison. It worked! I completed my Ph.D. in 1984. Read more detail in " Adventures of a Mathematical Biologist ."
It took me a while to acknowledge the fact that UW Madison's rejection turned out to be in my best interest. I was not ready for UW Madison graduate courses, but the idea of enrolling in undergraduate courses would have been unthinkable. (I have an ego, too.) I followed the road that was open to me. My journey on this path has taken me far beyond my own expectations. I hope that most students have thick-enough skins that they do not dwell on rejections. Fortunately, many entry points into the academic pipeline are available to us. We must move forward with the opportunities that come our way and make the best of them.
Finally, let's not forget that admission to a program, although important, is not the key. Completing a degree is a formidable task, and the key to success lies not with the school but on the work that is completed as part of a dissertation. Finding a good and supportive mentor increases the likelihood of completing a Ph.D. with a solid dissertation.
Carlos Castillo-Chavez is a Joaquin Bustoz Jr. Professor of Mathematical Biology at Arizona State University and may be reached at firstname.lastname@example.org .