PRIMES: Q&A with Chief Research Advisor
Dear prospective participant of PRIMES,
In this message I'd like to share my thoughts about our program and mathematics research by high school students in general. When I teach mathematics, I always tell my students to formulate the question they are answering before starting to answer it. So why not follow this principle here?
Q: What is the purpose of PRIMES?
A: To let mathematically talented high school students discover how it feels to be a research mathematician.
Q: Come on! To do math you just need pen and paper. Just start solving a problem, and you'll discover how it feels! Why do you need a program for this?
A: Actually, it's not that simple. Good mathematical research requires knowledge that takes years to acquire. It takes even longer to master the art of choosing the "right" problem. For this reason, mathematical research is typically done starting in the second year of graduate school, and it is quite difficult for a high school student and even for an undergraduate to really put himself or herself in the shoes of a research mathematician. In PRIMES, this challenge will be met by giving high school students an opportunity to collaborate with professional mathematicians - mathematics faculty and graduate students of MIT.
Q: I've been to math Olympiads, which have pretty tricky problems. Is it not a good enough way to discover what it feels like to be a mathematician?
A: While mathematical Olympiads and competitions are fun and undoubtedly useful, they are actually a poor model of mathematical research. Specifically, in Olympiads, one solves relatively simple problems that admit a short and elegant solution (known in advance to those administering the Olympiad), often using a small number of clever tricks, in a very short period of time (a few hours). On the contrary, in research one deals with much more complex unsolved problems, which often do not admit a short solution by a clever trick. On the other hand, the time allowed for mathematical research is much longer (months and sometimes years).
Q: I know that high school math research is done at RSI at MIT, PROMYS, and in some REU programs that admit high school students. Can't one get an adequate research experience in these programs? Why do you need a new one?
A: Summer math research programs (RSI, PROMYS, and REUs), unlike olympiads, do indeed give an excellent research experience. However, in them the time scale is still much more compressed than in a usual research environment (a few weeks rather than several months). On the contrary, PRIMES is designed in such a way that students are able to do mathematical research at a natural pace, i.e., work on a single project over about a year. Also, your participation in PRIMES does not rule out participation in the summer programs; in fact, it only improves your chances of doing well in them.
Q: You say that good math research requires knowledge that takes years to acquire. Then how can you expect high school students to achieve any substantial results? Clearly, your projects will have to be very low-tech, and have nothing to do with state-of-the-art deep mathematics! Wouldn't it be better to just read good math books instead?
A: First of all, "low-tech" does not necessarily mean "trivial." There are many problems in combinatorics whose solution, while elementary, is nevertheless deep and subtle, and leads to other fields of mathematics. Also, reading and learning are, of course, a big part of life of a good research mathematician, and they will be a huge part of the PRIMES experience as well. Besides, you will collaborate on projects with mentors and faculty, and they will help you with the more technologically advanced parts of the projects, so that your projects won't have to be necessarily very "low-tech."
Q: The Program website says that PRIMES students will work a lot on computers. Does this mean that they will simply write programs to do routine computations for some research projects?
A: Indeed, much of the mathematics in PRIMES will be done using computer algebra systems, such as Mathematica, MAGMA, GAP, and SAGE. But this does not mean just routine computation. Students will have to recognize patterns, form conjectures, and prove them with the help of mentors and faculty. This experience will allow students not only to learn how to use these systems (which is a very useful skill in itself), but also to acquire hands-on experience of dealing with abstract mathematical notions. This will allow for more "high-tech" and "state-of-the-art" projects.
Q: Participating in science competitions is a high priority for me. Should I participate in PRIMES?
A: Yes. We encourage participation in science competitions, since they are a valuable experience for students, and are an important factor in learning how to organize and communicate research results. In fact, our staff will help you prepare your project for competitions, and write a mentor letter for you. However, we feel that excessive focus on competitions and monetary awards is not a good idea in mathematical research, and we emphasize that while participation in competitions may be a priority for you, learning and doing mathematics should be an even higher priority. Also, we stress that intention to participate in science competitions is by no means necessary for being accepted to PRIMES.
Q: On the front page of the PRIMES website, there is a quotation by
von Neumann, who says, "In mathematics, you don't understand things. You
just get used to them." Does this mean that we won't really understand what
we do in our projects?
A: This means that understanding is a complicated thing which is not as perfect and blissful as one may want it to be. The road to understanding abstract mathematics is long and rough, and full of moments of frustration, when you feel that you are totally lost. Abstract things, since they initially don't correspond to images in our mind, present a challenge to our imagination, and it takes a long time to "get used" to them, which means to develop a system of verbal and visual images to think about them. And even after that, these images will necessarily be somewhat vague, never as vivid and tangible as those we have in ordinary life. So understanding is never perfect... But it will come, just be patient!
See also Pavel Etingof's advice How to Succeed in Mathematical Research.
If you come up with a question that is not answered here, please send us an email.