Mathematical Research in High School: The PRIMES Experience The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Etingof, Pavel, Slava Gerovitch, and Tanya Khovanova. “Mathematical Research in High School: The PRIMES Experience.” Notices of the American Mathematical Society 62, no. 08 (September 1, 2015): 910–918. © American Mathematical Society (AMS) As Published http://dx.doi.org/10.1090/noti1270 Publisher American Mathematical Society (AMS) Version Final published version Citable link http://hdl.handle.net/1721.1/110267 Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. Mathematical Research in High School: The PRIMES Experience Pavel Etingof, Slava Gerovitch, and Tanya Khovanova Consider a finite set of lines in 3-space. A joint is a point where three of these lines (not lying in the same plane) intersect. If there are L lines, what is the largest possible number of joints? Well, let’s try our luck and randomly choose k planes. Any pair of planes produces a line, and any triple of planes, a joint. Thus, they produce L := k(k − 1)/2 lines and J :=p k(k − 1)(k − 2)/6 joints. If k is large, J is 2 3=2 about 3 L . For many years it was conjectured that one cannot do much better than that, in the 3=2 sense that if L is large, thenp J ≤ CL , where C is a constant (clearly, C ≥ 2 ). This was proved 3 Figure 1. PRIMES student Christina Chen is by Larry Guth and Nets Katz in 2007 and was a showing a picture of a convex body which can breakthrough in incidence geometry. Guth also hide behind a tetrahedron of smaller volume showed that one can take C = 10. Can you do (PRIMES Conference, 2011). better? Yes! The best known result is that any number C > 4=3 will do. This was proved in 2014 by Joseph Zurier, an eleventh-grader from Rhode Island [Z]. the best currently known value ([Ch]; see Figure 1). Here is another problem. Let K and L be convex So, can it be arbitrarily large? No! Christina, Tanya bodies in space, and suppose that we can hide Khovanova, and Dan Klain showed that the volume K behind L no matter from where we look (we ratio is less than 3 in any dimension [CKK]. are allowed to translate the bodies but may not Seriously? Is it really possible for tenth- and rotate them). Is it true that the volume of K is at eleventh-graders to do original mathematical re- most the volume of L? Curiously, no! Christina search? Chen, a tenth-grader from Massachusetts, showed Yes! Christina and Joseph, as well as over a in 2011 that the volume ratio can be about 1.16, hundred other students, have done their research at PRIMES (Program for Research In Mathematics, Pavel Etingof is professor of mathematics at Massachusetts Engineering, and Science; web.mit.edu/primes), Institute of Technology and the PRIMES chief research ad- which we’ve been running in the MIT mathematics viser. His email is [email protected]. department since January 2011. Every year we Slava Gerovitch is a lecturer in history of mathematics receive numerous questions about our program at Massachusetts Institute of Technology and the PRIMES from prospective students and their parents and program director. His email is [email protected]. All also from academics who want to organize a article photos are courtesy of Slava Gerovitch. similar program. Here we’d like to answer some Tanya Khovanova is a lecturer in mathematics at Mas- of these questions, to share our experience, and sachusetts Institute of Technology and the PRIMES head to tell a wider mathematical community how such mentor. Her email is [email protected]. a seemingly impossible thing as mathematical DOI: http://dx.doi.org/10.1090/noti1270 research in high school can actually be done. 910 Notices of the AMS Volume 62, Number 8 What kind of students do you look for? My collaborators. This way PRIMES students learn the son was wondering—should he even bother to art of collaboration and teamwork. apply if he doesn’t have a perfect score at the P.E.: In short, we do have options for both IMO and hasn’t yet mastered Wiles’s proof of Fer- introverted mathematicians (who like to look at mat’s Last Theorem? their shoes while doing research) and extroverted P.E.: If he is in love with math, yes, by all means! ones (who prefer to look at the shoes of their Some background (such as calculus) is needed, collaborator). And, of course, all PRIMES students but generally he will learn along the way under are encouraged to look at the shoes of their his mentor’s guidance. Also, many gifted high- mentors as often as they like! schoolers do well at math competitions, but good How do you select projects? Can my student researchers are not always quick problem-solvers. be told to prove the Twin Primes Conjecture in It takes time, effort, and perseverance to learn the PRIMES? background and try different approaches, many P.E.: Famous open problems don’t usually make of which are doomed to fail. We look for students good projects, but we don’t assign “toy projects” with a talent for mathematical research and a with known solutions either. Students delve into stamina to carry it through, for avid learners, hard real research, with all its uncertainties, disap- workers, and imaginative explorers. And, above all, pointments, and surprises. Finding cutting-edge for those who are crazy about mathematics! projects requiring a minimal background is one How do you select students? My daughter of the trickiest tasks in running PRIMES. Here are asks: to get accepted, does she need to be a some features we want to see in a PRIMES project: machine that makes coffee into theorems? 1. Accessible beginning. Presence of simple initial P.E.: Mathematicians are mere humans who steps to get started. make coffee into theorems 10 percent of the time, 2. Flexibility. A possibility to think about several and into unsuccessful attempts to prove theorems related questions, switching from one to another the rest of their lives—and we welcome your if stuck, and to tweak the questions if they are too daughter to the club! hard or insufficiently interesting. S.G.: We carefully consider Olympiad scores, 3. Computer (experimental) component. A pos- statements of purpose, recommendations, and sibility of computer-assisted exploration aimed grades, but the pivotal part of the application is the at finding patterns and making conjectures. This entrance problem set. We post it in mid-September, way students, who often have strong programming due in two months. skills, can contribute to the project early, when T.K.: These are not the kinds of problems that they don’t yet have a working knowledge of the one can crack quickly: at first glance, some of them theoretical tools. It is also easier to learn new may puzzle even a math professor. Students are mathematical concepts, e.g., those from algebra expected to think about a problem, consult books and representation theory, through a hands-on and online sources, think again the next day, then experience with a computer algebra system. again and again, until one day they finally get it 4. Adviser involvement. Availability of a research and then write a full solution with detailed proofs. mathematician other than the mentor (usually the This protracted engagement with mathematical professor or researcher who suggested the project) problems resembles a research process. to advise the project through email and occasional P.E.: In fact, this is similar to my favorite hobby, meetings. Such meetings make a big difference. picking mushrooms. You may run around the 5. Big picture/motivation. Connection, at least at woods for hours seeing nothing, and then all of a the level of ideas, to a wider context and to other sudden you find a real treasure. You need patience people’s work. and ability to enjoy the process and forget about 6. Learning component. The project should everything else. There is a lot in common between a encourage the student to study advanced mathe- good mushroom hunter and a good mathematician. matics on a regular basis. Do PRIMES students work individually or in 7. Doability. A reasonable expectation that a groups? good student would obtain some new results in S.G.: Most projects are individual and involve several months to present at the annual PRIMES one-on-one mentoring. Freshmen and sophomores, conference in mid-May and produce publishable however, usually work in groups of two on joint results in one year. projects. Group discussions make research more 8. Relation to the mentor’s research program or exciting and stimulating for younger students area. and give them a gentler entry into the world T.K.: A crucial part of research is the art of of mathematics. Even in individual projects the asking your own questions, not just solving other students are not alone: they collaborate with their people’s problems. When the students realize that mentors and faculty who suggested projects. They it is in their power to move the project in a new form a team in which mathematicians of different direction, they get very excited and start feeling levels of experience and seniority become equal ownership of the project. The ability to trust September 2015 Notices of the AMS 911 themselves and ask their own questions is very In [DT] Fengning Ding and his mentor, Sasha important in their future lives, independent of Tsymbaliuk, considered representations of contin- their career choices.