Mathematical Researchin High School: ThePRIMES ExperiencePavel Etingof, Slava Gerovitch, and Tanya Khovanova

Consider a finite set of lines in 3-space. A joint is apoint where three of these lines (not lying in thesame plane) intersect. If there are L lines, what isthe largest possible number of joints? Well, let’s tryour luck and randomly choose k planes. Any pairof planes produces a line, and any triple of planes,a joint. Thus, they produce L := k(k− 1)/2 linesand J := k(k− 1)(k− 2)/6 joints. If k is large, J is

about√

23 L

3/2. For many years it was conjecturedthat one cannot do much better than that, in thesense that if L is large, then J ≤ CL3/2, where Cis a constant (clearly, C ≥

√2

3 ). This was provedby Larry Guth and Nets Katz in 2007 and was abreakthrough in incidence geometry. Guth alsoshowed that one can take C = 10. Can you dobetter? Yes! The best known result is that anynumber C > 4/3 will do. This was proved in 2014by Joseph Zurier, an eleventh-grader from RhodeIsland [Z].

Here is another problem. Let K and L be convexbodies in space, and suppose that we can hideK behind L no matter from where we look (weare allowed to translate the bodies but may notrotate them). Is it true that the volume of K is atmost the volume of L? Curiously, no! ChristinaChen, a tenth-grader from Massachusetts, showedin 2011 that the volume ratio can be about 1.16,

Pavel Etingof is professor of mathematics at MassachusettsInstitute of Technology and the PRIMES chief research ad-viser. His email is etingof@math.mit.edu.

Slava Gerovitch is a lecturer in history of mathematicsat Massachusetts Institute of Technology and the PRIMESprogram director. His email is slava@math.mit.edu. Allarticle photos are courtesy of Slava Gerovitch.

Tanya Khovanova is a lecturer in mathematics at Mas-sachusetts Institute of Technology and the PRIMES headmentor. Her email is tanya@math.mit.edu.

DOI: http://dx.doi.org/10.1090/noti1270

Figure 1. PRIMES student Christina Chen isshowing a picture of a convex body which can

hide behind a tetrahedron of smaller volume(PRIMES Conference, 2011).

the best currently known value ([Ch]; see Figure 1).So, can it be arbitrarily large? No! Christina, TanyaKhovanova, and Dan Klain showed that the volumeratio is less than 3 in any dimension [CKK].

Seriously? Is it really possible for tenth- andeleventh-graders to do original mathematical re-search?

Yes! Christina and Joseph, as well as over ahundred other students, have done their researchat PRIMES (Program for Research In Mathematics,Engineering, and Science; web.mit.edu/primes),which we’ve been running in the MIT mathematicsdepartment since January 2011. Every year wereceive numerous questions about our programfrom prospective students and their parents andalso from academics who want to organize asimilar program. Here we’d like to answer someof these questions, to share our experience, andto tell a wider mathematical community how sucha seemingly impossible thing as mathematicalresearch in high school can actually be done.

910 Notices of the AMS Volume 62, Number 8

What kind of students do you look for? Myson was wondering—should he even bother toapply if he doesn’t have a perfect score at theIMO and hasn’t yet mastered Wiles’s proof of Fer-mat’s Last Theorem?

P.E.: If he is in love with math, yes, by all means!Some background (such as calculus) is needed,but generally he will learn along the way underhis mentor’s guidance. Also, many gifted high-schoolers do well at math competitions, but goodresearchers are not always quick problem-solvers.It takes time, effort, and perseverance to learn thebackground and try different approaches, manyof which are doomed to fail. We look for studentswith a talent for mathematical research and astamina to carry it through, for avid learners, hardworkers, and imaginative explorers. And, above all,for those who are crazy about mathematics!

How do you select students? My daughterasks: to get accepted, does she need to be amachine that makes coffee into theorems?

P.E.: Mathematicians are mere humans whomake coffee into theorems 10 percent of the time,and into unsuccessful attempts to prove theoremsthe rest of their lives—and we welcome yourdaughter to the club!

S.G.: We carefully consider Olympiad scores,statements of purpose, recommendations, andgrades, but the pivotal part of the application is theentrance problem set. We post it in mid-September,due in two months.

T.K.: These are not the kinds of problems thatone can crack quickly: at first glance, some of themmay puzzle even a math professor. Students areexpected to think about a problem, consult booksand online sources, think again the next day, thenagain and again, until one day they finally get itand then write a full solution with detailed proofs.This protracted engagement with mathematicalproblems resembles a research process.

P.E.: In fact, this is similar to my favorite hobby,picking mushrooms. You may run around thewoods for hours seeing nothing, and then all of asudden you find a real treasure. You need patienceand ability to enjoy the process and forget abouteverything else. There is a lot in common between agood mushroom hunter and a good mathematician.

Do PRIMES students work individually or ingroups?

S.G.: Most projects are individual and involveone-on-one mentoring. Freshmen and sophomores,however, usually work in groups of two on jointprojects. Group discussions make research moreexciting and stimulating for younger studentsand give them a gentler entry into the worldof mathematics. Even in individual projects thestudents are not alone: they collaborate with theirmentors and faculty who suggested projects. Theyform a team in which mathematicians of differentlevels of experience and seniority become equal

collaborators. This way PRIMES students learn theart of collaboration and teamwork.

P.E.: In short, we do have options for bothintroverted mathematicians (who like to look attheir shoes while doing research) and extrovertedones (who prefer to look at the shoes of theircollaborator). And, of course, all PRIMES studentsare encouraged to look at the shoes of theirmentors as often as they like!

How do you select projects? Can my studentbe told to prove the Twin Primes Conjecture inPRIMES?

P.E.: Famous open problems don’t usually makegood projects, but we don’t assign “toy projects”with known solutions either. Students delve intoreal research, with all its uncertainties, disap-pointments, and surprises. Finding cutting-edgeprojects requiring a minimal background is oneof the trickiest tasks in running PRIMES. Here aresome features we want to see in a PRIMES project:

1. Accessible beginning. Presence of simple initialsteps to get started.

2. Flexibility. A possibility to think about severalrelated questions, switching from one to anotherif stuck, and to tweak the questions if they are toohard or insufficiently interesting.

3. Computer (experimental) component. A pos-sibility of computer-assisted exploration aimedat finding patterns and making conjectures. Thisway students, who often have strong programmingskills, can contribute to the project early, whenthey don’t yet have a working knowledge of thetheoretical tools. It is also easier to learn newmathematical concepts, e.g., those from algebraand representation theory, through a hands-onexperience with a computer algebra system.

4. Adviser involvement. Availability of a researchmathematician other than the mentor (usually theprofessor or researcher who suggested the project)to advise the project through email and occasionalmeetings. Such meetings make a big difference.

5. Big picture/motivation. Connection, at least atthe level of ideas, to a wider context and to otherpeople’s work.

6. Learning component. The project shouldencourage the student to study advanced mathe-matics on a regular basis.

7. Doability. A reasonable expectation that agood student would obtain some new results inseveral months to present at the annual PRIMESconference in mid-May and produce publishableresults in one year.

8. Relation to the mentor’s research program orarea.

T.K.: A crucial part of research is the art ofasking your own questions, not just solving otherpeople’s problems. When the students realize thatit is in their power to move the project in a newdirection, they get very excited and start feelingownership of the project. The ability to trust

September 2015 Notices of the AMS 911

themselves and ask their own questions is veryimportant in their future lives, independent oftheir career choices. That’s why we try to chooseprojects that develop this ability.

P.E.: Sounds easy? Well, if you have a bit of freetime or have nothing better to do (e.g., during anexcruciatingly boring math lecture that you can’tsneak out of), just try to come up with a projectsatisfying most of these conditions. And when youdo, please send it to us!

Is it true that in PRIMES mathematics equalselementary combinatorics? Do PRIMES studentswork on elaborate Olympiad problems insteadof learning about algebra, topology, geometry,analysis, number theory?

P.E.: Not really. We’ve had many projects in thesefields, especially in noncommutative algebra andrepresentation theory. Also, PRIMES students getexposed to these areas in PRIMES reading groups.

This said, it’s true that many PRIMES projectsare in discrete math. This is because in this field,it’s easier to find interesting projects requiringrelatively little initial background. However, theyare not just elaborate Olympiad problems. Manyof them are designed to touch upon fundamentalquestions and to encourage learning about otherareas with which discrete math has many deepconnections. In short, we try to show our studentsboth the breadth and the unity of mathematics.

Noncommutative algebra and representationtheory in high school? Touch upon fundamen-tal questions? No kidding? Can you give someexamples?

P.E.: You want me to get technical? All right,here you go.

One group of projects concerns representationsof rational Cherednik algebras. Let G be a finitegroup and V be its finite-dimensional representa-tion over a field k. Then one can define the rationalCherednik algebra H(G,V), which is a certainremarkable deformation of the algebra kGÎD(V),the semidirect product of the group algebra ofG with the algebra of differential operators onV . For example, if G = Z/2Z, V = k, and thegenerator s ∈ Z/2Z acts on the coordinate x onV by s(x) = −x, then H(G,V) is generated by s, xand the Dunkl operator ∂x− k

xs. Representations ofH(G,V) are currently a subject of active research.

In [DS] Sheela Devadas and her mentor, StevenSam, studied lowest-weight irreducible representa-tions ofH(G,V) forG being the complex reflectiongroup G(m, r, n) and V = kn (where chark = p)using methods of commutative algebra. They gaveconjectural character formulas for some of themand proved these formulas in a number of cases.In general, this is a difficult open problem. It isnot easy even in the case n = 2 and m = r (groupsof symmetries of a regular polygon); in this case,more definitive results were obtained by PRIMESstudent Carl Lian [Li].

In [DT] Fengning Ding and his mentor, SashaTsymbaliuk, considered representations of contin-uous Cherednik algebras, which are generalizationsof H(G,V) to the case when G is a reductive alge-braic group (rather than a finite group). Namely,they considered the case when G = GL(n,C)and V = Cn. They computed the center ofH(G,V), classified its finite-dimensional irre-ducible representations, and computed theircharacters.

In [KL] Shashwat Kishore and his mentor, GusLonergan, studied signature of the canonicalHermitian form on the space Hom(Mλ,Mλ1 ⊗· · ·⊗Mλn), where λ,λ1, . . . , λn ∈ R and Mλ is the Vermamodule for the Lie algebra sl 2. They classified thecases when this form is definite and also applied thesignature formula to solve a topological problem:give lower bounds for the number of real criticalpoints of the Gaudin model master function

F(t1, . . . , tm, z1, . . . , zn)

=∏

1≤i<j≤m(ti − tj)2

m∏i=1

n∏k=1

(ti − zk)−λk ,

where m = 12(λ1 + · · · + λn − λ). They also gener-

alized their results to the case of quantum groupUq(sl 2) (where |q| = 1).

We’ve also had some other algebraic projects.With Yongyi Chen, Michael Zhang, and their mentor,David Jordan, we studied trace functions on thealgebra AP := k[x, y, z]/(P), where P is a generichomogeneous polynomial of degree d and k isa field of characteristic p [CEJZ]. By definition, atrace function is a linear function on AP whichvanishes on Poisson brackets

{f , g} := ∂(P, f , g)∂(x, y, z)

.

The problem was to compute the Hilbert series ofthe space of trace functions, i.e., h(z) :=

∑n≥0 hnzn,

where hn is the dimension of the space of tracefunctions of degree n. It turns out that for largeenough p, the function h(z) is given by thefollowing peculiar formula:

h(z) = (1− zd−1)3

(1− z)3

+ zd−3

(1− zpd(1− zp)3 +

d(d − 3)zp

1− zp − 1

).

We found this formula empirically on a computerand then proved it (and generalized to the quasi-homogeneous case) using some algebraic geometryand the theory of D-modules.

Another algebraic project concerned the lowercentral series of an associative algebra A: L1 = A,L2 = [A, L1], L3 = [A, L2], and so on. Feigin andShoikhet showed in 2006 that if A is free in ngenerators over Q, then B2 = L2/L3 is the space ofclosed differential forms of positive even degree inn variables. With Surya Bhupatiraju, Bill Kuszmaul,

912 Notices of the AMS Volume 62, Number 8

Jason Li, and their mentor, David Jordan, wegeneralized this result to the case of integercoefficients, expressing B2 in terms of the de Rhamcohomology over the integers [BEJKL]. In anotherproject, Isaac Xia and his mentor, Yael Fregier,studied quotients Ni := ALi/ALi+1 and showedthat if A is a free algebra in x1, . . . , xn over a fieldof characteristic p modulo relations written in

terms of xpm1

1 , . . . , xpmnn and if the abelianization of

A is finite-dimensional, then Ni have dimensionsdivisible by p

∑mi [FX]. The proof is based on the

representation theory of algebras of differentialoperators with divided powers.

Tired of algebra? Here is a project in combina-torics. A linear equation is r -regular if for everyr -coloring of the positive integers, there existpositive integers of the same color which satisfythe equation. In 2005 Fox and Radoicic conjecturedthat the equation

x0 + 2x1 + · · · + 2n−1xn−1 − 2nxn = 0,

for any n ≥ 1, has a degree of regularity of n,which would verify a conjecture of Rado from 1933.While Rado’s conjecture was later verified witha different family of equations, the Fox-Radoicicconjecture remained open. This conjecture (in ageneralized form) was proved by Noah Golowich[Go] under the mentorship of László Lovász.

S.G.: This is beautiful math, but sounds like the“prior results” section of our grant proposal. Didyou copy-paste it here? This will put the readers tosleep! Tell them what our students do in the formof an exciting game or an engaging story.

P.E.: OK, let me try my best. Every day eachMartian gives each of his friends one Martian pesoif he is sufficiently rich to do so. What will happen?

This process is called “the parallel chip-firinggame” (see Figure 2) and is an important modelof dynamics on graphs. Clearly, it is eventuallyperiodic, but as it is nonlinear, one could a prioriexpect complicated behavior. Yet, Ziv Scully with hismentors, Damien Jiang and Yan Zhang, were ableto completely characterize the possible periodicpatterns [JSZ]. This is a truly beautiful result!

S.G.: Perhaps this is a good model for fundingPRIMES? I suppose this model applies not only toMartian pesos but equally to earthly hundred-dollarbills? Then we just need to make sure that PRIMEShas enough sufficiently rich friends.…

P.E.: Well, there is a small catch: according tothis model, PRIMES would also have to give outhundred-dollar bills. The total amount of moneyin the system is preserved, so the salary of theprogram director would unfortunately have to bezero!

But surely not all your projects are at thishigh level. I saw that one of them is about“dessins d’enfants.” Unless I am forgetting myFrench, this means “child’s drawings.” Can thispossibly involve serious mathematics?

Figure 2. PRIMES project, 2011.

P.E.: In fact, this is one of our more advancedprojects! The child here is Alexandre Grothendieck(1928–2014), one of the greatest mathematiciansof the twentieth century. In 1984 in his famous“Esquisse d’un Programme,” he proposed to studythe Galois group Gal(Q/Q) through its action onthe set of finite covers of the complex plane thatbranch at 0,1,∞. He represented such covers bycertain planar graphs which he called “dessinsd’enfants.” An important problem is to find in-variants of covers (or, equivalently, Grothendieck’sdessins) that allow one to show that two givencovers are not equivalent under the action ofGal(Q/Q). Ravi Jagadeesan (mentored by AkhilMathew) found a new invariant of covers which ismore powerful than the previous invariants andused it to prove a new lower bound on the numberof Galois orbits of a certain type [Ja]. Even thoughit is about “dessins d’enfants” and the author wasin eleventh grade, this result is of real interest togrown-up mathematicians!

How do you match students to projects? Is thematching theory relevant here?

P.E.: Yes. We always manage to find a goodmatching, all thanks to the counterintuitive mathe-matical fact that good matchings exist and are easyto find: the Gale-Shapley Stable Marriage Theorem,which says that in the ideal world all marriagesare stable. In fact, the only reason the real world isshort of ideal is that people don’t know enoughmathematics!

T.K.: However bright, PRIMES students rarelyhave an idea of what a suitable project would looklike. Many applicants, for example, declare on theirapplication that they want to work on the Riemannhypothesis. Most list number theory as their topinterest, which could be the result of PROMYS,Canada-USA MathCamp, Ross, and other programsteaching students advanced number theory. Yetstate-of-the-art projects in number theory areusually too advanced for high school students.For this reason, it does not always make sense tofollow the applicant’s preferences literally.

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But we try to glean from the application the trueinclinations and strengths of the student and finda project that would let us play on those as muchas we can.

Also, an average PRIMES student is a betterprogrammer than an average MIT math professor.And many of our projects have a computationalcomponent. So we look at the programmingbackground in addition to the math backgroundwhen matching projects.

S.G.: Most students end up working in areasthey’ve never heard of before, because that’s wherethe good projects are. Also, we adjust the difficultyof each project along the way, depending on thestudent’s abilities, preparation, and progress. Thisway every student discovers the joy of proving anew theorem.

How do you find mentors and match themto projects and students? Does PRIMES distractmentors from their research and ruin theircareers?

S.G.: PRIMES mentors are typically math graduatestudents and sometimes postdocs or faculty whoshow desire and ability to work with high schoolstudents. We look for mentors with a knack forteaching and an inspiring personality who can beeffective role models. We also try to match projectsand students with the mentor’s research style,whether conceptual or oriented toward problemsolving.

Finally, to make sure that PRIMES does not takea toll on the mentor’s career and research, weencourage mentors to suggest projects relatedto their own work. This not only improves thequality of mentoring but also allows mentors tocombine mentoring with research, often leading tojoint papers with students. Thus mentors not onlyreceive a supplement to their stipend and acquirevaluable advising experience but also add jointpapers to their publication record, strengtheningtheir position in the job market.

T.K.: Some mentors say that after teaching highschool students how to do research, they finallyunderstand it themselves!

Do you admit students from other states?Other countries? Other planets, planetarysystems, galaxies?

S.G.: For the first two years, PRIMES operatedas a program for local students. In 2013 wedecided to do an experiment. We selected fivestudents in a nationwide search and mentored theirresearch projects over the Internet, using softwareand hardware tools for online collaboration. Theexperiment proved to be a success, and thefollowing year we expanded the PRIMES-USA sectionto thirteen students, including two supervised byfaculty from the University of Illinois at Urbana-Champaign. This year the number of out-of-statestudents rose to fifteen, and to meet the demand we

collaborate with faculty from several universities,including CUNY and SUNY at Stony Brook.

The PRIMES-USA section not only providesresearch opportunities to talented math studentsacross the nation but also serves as a laboratoryfor testing new methods of distance researchmentoring, as well as helps spread the PRIMESapproach to other universities in the United Statesand beyond. Arrangements to open a section ofPRIMES in Europe are currently under way.

P.E.: And, yes, PRIMES-Extraterrestrial ispresently under construction. Originally ourdown-to-earth program director was reluctant, butI’ve convinced him to go ahead. However, alienhigh school students may well turn out to knowmore math than MIT professors. So I envisionworking on proving the Riemann hypothesis underthe guidance of a high school mentor from the farend of the Milky Way.…

Is it good for a math student to start researchso early? Isn’t it better to spend time reading andlearning new mathematics?

P.E.: In many cases, it is better, yes. Readingmathematical literature and learning are vital partsof the professional life of every mathematician.They are of key importance at all ages. In fact, oneof the greatest mathematicians of the twentiethcentury, I. M. Gelfand, said at his ninetieth birthdaycelebration: “I am a student of mathematics.”

It is a tautology that learning is especially vitalfor students. For many students, guided readingis more intellectually stimulating and beneficialthan an immediate plunge into research. We care-fully evaluate PRIMES applicants and recommendresearch only for those who are ready. For mostyounger students we set up reading groups of 2–3students who study an advanced mathematicalbook with a mentor. Devoting the first year ofPRIMES to guided reading helps students build afoundation for attacking research problems in thefollowing years.

PRIMES research projects are also designedto require learning new mathematics. Before theproject starts, students devote one month tobackground reading, and they continue readingalong with research. A research problem providesexcellent motivation and environment for learninga new area of math.

S.G.: We also recommend reading groups toseniors, who spend only half a year in PRIMESbefore going to college. After doing a researchproject in junior year, PRIMES students often stayin reading groups to expand their mathematicalknowledge. We encourage them to explore areasbeyond the topic of their research project. Thisyear, 40 percent of local math students at PRIMESare in reading groups, while 60 percent work onresearch projects.

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What’s the timeline of your program? Isn’t awhole year too long? Do PRIMES students everget a life?

P.E.: A year is too short! We all know it takesmonths and sometimes years to prove a goodtheorem and to write a good paper. And you geta life by getting to do math all this time! Onefamous mathematician said that to come up witha theorem that’s any good, you have to become asleepwalker for at least several weeks. That’s whatI’ve been trying to explain to my wife, admittedlywithout much success.…

S.G.: The PRIMES cycle runs through a calendaryear.

January is the reading period: mentors givetheir students background reading and exercisesvia email or Skype. In early February we inviteall local students and their parents to campusfor an orientation meeting, where students meettheir mentors and mingle with other students.At this meeting, Pavel shares his tips on doingmathematical research, also available at the PRIMESwebsite.

P.E. This is a really exciting speech, especiallyfor continuing students (given that the tips don’tchange from year to year, as they have eternalvalue).

S.G.: The spring semester is the active researchperiod. At weekly meetings students and mentorsdiscuss progress and set goals for next week.Meetings nominally last one and a half hours, butoften run longer, as students and mentors getexcited about new ideas. Students are encouragedto get in touch with their mentors over emailmidweek or any time they have a question or getstuck.

T.K.: During this period, I periodically check onevery project, suggesting adjustments if necessary.

S.G.: The active research period culminates witha presentation at an annual PRIMES conference,held at MIT in May on the last weekend beforeMemorial Day (Figure 1, Figure 3). Prior to theconference, students prepare a research report,which includes preliminaries, previous results,statement of the problem, and new results. Sincethe conference comes in the middle of the annualcycle, the students present work in progress. Theyincorporate feedback received at the conferenceinto their future work. The conference lasts fortwo days, with talks on mathematics, computerscience, and computational biology before a livelyaudience that includes grad students, postdocs,and faculty. PRIMES students’ parents, many ofwhom are academics or industry researchers, arealso invited; they often ask interesting questionsand invariably end up thoroughly impressed.

The summer break is the independent study pe-riod. The student and the mentor coordinate theirschedules, meeting when in town, communicatingby email when away, or taking a beach break when

Figure 3. Pavel Etingof and Tanya Khovanovawith PRIMES Conference 2013 participants.

the weather is good. We also encourage PRIMESstudents to take advantage of other opportunities,such as attending summer math camps, whichallows them to expand their scope and take abreak from their project, only to return to it withrenewed vigor in the fall.

Fall is the write-up period. Students meet withtheir mentors as needed, finalize their project, andwrite a final paper summarizing their results. Thisis the time when we can teach our students to writemathematics, which is one of the important goalsof PRIMES. Many PRIMES papers are submitted tonational science competitions and the MAA-AMSundergraduate student poster session at the JointMeetings in January.

T.K.: Sometimes by the end of the projectthe student and the mentor see a big, beautifulconjecture that generalizes their results. Thisconjecture is like a star shining ahead of them.When the PRIMES year is over, they can’t stop, andcontinue working until they prove their conjecture.

Can students stay for a second year?S.G.: Yes, every year a number of students stay

for another year. This allows younger students tomature as researchers.

An example: Bill Kuszmaul was in PRIMES forfour years. Having entered PRIMES in ninth grade,he did two joint projects in years 1 and 2, anindividual project in year 3, and a reading group inyear 4. Bill authored four papers posted on arXiv.org(two of them published in the Journal of Algebraand the Electronic Journal of Combinatorics), was aSiemens regional finalist in 2011 and 2012, a 2013Davidson Fellow, and won Third Prize in the 2014Intel STS. He is now a sophomore at Stanford.

P.E.: In fact, besides proving many cool theo-rems, Bill introduced a new English word. In histestimonial, he wrote: “It gave me an incrediblefeeling to have the paper come together in the finaldays of it being written, and I came to cherish thefeeling of just putting everything in life aside and“primesing” for the rest of a day.”

September 2015 Notices of the AMS 915

Is high school math research possible outsideof PRIMES?

P.E.: Sure. One option is for students to workby themselves, supervised by mentors active inresearch. Also, there are summer programs offeringsuch opportunities: RSI (for individual projects),PROMYS, Canada-USA MathCamp, and others (forgroup projects).

Sounds like you have competitors. Is your goalto put them out of business?

P.E.: In fact, our goal is to put as many of themas possible into business, and that’s exactly whywe are answering these questions here. Each yearwe have to turn down a growing number of strongapplicants, which is a pity. We hope that soonthere will be more opportunities like PRIMES. Thesestudents ought to have a chance to achieve theirdream!

How is your program different from RSI,PROMYS, Canada-USA MathCamp, Ross, andother summer programs? Which one should mychild choose?

S.G.: The main difference is that summer pro-grams are compressed into a few weeks, whilePRIMES operates for an entire year. This allowsresearch at a natural pace, with sufficient time fortrial and error, gaining additional background, andwriting a detailed text according to professionalstandards.

T.K.: A few weeks are not enough. My best ideascome to me in the shower. I wouldn’t be able tofinish research in a summer program, as therearen’t enough showers!

P.E.: Exactly. And there isn’t enough hot waterin the boiler. My wife complains that I leavenone for anyone else, and this is expensive andenvironmentally unhealthy. Perhaps we shouldfigure out a way to balance family and ecologicalneeds with the need to do good mathematics!

S.G.: Well, Archimedes’s example clearly showsthat a bath, while much less wasteful, can beequally stimulating for a mathematician.

But bathroom issues aside, summer programsgive students an excellent experience. They takea variety of short courses and are exposed toa wide range of mathematical topics, useful forfurther research. And your child doesn’t reallyhave to choose! PRIMES has a flexible schedule inthe summer, which allows our students to attendsummer programs, and we strongly encouragethem to do so, as the two experiences reinforceeach other. For instance, PRIMES students attendingRSI often work there on projects related to theirPRIMES projects, which magnifies the effects ofboth programs and often results in much strongerfinal papers.

How do you measure success?S.G.: Every year PRIMES students win many

prizes at national science competitions, includingthe very top ones. For example, in four years PRIMES

has claimed twenty-four Siemens and fifteen IntelSTS finalist awards. The first and second prizesat Siemens 2014, as well as the first, second, andthird prizes for basic research at Intel STS 2015,went to PRIMES. Yet this is not our main criterionof success. A more important one is publications:our students have completed seventy-one papers,posted forty of them on arXiv.org, and publishedfifteen in high-level academic journals. Anothercriterion is matriculation record: virtually all ourgraduates go to top universities, where they areamong the best students. Finally, the number ofapplications: in PRIMES-USA, it has tripled in thelast two years. But above all, we feel that ourmission is accomplished when our students get ataste of genuine mathematical research and fall inlove with mathematics.

T.K.: The ultimate measure of success will comein a few years when these kids grow up. They arejust amazing! I feel honored to work with the bestmathematicians of the future generation.

Is your goal to win the largest possible num-ber of prizes?

P.E.: Not really. We tell our students not tohyperfocus on winning science competitions andexplain that mathematical research is about collab-oration rather than competition. Yet, competitionsare useful as an organizing and motivating factor.They need to write a paper which will be read byjudges by a certain deadline, and this makes adifference. Also the Siemens and Intel STS com-petitions do a great job organizing activities forfinalists. They meet and discuss their work withvery competent judges, some of them top-levelprofessional researchers in the field of their project.They also learn a lot from each other. And, last butnot least, they have a lot of fun!

Do you expect all PRIMES math students tobecome research mathematicians? If they don’t,do you view this as failure?

T.K.: Not necessarily. Some of them may wantto do computer science, law, business, medicine,and so on. They come to us because they want tochallenge their minds and try to see what mathresearch is like, and this experience is valuable tothem whatever career they choose. We had caseswhen students enjoyed math research so muchthat they changed their life plans and decidedto become mathematicians. And we had otherstudents who realized that they do not want to bemathematicians. They have a gift for mathematics,but their hearts are not there. And it is very usefulto discover this before college.

So being sure that one wants to become a math-ematician is not a requirement for our program.Intellectual curiosity and willingness to exploreare way more important.

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Figure 4. Primes Circle Conference 2013: PRIMESCircle students Omotoyosi Oyedeji and TyreikSilva are giving a talk about probability theory.

What do you do to help diversify the mathe-matical community?

S.G.: In 2013 we set up PRIMES Circle, a mathenrichment program for talented sophomoresand juniors from local urban public high schools.Working in small groups under the guidance of MITundergraduate students, PRIMES Circle participantsdiscover the beauty of the mathematical way ofthinking and the thrill of solving a challengingproblem. Circle students study advanced topics ingeometry, probability, combinatorics, knot theory,and so on; prepare expository papers; and makepresentations at a miniconference at MIT. PRIMESCircle has expanded from eight students in 2013to fifteen in 2015. Of current Circle students 60percent are female, 27 percent are Hispanic, and13 percent are African-American (see Figure 4).

In 2015 we organized a new section, MathROOTS,a twelve-day summer camp hosted by MIT fornationally selected promising high school studentsfrom underrepresented backgrounds interested increative mathematical experiences. At MathROOTSstudents discover new mathematical ideas andlearn problem-solving skills through a series ofclasses, group activities, and invited lectures ledby a team of instructors with diverse experiencesdoing and teaching both research and competitionmath. (See Figure 5).

The mission of PRIMES Circle and MathROOTSis to increase diversity in the mathematical com-munity by helping strong students from under-represented backgrounds develop their interest inmathematics and to set them on a path towardpursuing a math-based major in college.

Do students enjoy your program?S.G.: Every year we collect student impressions

of the program and post them on the PRIMESwebsite on the “Testimonials” page. Here are acouple of excerpts:

“At the beginning of the PRIMES program inJanuary last year, I was mildly nervous that

Figure 5. MathROOTS students: (l to r) JosueSican, Ben Bennington-Brandon, TrajanHammonds, Adedoyin Olateru-Olagbegi.

I would not be able to discover anythingnew. However, such fears were certainlyunmerited. During the first few meetings,my mentor provided my partner and me withbackground readings to become familiarwith the common techniques. Within twomonths, we were formulating some ofour own conjectures based on computersimulations, and before long, we wereeven able to find proofs of some of theseconjectures.”

“I loved the feeling of being able to sitand think about problems without havinganything else in my mind. It was a stress-freeenvironment, and I thrived here. PRIMESis an excellent program—it’s a remarkableway to start research at a young age withthe help of incredible professionals andmentors who love the math and sciencethat you do and will help you learn moreand more. I’m very glad I chose to come toPRIMES, and it has truly changed my life asa student and a mathematician.”

T.K.: Many high schools are worried about failingstudents and do not worry about bright studentsbeing bored. In our program no one is bored.

S.G.: Not even the program director! PRIMES hasgrown almost four times since its creation andcurrently has well over a hundred affiliates. Itsadministration and accounting have become aschallenging as a PRIMES project!

Who pays for PRIMES?S.G.: PRIMES is free for students, which is why

it is not at all free for MIT. But it is paid for bygenerous people with big hearts. The biggest heartsbelong to the NSF Department of MathematicalSciences and the MIT math department (personally,its former Head Mike Sipser, currently MIT’s Deanof Science, and its current Head Tom Mrowka),

September 2015 Notices of the AMS 917

who have provided crucial support since the incep-tion of PRIMES. NIH, Clay Mathematics Institute,Simons Foundation, Rosenbaum Foundation, somecompanies and private donors have also mademajor contributions. Notably, George Lusztig, MITmathematics professor and the recipient of the2014 Shaw Prize in mathematics, used part of hisprize to make a very significant gift to PRIMES asthe first contribution to its endowment. This madeit possible to establish George Lusztig PRIMES men-torships. Several such mentorships are awardedeach year to continuing mathematics mentors forexceptional mentor service in past years.

P.E.: In fact, while we all think hard about math,our program director has to think hard how to findmore people with big hearts. And he will definitelyappreciate your help!

One of the PRIMES research papers is called“Cookie Monster Plays Games.” Is this a seriousmathematical paper or is it really about cookies?Do you supply cookies for your research?

Figure 6. PRIMESstudent Leigh Marie

Braswell with theCookie Monster

(PRIMES Conference,2013).

T.K.: It is entirely serious! TheCookie Monster (Figure 6) is just afun way to represent a certain class ofcombinatorial problems. Namely, thereare several jars with cookies. In onemove the Cookie Monster is allowed tochoose some of the jars and take thesame number of cookies from all ofthem. The question “Given the numberof cookies in jars, what is the smallestnumber of moves needed to empty allthe jars?” was studied by Leigh MarieBraswell, a PRIMES 2013 student ([BK1];[BK2]: see Figure 6). With another stu-dent, Joshua Xiong (PRIMES 2014), weconverted the Cookie Monster probleminto a game and made some interestingdiscoveries [KX].

P.E.: After attending our conference,MIT mathematics professor Richard

Stanley observed that a certain breed of CookieMonster (the one that eats cookies only fromconsecutive jars) corresponds to the combinatoricsof the root system An−1 attached to the simple Liealgebra sl (n). In fact, there are rumors that thisbreed was genetically engineered at my request atthe MIT biology department to encourage studentsto learn about Lie algebras!

S.G.: Even though the Cookie Monster researchdoes not really require cookies, we do supplythem during the annual PRIMES conference in May,which you are welcome to attend. In fact, everyyear, the night before the conference Pavel drivesto Costco and fills up his van with cookies forparticipants. This allows all of us to enjoy manykinds of delicious cookies. They serve as a catalystfor making coffee into theorems!

References[BEJKL] S. Bhupatiraju, P. Etingof, D. Jordan, W. Kusz-

maul, and J. Li, Lower central series of a freeassociative algebra over the integers and finitefields, J. Algebra 372 (2012), 251–274.

[BK1] L. M. Braswell and T. Khovanova, On the CookieMonster Problem, arXiv:1309.5985.

[BK2] , Cookie Monster devours naccis, CollegeMathematics Journal 45 (2014), no. 2.

[CEJZ] Y. Chen, P. Etingof, D. Jordan, and M. Zhang,Poisson traces in positive characteristic,arXiv:1112.6385.

[Ch] C. Chen, Maximizing volume ratios for shadowcovering by tetrahedra, arXiv:1201.2580.

[CKK] C. Chen, T. Khovanova, and D. A. Klain, Volumebounds for shadow covering, Trans. Amer. Math.Soc. 366 (2014), 1161–1177.

[DS] S. Devadas and S. Sam, Representations of ratio-nal Cherednik algebras of G(m, r, n) in positivecharacteristic, J. Commut. Algebra (online).

[DT] F. Ding and A. Tsymbaliuk, Representations of in-finitesimal Cherednik algebras, Represent Theory(electronic) 17 (2013), 557–583.

[Li] C. Lian, Representations of Cherednik algebrasassociated to symmetric and dihedral groups inpositive characteristic, arXiv:1207.0182.

[FX] Yael Fregier and Isaac Xia, Lower Central SeriesIdeal Quotients Over Fp and Z, arXiv:1506.08469.

[Go] N. Golowich, Resolving a Conjecture on degreeof regularity of linear homogeneous equations,Electro. J. Combin. 21 (2014), no. 3.

[Ja] R. Jagadeesan, A new Gal(Q/Q)-invariant ofdessins d’enfants, arXiv:1403.7690.

[KL] Shashwat Kishore and Augustus Loner-gan, Signatures of Multiplicity Spaces in tensorproducts of sl 2 and Uq(sl 2) Representations,arXiv:1506.02680.

[KX] T. Khovanova and J. Xiong, Cookie Monsterplays games, arXiv:1407.1533.

[JSZ] T.-Y. Jiang, Z. Scully, and Y. X. Zhang, Motorsand impossible firing patterns in the parallel chip-firing game, SIAM J. Discrete Math. 29 (2015), 615–630.

[Z] J. Zurier, Generalizations of the Joints Problem,math.mit.edu/research/highschool/primes/materials/2014/Zurier.pdf.

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