www.msri.org

EMISSARYM a t h e m a t i c a l S c i e n c e s R e s e a r c h I n s t i t u t e

Spring 2017

Analytic Number TheoryEmmanuel Kowalski

The basic problem in analytic number theoryis often to compute how many integers, orprimes, or other arithmetic objects (from L-functions to modular forms to elliptic curvesto class groups to . . . ), satisfy certain condi-tions. Such questions have fascinated math-ematicians for centuries; for instance, onecan ask:

(a) How many prime numbers are thereless than some large given number x?How many n 6 x are there such thatthe interval from n to n+246 containsat least two prime numbers?

(b) Fixing integers k > 2 and r > 1, for alarge integer n, how many integers n

1

,. . . , n

r

are there such that nk

1

+ · · ·+n

k

r

= n (Waring’s problem)?

(c) How many integers in an interval x <

n 6 x + x

1/2 are only divisible byprimes smaller than x

1/20? or largerthan x

1/1000? or both?

A Kloosterman path joins the partial sums of Kl2

(1;10007) over 16 x6 k when

k varies from 0 to 10006. (The figure appears with axes on page 5.)

(d) How many integers n6 x are such thatn

4+2 is not divisible by the square ofany prime?

The techniques that are used in analytic num-ber theory are remarkably varied and cun-ning (combinatorics, harmonic and complexanalysis, probabilistic ideas, ergodic theory,

algebraic geometry, representation theory,all — and others — find their use). And theinfluence of such problems on mathematicsoften goes well beyond the original moti-vation of solving them. For instance, com-plex analysis benefited in the 19th and early20th century from the efforts devoted tothe proof of the Prime Number Theorem,from Riemann to Hadamard and de la Val-

lée Poussin. Similarly, representation theorytook some of its inspiration from Dirichlet’sstudy of primes in arithmetic progressions;much more recently, the ideas of higher-order Fourier analysis were to a large extentpursued in connection with problems aboutlinear equations in primes, in particular inthe recent work of Green, Tao, and Ziegler.

(continued on page 4)

ContentsThe View from MSRI 2CME Group–MSRI Prize 2Uhlenbeck Fellowship 3Named Positions 3ANT Program (cont’d) 4Jill Pipher 6Upcoming Workshops 6Public Understanding 7Math Circles 7HA Program 8

Kaisa Matomäki 9Spring Postdocs 11Poem: A Kind of Silence 12Call for Proposals 12Pacific Journal of Math. 12Puzzles Column 13Call for Membership 13Clay Senior Scholars 132016 Annual Report 14Staff Roster 16

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Festivals, Concerts, Contests, Circles. For a Public Under-standing update and Math Circles news — including TatianaShubin’s visit to the Tulsa Girls’ Math Circle — visit page 7.

1

The View from MSRIDavid Eisenbud, Director

Time Scales of the ProgramsLike annual flowers, new programs and workshops bloom at MSRIeach spring and fall. Other Institute activities have different rhythms.Here are some reflections on the long and short time scales.

The Scientific Advisory Committee (SAC) meets twice a year inperson, each time for two days, and one agendum each time is adiscussion of potential semester-long programs, either submittedfrom outside or generated from within, that MSRI might run in thefuture. It’s a surprisingly distant future: because of the planningand resources that are necessary, we generally approve programsin place three years ahead of when they will run, and the processof vetting and accepting a program, from letter of intent to finalsignoff, can add a year or more.

We also take account of “hot topics” with workshops planned a yearor less in advance. This spring’s Hot Topics workshop, on MotivicPeriods, was planned last year, and the Insect Navigation workshopwe ran in the fall with the Howard Hughes Medical Institute wasplanned at even shorter range. The conferences on Critical Issuesin Math Education, the Summer Graduate Schools . . . all these areplanned more quickly than the semester-long programs.

One planning operation takes surprisingly long (surprising, anywayto me): the National Math Festival. The “first ever” took place inApril 2015. We are organizing the second Festival for April 22,2017, and we’ve been working on it ever since the close of thefirst Festival! I believe the work has paid off: You can see what’scoming at nationalmathfestival.org. About 25,000 people attendedthe public day of the first Math Festival — we expect a still largercrowd this time. Come and join the fun if you can!

Time Scales of the InstituteMSRI held its first programs nearly thirty-five years ago, in thefall of 1982 (they were Nonlinear Partial Differential Equations,organized by A. Chorin, I. M. Singer, S.-T. Yau, and MathematicalStatistics, organized by L. LeCam, D. Siegmund, C. Stone). This isnot quite a mathematical lifetime ago: many of the members andpostdocs of the programs are now leaders in the field (and such

a postdoc, David Donoho, has recently joined MSRI’s Board ofTrustees). These thirty-five years have been long enough for MSRIto have established itself as an essential part of the mathematicalestablishment of the United States and of the world. In recent timethe number of applications for MSRI’s semester programs, work-shops and schools has risen steadily, and now far outstrips what wecan provide.

On the other hand, thirty-five years is short in the lives of greatinstitutions. MSRI is still highly dependent on just a few sources offunding, and thus vulnerable to changes outside the mathematicalcommunity. Given the importance that MSRI has for the commu-nity, it seems to me of the greatest importance to lay the groundworkfor a broader base of support, one that will ensure the Institute’scontinued health over the next 35 years and beyond.

Personal Time Scales — Ars Longa, Vita BrevisI first came to be Director at MSRI in 1997 — 20 years ago thissummer. I retired from the job (and Robert Bryant became Direc-tor) in 2007, and after which I taught, and worked for the SimonsFoundation, until taking a new term as Director in 2013. Now I’veaccepted a renewal until 2022. I’m enjoying the job, and there’smuch I hope to accomplish. But at the end of this next term I willbe 75, and, both for MSRI’s sake and my own, it will be time toturn over the reins!

My aspirations for the next five years? First and foremost, to con-tinue the breadth and excellence that has characterized MSRI’sprograms. But, also, to plan for the long-range future by increasingthe diversity of MSRI’s funding. We’ve already made progress: Inthe 2017–18 budget, just passed by the Board of trustees, 48% of thefunding will come from sources other than the federal government.This is exactly double the percentage of 2012–13, five years ago,and the total is up as well.

The new funding is made up of endowment, grants from privatefoundations, and individual giving. Thanks to all those whose giftsand help have made this expansion a reality! Your generosity servesmathematics in a crucial way. I believe that this diversification, ifcultivated and developed, will make MSRI sustainable in the longrun, so that the Institute can continue to play a central role in themathematical life of the country and the world.

CME Group–MSRI Prize in Innovative Quantitative Applications

Robert B. Wilson

The eleventh annual CME Group–MSRI Prize in Innovative Quantitative Applications was awardedto Robert B. Wilson, Adams Distinguished Professor of Management, Emeritus, and Professor ofEconomics (by courtesy), School of Humanities and Sciences, Stanford Graduate School of Business,at a luncheon in Chicago on February 2, 2017.

Dr. Wilson’s research and teaching are on market design, pricing, negotiation, and related topicsconcerning industrial organization and information economics. He is an expert on game theory and itsapplications. A panel discussion on “Frontiers of Game-theoretic Applications in Economics” washeld in conjunction with the award ceremony.

The annual CME Group–MSRI Prize is awarded to an individual or a group to recognize originalityand innovation in the use of mathematical, statistical, or computational methods for the study of thebehavior of markets, and more broadly of economics. You can read more about the CME Group–MSRIPrize at tinyurl.com/cme-msri.

2

The New Uhlenbeck Fellowship

Karen Uhlenbeck

As we announced in the previous is-sue of the Emissary, an anonymousdonor has made possible two newnamed fellowships at MSRI, cre-ated in honor of Dr. Karen Uhlen-beck and Dr. Dusa McDuff, bothcurrent MSRI trustees.

Postdoctoral fellows at MSRI havethe unique opportunity to work to-gether with leading researchers intheir fields in an environment thatpromotes creativity and the effec-tive interchange of ideas and tech-niques. Privately funded fellow-ships play an important part in providing these opportunities. Thefirst McDuff Fellowship was awarded last fall, and this spring marksthe first awarding of the new Uhlenbeck Fellowship.

Karen Uhlenbeck, currently a trustee of MSRI, received her Ph.D.from Brandeis University in 1968. Following appointments atMIT, UC Berkeley, the University of Illinois, and the University ofChicago, she has held since 1987 the Sid W. Richardson Founda-tions Regents Chair III in Mathematics at the University of Texas atAustin. From 2014–17, she is a visiting Professor at the Institutefor Advanced Study in Princeton. She also served as Vice Presidentof the American Mathematical Society from 1987–90.

Karen is a highly distinguished mathematician specializing in dif-ferential geometry, nonlinear partial differential equations, andmathematical physics. At the same time, her efforts across theeducational spectrum, especially her role as a founder of the IAS/Park City Mathematics Institute, have added vitality to the mathe-matical scene. Karen’s mentoring, both formal (she co-founded theAnnual Women in Mathematics Program at the IAS) and informal,is legendary.

Karen has been awarded a MacArthur “Genius” Fellowship, theSteele Prize of the American Mathematical Society, and the U.S.National Medal of Science. In 1990, she became the second woman(after Emmy Noether in 1932) to present a plenary lecture at anInternational Congress of Mathematics. She is a fellow of theAmerican Academy of Arts and Sciences and the U.S. NationalAcademy of Sciences. Karen is also the recipient of seven hon-orary degrees, most recently from Harvard, Princeton and BrandeisUniversities.

. . . and First Uhlenbeck Fellow

Naser Talebizadeh Sardari

Naser Talebizadeh Sardari is thefirst recipient of the Uhlenbeckpostdoctoral fellowship. He is amember of this spring’s AnalyticNumber Theory program.

In his Ph.D. thesis, Naser studiedthe distribution and lifting prop-erties of solutions of quadraticequations, which have appli-cations in a number of areasof analysis and mathematicalphysics, and proved a uniformquantitative form of strong ap-proximation for a quadratic formin at least 4 variables that is op-timal for at least 5 variables. Currently, Naser’s favorite resultstates that whereas there is a polynomial time algorithm dueto Schoof to express integers as sums of two squares, whenpossible, the analogue problem with imposed congruence condi-tions is NP-complete, assuming some widely-believed arithmeticconjectures.

Named Positions for Spring 2017MSRI is grateful for the generous support that comes from endowments and annual gifts that support faculty and postdoc members of itsprograms each semester. The Clay Mathematics Institute awards its Senior Scholar awards to support established mathematicians to playa leading role in a topical program at an institute or university away from their home institution.

Eisenbud and Simons ProfessorsAnalytic Number TheoryTim Browning, University of BristolChantal David, Concordia UniversityAndrew Granville, Université de MontréalPhilippe Michel, École Polytechnique Fédérale de LausanneTamar Ziegler, Hebrew University of Jerusalem

Harmonic AnalysisGuy David, Université Paris-SudCiprian Demeter, Indiana UniversityJill Pipher, Brown UniversityElias Stein, Princeton University

Chancellor’s ProfessorHarmonic AnalysisTatiana Toro, University of Washington

Named Postdoctoral FellowsAnalytic Number TheoryViterbi: Julia Brandes, University of GothenburgBerlekamp: Maria Nastasescu, Brown UniversityUhlenbeck: Naser Talebizadeh Sardari, Princeton University

Harmonic AnalysisHuneke: Max Engelstein, Massachusetts Institute of TechnologyStrauch: Marina Iliopoulou, University of Birmingham

Clay Senior ScholarsAnalytic Number TheoryManjul Bhargava, Princeton UniversityDavid (Roger) Heath-Brown, University of Oxford

Harmonic AnalysisAlexander Volberg, Michigan State University

3

Analytic Number Theory (continued from page 1)

A StrategySuppose we want to count, at least approximately, the solutions toa problem of analytic number theory. Let N be this number; thebasic approach to evaluating N often proceeds in the three followingsteps:

(a) Try to guess the answer, using various heuristic techniquesand expected properties of the objects that are considered (forexample, since Gauss, we expect that an integer n of large sizeis prime with probability about 1/(logn)). Let G denote thisguess.

(b) Try to express naturally the desired number N as the sum ofG and some other expression R that might be tractable. If thisfails, attempt, at least, to obtain some lower (or upper) boundsof the kind N > aG+R or N 6 aG+R for some numbera > 0.

(c) If step (b) has succeeded, the remainder R will often look likea sum of quantities that change sign, and the remaining task isto show that these changes of sign are sufficiently chaotic toensure that R is of size smaller than the guess G.

Very frequently, it is the last step that contains the core of a proof,and for this reason, much work in analytic number theory is devotedto proving estimates for oscillating sums of various origins. Theexamples we discuss below are also of this type. This does notmean that the first two steps are easy — for instance, much of therecent spectacular progress on gaps between primes is based onexceptionally clever arguments related to these first two stages, dueto Goldston–Pintz–Yıldırım, and more recently to Maynard andTao.

Multiplicative FunctionsIn the study of the distribution of prime numbers, and of the multi-plicative properties of integers, many fundamental questions involvethe interplay between addition and multiplication. The Möbius func-tion µ(n) and its close cousin the Liouville function �(n) illustratevery well how little we understand of this interaction! For a positiveinteger n> 1, one defines �(n) = (-1)k, where k is the number ofprime factors of n, counted with multiplicity. This is an example ofan oscillating function. The sum

M(x) =X

16n6x

�(n)

then measures the difference between the number of integers up to x

with an even and with an odd number of prime factors. The simplequestion of estimating its size as x gets large is already one of thegreat mysteries of number theory. The conjecture that M(x)/x�

tends to 0 for any fixed � > 1/2 is equivalent to the Riemann hy-pothesis — but this is only known for � = 1! However weak thisseems, the fact that M(x)/x tends to 0 has rich consequences: it isequivalent to the Prime Number Theorem, that the number ⇡(x) ofprime numbers p6 x is asymptotic to x/(logx).

Even the analogue of such a weak statement as M(x)/x! 0 is outof reach if one attempts to study correlations of values of the Möbiusor Liouville function. For instance, if k> 2 and 06 h

1

< · · ·< h

k

are fixed integers, one conjectures that the sums

X

16n6x

�(n+h

1

) · · ·�(n+h

k

) (⇤)

should also be o(x) as x gets large. In other words (for k = 2,(h

1

,h

2

) = (0,1)) to know the value of �(n) should not allow us toguess anything about the value of �(n+1). What could look morenatural? As before, even a small amount of cancellation wouldhave important consequences, and yet this remains open as soon ask> 2. However, a recent breakthrough of Matomäki and Radziwiłłin the study of short sums of multiplicative functions (see the articleof K. Soundararajan) has led to spectacular progress — even formuch more general functions, thanks to the idea of “pretentious”functions mimicking the Möbius function, due to Granville andSoundararajan.

In particular, in joint work with Tao, Matomäki and Radziwiłł haveproved that the quantities in (⇤) are o(x) as soon as one can average,even in the slightest way, over the parameters (h

1

, . . . ,h

k

). Anotherresult of Tao, restricted to k = 2, is the logarithmically averagedform of (⇤); among other ingredients, it has led to the solution ofthe Erdos discrepancy problem.

Visualizing the peaks in the circle method: The circle method

extracts a precise guess to the number of solutions to Waring’s

problem by analyzing these peaks.

The Circle Method

When attempting to solve polynomial equations with integral coeffi-cients, one of the basic methods is the circle method, which uses

4

the elementary formula

Z1

0

exp(2i⇡↵(n-m))d↵=

�0 if n 6=m

1 if n=m

to detect the equality of two integers. From there, the number N ofsolutions to Waring’s problem of representing n> 1 as a sum of rinteger k-th powers is

N=

Z1

0

S(↵)r exp(-2i⇡n↵)d↵ ,

where S(↵) is the oscillatory sumX

16m6n

1/k

exp(2i⇡↵mk) .

The circle method, first developed by Ramanujan, Hardy, and Lit-tlewood, extracts from this formula a very precise guess G byevaluating the part of the integral corresponding to values of ↵ thatare close to a rational number a/q with small denominator. Thisis illustrated by the figure on the preceding page, where the peaksalong the circle correspond to such values of ↵, where the functionS(↵) is much more predictable.

To show that N is indeed close to G is relatively easy if the numberr of variables is allowed to be large, since a large r accentuatesthe contributions of those peaks. When r gets smaller, it becomesincreasingly difficult to control the contribution of the other valuesof ↵. In recent years, striking progress has been made in this area byWooley through his method of “efficient congruencing.” Of specialnote, a crucial conjecture of Vinogradov on the optimal size ofZ

[0,1]n

���X

16m6x

exp(2i⇡(↵1

m+ · · ·+↵

n

m

n))���2s

d↵

1

· · ·d↵n

was proved by Wooley for n= 3, and in general by Bourgain, Deme-ter, and Guth, who used methods of harmonic analysis — illustratingonce again the vitality of the connections between analytic num-ber theory and other areas of mathematics. The circle method andthis result are discussed in more detail in the Harmonic Analysisprogram article by L. Guth on page 8 of this issue.

Trace FunctionsAs our last example, we wish to highlight the truly remarkable inter-play between algebraic geometry and exponential sums over finitefields. This is currently studied extensively (by Fouvry, Kowalski,Michel, and Sawin, among others) in the general context of “tracefunctions,” which are functions over finite fields similar to the clas-sical functions of mathematical physics. They appear extremelyfrequently in problems of analytic number theory, and their algebro-geometric origin often holds the key to analytic applications.

A typical example, which is ubiquitous in the theory of quadraticforms, modular forms, and L-functions, is the Kloosterman sumKl

2

(a;p), defined for a prime number p and an integer a coprimeto p by

Kl2

(a;p) =1

pp

X

16x6p-1

exp⇣2i⇡

p

(ax+ x)⌘

,

where x denotes the inverse of x modulo p.

The example of a Kloosterman path in the figure below (also shownon the cover) illustrates the oscillatory nature of this sum: it showsthe polygonal path joining the partial sums of Kl

2

(1;10007) over16 x6 k when k varies from 0 to 10006.

–1.5 –1.0 –0.5

–0.1

0.1

0.2

0.3

0.4

0.5

A Kloosterman path: The polygonal path joins the partial sums

of Kl2

(1;10007) over 16 x6 k when k varies from 0 to 10006.

Obtaining a good estimate for the Kloosterman sums is crucialto many applications. The best possible individual bound is|Kl

2

(a;p)|6 2, which was proved by A. Weil in the 1940s using theRiemann hypothesis for curves over finite fields. Deeper properties,concerning the variation of the sum with a, involve much moresophisticated algebraic geometry, and especially Deligne’s mostgeneral form of the Riemann hypothesis over finite fields.

Zhang’s sensational proof of the existence of infinitely many primeswithin intervals of bounded length hinged, for instance, on the proofof estimates like

���X

16y6p-2

Kl2

(y;p)Kl2

⇣y

y+1

;p⌘���6 Cp

1/2 ,

which first appeared, with a proof by Birch and Bombieri, in a paperof Friedlander and Iwaniec. In fact, we can illustrate the breathtak-ing efficiency of algebraic geometry with the following estimate,following from the Riemann hypothesis over finite fields and workof Katz: for any k> 2, if the shifts (h

i

) are distinct modulo p, wehave

���X

16a6p-1

Kl2

(a+h

1

;p) · · ·Kl2

(a+h

k

;p)���6 C

k

p

1/2

for some constant Ck

> 0— compare this statement to the conjec-tures concerning the sum (⇤). Estimates of “sums of products” likethese were used, for instance, by Kowalski and Sawin to “explain”the statistical shape of paths like those in the Kloosterman pathfigure.

5

Focus on the Scientist: Jill PipherThis spring, Jill Pipher is one of the Eisenbud Professors in theHarmonic Analysis program at MSRI. Jill was born in Harrisburg,PA. An interest in music led her to Indiana University to studypiano. She transferred to UCLA, where her coursework rangedfrom Russian literature to epistemology. A course in set theoryand logic set her on the path to a career in mathematics.

Jill Pipher

At UCLA, Jill completed her B.A.and continued for a Ph.D. witha dissertation supervised by JohnGarnett. After serving as anL.E. Dickson instructor and As-sistant Professor at the Universityof Chicago, she moved to BrownUniversity, where she is the ElishaBenjamin Andrews Professor ofMathematics.

Her fundamental contributions toboth analysis and cryptographyhave been widely recognized: shewas elected a fellow of the Amer-

ican Mathematical Society (inaugural class), was an invitedspeaker at the International Congress of Mathematicians in 2014,and was elected Fellow of the American Academy of Arts andSciences in 2015. She has been named Vice President for Re-search at Brown, effective Summer 2017.

Jill is an applied mathematician. In 1999, together with JeffHoffstein and Joe Silverman, she co-founded NTRU Cryptosys-tems, Inc., to market cryptographic algorithms, NTRUEncryptand NTRUSign. These are based on the shortest vector problemin lattices. Today, NTRU is a part of Security Innovation, Inc.Jill holds several related patents.

She is also a theoretician. In mathematical analysis, Jill’s work

lies at the heart of the theory of solutions of elliptic partial differ-ential equations (PDE) on non-smooth domains. Her expertisein harmonic analysis has been essential in the development ofthe modern perspective on the study of boundary regularity ofsolutions of elliptic PDEs on the natural class of domains whoseboundaries have Lipschitz regularity. One of the main themes ofthis spring’s Harmonic Analysis program is rooted in these con-tributions. One of her lectures delivered a few years ago vividlyillustrated how Carleson measures, a classical tool in harmonicand complex analysis, had transformed the field. A member ofthe audience summarized the talk: “Give Jill a Carleson measure,and she will change the world.”

Jill has also exercised a remarkable influence on the mathematicalcommunity at large. She was a driving force behind the formationof the Institute for Computational and Experimental Researchin Mathematics (ICERM) at Brown University and served as itsfounding director from 2010 to 2016. In the fall of 2011, asICERM was launching its first semester program, Jill simultane-ously began a term as president of the Association for Women inMathematics (AWM). Her energy and commitment are incredible.As AWM president, she guided a renewed focus on supportingresearch of women in Mathematics, and was instrumental in thelaunching of a new series of Biennial Research Symposia.

Jill has inspired so many of us, both professionally and person-ally. She is a powerful and innovative researcher, whose scientificwork spans the gamut from overarching theory to industrial ap-plication. Her leadership has sparked the creation of a scientificinstitute and a private company, and has guided the developmentof other institutions. She has done all this, while raising a familythat includes three children and two grandchildren. The term“gamut” — whose origins lie in the musical scale with which Jill’sodyssey began — could not be more apt.

— Tatiana Toro and Michael Christ

Forthcoming WorkshopsMay 1–5, 2017: Recent Developments inAnalytic Number Theory

May 15–19, 2017: Recent Developmentsin Harmonic Analysis

Jun 7–9, 2017: Careers in Academia (atthe American Institute of Mathematics, SanJose)

Aug 17–18, 2017: Connections for Women:Geometry and Probability in HighDimensions

Aug 21–25, 2017: Introductory Workshop:Phenomena in High Dimensions

Aug 31–Sep 1, 2017: Connections forWomen Workshop: Geometric andTopological Combinatorics

Sep 5–8, 2017: Introductory Workshop:Geometric and Topological Combinatorics

Oct 9–13, 2017: Geometric andTopological Combinatorics: ModernTechniques and Methods

Nov 13–17, 2017: Geometric FunctionalAnalysis and Applications

Nov 29–Dec 1, 2017: Women in Topology

Summer Graduate Schools

Jun 12–23, 2017: Subfactors: PlanarAlgebras, Quantum Symmetries, andRandom Matrices

Jun 26–Jul 7, 2017: Soergel Bimodules

Jul 10–21, 2017: Séminaire deMathématiques Supérieures 2017:Contemporary Dynamical Systems(Montreal)

Jul 10–21, 2017: Positivity Questions inGeometric Combinatorics

Jul 17–28, 2017: Nonlinear DispersivePDE, Quantum Many Particle Systems, andthe World Between

Jul 24–Aug 4, 2017: Automorphic Formsand the Langlands Program

To find more information about any of theseworkshops or summer schools, as well asa full list of all upcoming workshops andprograms, please visit msri.org/scientific.

6

Public UnderstandingNational Math Festival

The 2017 National Math Festival ar-rives Saturday, April 22, in Washing-ton, DC, hosted by MSRI in cooper-ation with the Institute for Advanced

Study (IAS) and the National Museum of Mathematics (MoMath).This free, public event features a full day of math activities forall ages from 10am–7pm, including lectures, hands-on demos, art,films, performances, athletics, games, and stories at the WashingtonConvention Center. The 2017 Festival is supported by donationsfrom private foundations, corporations, and individuals, includingthe Simons Foundation.

A full schedule of all 2017 Festival events is online at nationalmath-festival.org or as a mobile app, NMF 2017.

Featured PresentersStephon Alexander • American Mathematical Society • Association forWomen in Mathematics (AWM) • The Bridges Organization • Eugenia

Cheng • Alissa Crans • Christopher Danielson • Emille Davie Lawrence •Robbert Dijkgraaf • Maria Droujkova • Marcus du Sautoy • Elwyn and

Jennifer Berlekamp Foundation • Events DC • FIRST • First 8 Studios atWGBH • James Gardner • Gathering 4 Gardner • The George WashingtonUniversity, Department of Mathematics • Paul Giganti • Herbert Ginsburg •Rebecca Goldin • Brady Haran • George Hart • Elisabeth Heathfield • Joan

Holub • Ithaca College, Department of Mathematics • Julia RobinsonMathematics Festival • Scott Kim • Marc Lipsitch • Po-Shen Loh • Math for

Love • MSRI • MathPickle • Talea L. Mayo • Mark Mitton • MichaelMorgan • Colm Mulcahy • National Aeronautics and Space Administration(NASA) • National Museum of Mathematics (MoMath) • Natural Math •NOVA • NOVA Labs • Laura Overdeck • Stephanie Palmer • Matt Parker •Andrea Razzaghi • Mark Rosin • Science Cheerleader • Raj Shah • TravisSperry • Clifford Stoll • James Tanton • Richard Tapia • ThinkFun • Janel

Thomas • Mariel Vazquez • Talitha Washington • The Young People’sProject • Zala Films • Mary Lou Zeeman

Bay Area Mathematical OlympiadEach spring, MSRI hosts the award cer-emony for the Bay Area MathematicalOlympiad, following two exams (one for stu-dents in 8th grade and under, and one forstudents in 12th grade and under), each taken

by hundreds of Bay Area students, with five proof-type math prob-lems to be solved in four hours. This year, Matthias Beck (SanFrancisco State University) spoke to the competitors on reciprocityinstances in combinatorics.

Harmonic Series ConcertsMSRI hosts the free, public Harmonic Series concertsthroughout the year, highlighting the links betweenmusic and mathematics. In 2016–17, performanceswere held featuring the Del Sol String Quartet; violin-

ist (and University of Kansas math professor) Purnaprajna Bangerewith Amit Kavthekar, tabla; pianist David Saliamonas; and jazzvocalist Torbie Phillips with Jason Martineau (piano), Jeff Denson(bass), and Kelly Park (percussion).

Math Circles NewsDiana White, Director

New Associate Director, Dr. Jane LongThe National Association of Math Circles(NAMC) is excited to formally announceDr. Jane Long, an associate professor fromStephen F. Austin State University, as the newAssociate Director. Her primary focus is onleading the Math Circle–Mentorship and Part-nership program, with which she has served as

a mentor since its inception in 2015. She is founder and director ofthe East Texas Math Teachers’ Circle, which began meeting in 2013.Dr. Long regularly visits Math Circles for students and teachers asan invited facilitator and frequently engages in outreach activitiesrelated to problem solving for people of all ages and levels. Dr.Long earned a Ph.D. in algebraic topology from the University ofMaryland at College Park in 2008.

Tatiana Shubin Shares Navajo Math CirclesAt the invitation of the Tulsa Math Teachers’ Circle and the TulsaGirls’ Math Circle, mathematician Tatiana Shubin, a co-founder andco-director of the Navajo Nation Math Circles program, screenedthe film, Navajo Math Circles, to an enthusiastic Circle Cinemaaudience in early February. Members of the Tulsa Girls’ MathCircle worked with Shubin to sort various polyhedra into threegroups — finding Platonic solids, irregular solids, and hypercubes —and derive Euler’s constant (See the photo on the cover.)

Since its premiere in January 2016, Navajo Math Circles hasscreened at venues nationally and in Canada, and recently at Stan-ford University, and at seven film festivals, including Sedona,Arivaca, Arizona International, the American Indian FF and Vi-sion Maker. The film, sponsored by MSRI and Vision MakerMedia, premiered on PBS in September 2016 and will continueto be broadcast on local public television stations through 2020(www.navajomathcirclesfilm.com).

What are Math Circles? and NAMC?Math Circles are a unique form of K–12 outreach designed to im-prove students’ content knowledge and spark their mathematicalinterest and identity. In authentic, outside-the-classroom experi-ences, students engage with mathematics as an open-ended disci-pline involving exploration, conjecture, and communication withothers. Students work on problems — often tackled in groups witha mentor — that can be approached with minimal mathematicalbackground, but lead to deep and advanced mathematical concepts.The number of active Math Circles is ever-changing; currently thereare about 180 across the United States.

The National Association of Math Circles (NAMC) is a program ofMSRI designed to seed the creation of new Math Circles; provideresources, training, and support to Math Circles across the UnitedStates; and catalyze rigorous research and evaluation efforts forMath Circles. To learn more, and to be added to a list to receiveupdates on future training programs and grant opportunities, pleaseemail namc.manager@msri.org.

7

Fourier Analysis andDiophantine Equations

Larry Guth

Fourier analysis was initially developed in the early 1800s to studyproblems in mathematical physics. Fourier’s fundamental insightwas that a basically arbitrary function could be decomposed as alinear combination of complex exponentials with different frequen-cies. More informally, an arbitrary function can be written as a sumof sine waves with different frequencies. Fourier and others usedthis insight to study the time evolution of solutions to physicallymotivated partial differential equations, like the heat equation or thewave equation. This direction of research has remained fruitful eversince, and it is one of the active topics of the Spring 2017 HarmonicAnalysis semester at MSRI.

In the early 1900s, mathematicians realized that Fourier analysis isalso relevant to Diophantine equations in number theory. Just re-cently, Jean Bourgain and Ciprian Demeter developed a new theorycalled decoupling, which makes important progress on Diophantineequations. This theory uses insights that developed (partly) fromtrying to understand problems about physically motivated partialdifferential equations from the Fourier analysis angle. This decou-pling approach to Diophantine equations is another active topic inthis semester’s program.

The Fourier–Diophantine ConnectionThe connection between Fourier analysis and Diophantine equa-tions begins with the following observation. If m is an integer, thenthe integral

Z1

0

e

2⇡imx

dx

is equal to 1 if m= 0, and vanishes if m is not zero. We can use thisfact to express the number of solutions to a Diophantine equationas an integral. Suppose that P(a

1

, . . . ,a

r

) is a polynomial in r

variables with integer coefficients. Then the integral

Z1

0

X

16a

i

6A

e

2⇡iP(a1

,...,a

r

)xdx (⇤)

is equal to the number of solutions to the Diophantine equationP(a

1

, . . . ,a

r

) = 0 with 1 6 a

i

6 A. The last formula involves asum over many variables, but in some special cases, it can be sim-plified and written as a moment of a sum in fewer variables. Forexample, consider the polynomial

P(a1

, . . . ,a

6

) = a

3

1

+a

3

2

+a

3

3

-a

3

4

-a

3

5

-a

3

a

.

In this special case, the integral (⇤) reduces to

Z1

0

������

X

16a6A

e

2⇡ia

3

x

������

6

dx . (†)

This integral is the sixth moment of the trigonometric sum

AX

a=1

e

2⇡ia

3

x ,

which is a sum of waves of different frequencies (1, 8, 27, . . . ).To understand the sixth moment, we need to understand how thesewaves interact: How much constructive interference is there, andhow much do the different waves cancel each other? Let us make afew elementary observations to start to get a sense of the problem.

By the triangle inequality, we have immediately

�����

AX

a=1

e

2⇡ia

3

x

�����6A .

This bound is sharp when x= 0, because at this point, all the com-plex exponentials are positive real. In the physics literature, thiswould be called “pure constructive interference.” But we wouldlike to show that at most points x, the trigonometric sum is muchsmaller than A. The most classical tool to do this is the idea oforthogonality, which dates back to even before Fourier’s work. Thefunctions {e2⇡inx}

n2Z are orthogonal on [0,1], and so

Z1

0

������

X

16a6A

e

2⇡ia

3

x

������

2

dx=X

16a6A

Z1

0

���e2⇡ia3

x

���2

dx=A .

We can get a first bound on the sixth moment of our trigonometricsum by applying the triangle inequality and then orthogonality:

Z1

0

������

X

16a6A

e

2⇡ia

3

x

������

6

dx6A

4

Z1

0

������

X

16a6A

e

2⇡ia

3

x

������

2

dx=A

5 .

Now let us switch to the number theory side and make an educatedguess of the order of magnitude of this integral. Recall that theintegral (†) is equal to the number of solutions of the Diophantineequation

a

3

1

+a

3

2

+a

3

3

= a

3

4

+a

3

5

+a

3

6

,

with 1 6 a

i

6 A. Each side of this equation lies somewhere be-tween 3 and 3A

3. Both sides are complicated. In order to make arough guess of the number of solutions, let us imagine that eachside is a random integer in the range [3,3A3]. The probability thatthe two sides are equal is then on the order of A-3. There are A

6

possible choices of a1

, . . . ,a

6

, and so we might guess that there areon the order of A3 solutions.

8

Focus on the Scientist: Kaisa MatomäkiKaisa Matomäki is a key member of this spring’s Analytic Num-ber Theory program at MSRI. Kaisa is part of the exciting newgeneration of analytic number theorists who are radically trans-forming the field.

Kaisa Matomäki

Kaisa was born in Nakkila, Fin-land, and completed her under-graduate and masters degrees atthe nearby University of Turku.She obtained her doctorate in2009 (for work on applications ofsieve methods) under the super-vision of Glyn Harman at RoyalHolloway, University of London,before moving back to the Univer-sity of Turku, where she currentlyholds an Academy Research Fel-lowship.

Kaisa’s work is characterized by a virtuosic technique whichallows her to brush aside formidable technical difficulties andget straight to the heart of a problem. While she has a numberof impressive publications, perhaps her standout result is the re-cent spectacular work with Radziwiłł on multiplicative functionsin short intervals (Annals of Math., 2016). A central object in

number theory is the Liouville function which is a completelymultiplicative function taking the value 1 on numbers with aneven number of prime factors, and -1 on numbers with an oddnumber of prime factors. The Liouville function is expectedto behave much like a random sequence of signs ±1, but littleis known in this direction. A special case of their work estab-lishes that for almost all starting points n, the Liouville functiontakes about as many positive values as negative in the interval[n,n+h], provided only that h tends to infinity with n.

This milestone result has already spurred further progress in thefield — it forms a crucial ingredient in Tao’s proof of a logarith-mic version of the Chowla conjecture, and which in turn led tohis resolution of the Erdos discrepancy problem. The Chowlaconjecture, which asserts that consecutive values of the Liouvillefunction are uncorrelated, appeared as intractable as the twinprime conjecture; now, after the work of Kaisa, Radziwiłł, andTao, it no longer seems such a distant dream.

Kaisa and Radziwiłł were recognized for their breakthrough withthe 2016 SASTRA Ramanujan Prize. Kaisa has also receivedthe 2016 Academy of Finland’s Award for Scientific Courage,and the 2016 Väisälä Prize. She lives in Lieto, Finland, with herhusband Pekka and two young children, ages one and four.

— Kannan Soundararajan

This is an open problem of number theory (and of Fourier analy-sis). Conjecturally, the integral (†) is bounded by CA

3. The bestknown bounds are better than the A

5 that we obtained, but stillsignificantly larger than A

3. In order to do better, we need betterestimates about the cancellation in the trigonometric sum. Theestimate from orthogonality tells us that the set of x 2 [0,1] where

������

X

16a6A

e

2⇡ia

3

x

������>A

3/4

has measure at most A-1/2. But the conjectured bound on the sixthmoment implies that this set has measure at most A-3/2. So weneed to show that constructive interference is actually much rarerthan what orthogonality tells us.

Constructive Interference and OrthogonalityThe issue of constructive interference plays an important role inpartial differential equations from physics. Let us discuss the waveequation as an example. The wave equation models a variety ofkinds of waves, including sound waves traveling in air. UsingFourier’s insight, we can write any solution to the wave equationas a linear combination of pure sine waves, traveling in differentdirections. The places where these waves interfere constructivelyare places where the original solution is big: places where the soundwaves are focusing. Estimating how much focusing occurs is animportant problem in the study of the wave equation. The Strichartzinequality is a fundamental result in this area from the 1970s.

Suppose we consider a solution to the wave equation in three spatialdimensions, u(x,t), with initial data u

0

(x). The statement of theStrichartz inequality involves the frequencies of the waves in thedecomposition. To simplify the statement of the inequality, let usassume that the Fourier transform of the initial data, u

0

, is sup-ported in the unit ball and also that

RR3

|u0

(x)|2dx= 1. In this case,Strichartz’s inequality says that

Z

R3⇥R|u(x,t)|4 6 C ,

where C is a constant independent of the solution. This tells usthat the set of (x,t) 2 R3⇥R where |u(x,t)| > � has measure atmost C�-4. This bound is much stronger than what follows byorthogonality alone. (In fact, just using orthogonality does not giveany finite bound for the measure of this set.)

The Strichartz inequality grew out of work by Stein, Fefferman,and others on the “restriction” theory of the Fourier transform.This theory involves the Fourier transform on Rd for d > 2, andit involves geometric ideas in order to get further improvementbeyond just orthogonality. We’ll explain a little more about it be-low. The first number theory problem we described involves aone-dimensional integral, but there are similar problems that leadto higher-dimensional integrals. Decoupling theory leads to an im-provement in this higher-dimensional setting using geometric ideasthat come out of restriction theory.

(continued on next page)

9

Harmonic Analysis (continued from previous page)

The Approach to Higher DimensionsHigher dimensional integrals appear whenever we study a systemof Diophantine equations instead of a single equation. Suppose that~P = (P

1

, . . . ,P

n

) is an n-tuple of polynomials with integer coef-ficients. The number of solutions of the system P

1

(a1

, . . . ,a

r

) =· · · = P

n

(a1

, . . . ,a

r

) = 0 with 1 6 a

i

6 A is given by an n-dimensional version of the integral (⇤):

Z

[0,1]n

X

16a

i

6A

e

2⇡i

~P(a

1

,...,a

r

)·~xdx

1

. . .dx

n

(⇤⇤)

In special cases, this expression can be written as the momentof a simpler trigonometric polynomial. For instance, Vinogradovproposed the following Diophantine system:

a

1

+ · · ·+a

s

= b

1

+ · · ·+b

s

,

a

2

1

+ · · ·+a

n

s

= b

2

1

+ · · ·+b

2

s

,...

a

n

1

+ · · ·+a

n

s

= b

n

1

+ · · ·+b

n

s

.

The number of solutions to this system with 16 a

i

,b

i

6A is givenby a moment integral:

Z

[0,1]n

������

X

16a6A

e

2⇡iM(a)·x

������

2s

dx , (‡)

where M(a) = (a,a2

, . . . ,a

n). In other words, the number ofsolutions is the 2s-moment of the trigonometric sum f(x):

f(x) =X

16a6A

e

2⇡iM(a)·x .

The case n= 2 was understood in Vinogradov’s time. Only recently,Trevor Wooley proved nearly sharp estimates for n= 3. Even morerecently, decoupling has led to a good understanding of this problemfor all n. Bourgain, Demeter, and I used the decoupling approachto give nearly sharp estimates for the number of solutions to theVinogradov system by estimating the moment integral (‡).

The First Three Stages of DecouplingThe decoupling approach considers many different pieces of thetrigonometric sum f(x). For any interval I⇢ [1,A], we let

f

I

(x) =X

a2I

e

2⇡iM(a)·x .

In the first stage, we consider intervals I of some small length L

1

,much smaller than A. One can prove some rough bounds for f

I

using orthogonality as above. At the second stage, we considerlarger intervals J of length L

2

, much bigger than L

1

but still muchsmaller than A. We express f

J

=P

I⇢J

f

I

, and we study how the

different functions fI

interact with each other. At this stage, somegeometric reasoning comes into play which gives an improvementover just orthogonality.

If the different functions fI

were supported in disjoint regions, thenwe would immediately get an estimate much stronger than orthogo-nality:

Z|fJ

|2s 6X

I⇢J

Z|fI

|2s .

A bit more generally, if the functions fI

are concentrated in regionsthat overlap only a little, one still gets good estimates.

Restriction theory provides tools to help understand when the func-tions f

I

are concentrated in regions that don’t overlap much. Foreach I, there is a tiling of space, T

I

, and |fI

| is roughly constant oneach tile of the tiling. The tiles of T

I

are highly eccentric rectan-gles. For different intervals I of the same length, the tilings T

I

arecongruent to each other, but they are oriented at different angles. Inparticular, if I

1

and I

2

are intervals that are not too close together,then a single tile of T

I

1

and a single tile of TI

2

do not intersect eachother very much.

Now the previous analysis by orthogonality would be sharp in asituation where the mass of f

I

is concentrated into a small numberof tiles. If the mass of f

I

is spread among many tiles of TI

, thenwe get a better estimate at the first stage. But if the mass of f

I

isconcentrated in just a few tiles for each I, then these tiles do notintersect very much, and the geometry gives us a better estimate atthe second stage. Either way, this gives enough leverage to improveon the estimate that comes just from orthogonality.

Next one goes to the third stage, considering intervals K of lengthL

3

, much bigger than L

2

but still much smaller than A. We canwrite f

K

=P

J⇢K

f

J

. If the mass of each function f

J

is concen-trated into a small number of tiles of T

J

, then the same geometryargument as above gives an improvement over the basic orthog-onality estimate. But decoupling uses more than this. Bourgainand Demeter consider both the geometry of the functions f

J

, J⇢ K,and the geometry of the functions f

I

, I⇢ K. Combining geometricinformation from the first stage and the second stage, they are ableto get an even better estimate at the third stage.

Surprises, Open QuestionsThe decoupling approach considers many different scales of inter-vals. At the s-th stage, the argument combines geometric informa-tion from all of the previous s-1 stages. For the argument to workwell, it is necessary to use a large number of different stages, andthe novelty of decoupling is the way that it combines informationfrom these many different scales. The proof is intricate, but it onlyuses orthogonality, the shapes of the tiles in the tiling, and this newway of combining information from many scales.

It came as a big surprise to me, and I think to many others, thatsuch strong estimates could be proven using only these ingredients.It is not yet clear how far one can push these methods and whichproblems one can solve using just these tools.

10

Named Postdocs — Spring 2017Berlekamp

Maria Nastasescu

Maria Nastasescu, a memberof the Analytic Number The-ory program, is the Spring 2017Berlekamp Endowed Postdoc-toral Scholar. Maria studies ana-lytic and algebraic aspects of au-tomorphic forms and especiallynon-vanishing theorems for L-functions of different types. Dur-ing her graduate studies, sheproved, for instance, that ellip-tic curves over the rationals thathave semistable reduction at aprime p are characterized, up to isogeny and quadratic twist,by their adjoint p-adic L-function. This algebraic result is ob-tained by an analytic statement concerning the determinationof automorphic forms on GL(3) from a sparse set of specialvalues of twisted L-functions, which improves significantlyearlier work of Luo and Ramakrishnan. The Berlekamp Post-doctoral Fellowship was established in 2014 by a group ofElwyn Berlekamp’s friends, colleagues, and former studentswhose lives he has touched in many ways. He is well knownfor his algorithms in coding theory and has made importantcontributions to game theory. He is also known for his love ofmathematical puzzles.

Huneke

Max Engelstein

Max Engelstein is theHuneke Postdoc this springin the Harmonic Analysis pro-gram. Max obtained his Ph.D.at the University of Chicago inJune 2016, under the directionof Carlos Kenig. He took upa C.L.E. Moore PostdoctoralInstructorship at MIT, begin-ning Fall 2016, working withDavid Jerison. Max workson problems at the interfacebetween PDE and geometricmeasure theory, often using techniques of harmonic analysis.Most of his work to date concerns free boundary problems,in which information about the regularity of some solutionto a PDE (or to some variational problem) may be used toglean information about the regularity of the free boundary.Max has already obtained some striking results in this area,concerning both 1-phase and 2-phase problems, and in boththe elliptic and parabolic settings. The Huneke postdoctoralfellowship is funded by a generous endowment from ProfessorCraig Huneke, who is internationally recognized for his workin commutative algebra and algebraic geometry.

Strauch

Marina Iliopoulou

Marina Iliopoulou is thisspring’s Strauch PostdoctoralFellow as part of the HarmonicAnalysis program. Marina wasborn in Greece and finished herBachelor degree in Mathemat-ics at the University of Athensin 2009, earning multiple aca-demic awards along the way.She completed her Ph.D. atthe University of Edinburgh in2013. After several years as Re-search Fellow at the Universityof Birmingham (UK), she joined UC Berkeley in Fall 2016 asa Morrey Visiting Assistant Professor. Marina’s research hasfocused on the exciting interface between Euclidean harmonicanalysis and extremal combinatorics. She and collaborators areworking to develop combinatorial and algebraic methods suitedto the analysis of inequalities connected to the Fourier transformand to Schrödinger equations. She has also investigated incidencecounting problems and their connection with Fourier analysis.The Strauch Fellowship is funded by a generous annual gift fromRoger Strauch, Chairman of The Roda Group. He is a member ofthe Engineering Dean’s College Advisory Boards of UC Berkeleyand Cornell University, and is also currently the chair of MSRI’sBoard of Trustees.

Viterbi

Julia Brandes

Julia Brandes is the ViterbiPostdoctoral Fellow in thisspring’s Harmonic Analysis pro-gram. Julia’s research concernsDiophantine equations and inparticular the counting of thenumber of solutions of certainsystems of polynomial equa-tions with special structure. Inher Ph.D. thesis, and later, shestudied the linear subspaces con-tained in a hypersurface, exploit-ing the geometric nature of thiscondition to go beyond earlier estimates. In recent work, partlyjoint with S. Parsell, she has considered systems of diagonalequations in much greater generality than had been done before,obtaining in some cases the optimal result expected from generalheuristics. The Viterbi Endowed Postdoctoral Scholarship isfunded by a generous endowment from Dr. Andrew Viterbi,well known as the co-inventor of Code Division Multiple Ac-cess based digital cellular technology and the Viterbi decodingalgorithm, used in many digital communication systems.

Naser Talebizadeh Sardari, this semester’s first Uhlenbeck Post-doctoral Fellow, is profiled on page 3.

11

A Kind of SilenceEd Baker

November 1, 2016Something divided by nothingBecomes everything;But only in the limitA limit that’s never reached.Expanding without endNot as cause and effect;As the air, swept along imperceptiblyBy the butterfly’s shimmering wings.What is this asymptote beyond our reach?The end of time? God?That nothing in everything,Empty space, whiteness, chaos —Silence.

That something pervades the emptiness;The emptiness as emptiness is.Art moulds it through a formA formulaA beginning, an end, a middle;An audience,Reduced to its bare essentialsIts barenessNaked, vulnerable and contained.Like a thought, it floatsOut of the airWe hear an invigorated freshness,A disenchanted emptiness,A teased and tormented aloofness;We hear it — that confoundingFiddle-about silence.

As a mirrorAs a page of whitenessSuggestively gleaming back at usThe point of view has shiftedBeauty, if there beauty be,Can only be — well, beheld;The beauty of the moment.When we put that conch shell to our earThe sound of the eternalIs surely what we hearEternal emptiness.She sells seashells, for profitThis was another point of objectionAs each tick-tock ticks byWe sense the freedom —The ultimate anarchy of silence.

Can you bear it any longer?The allure of ice cream,Or a martiniAnd yet it moves alongEverything is tickety-boo.Perhaps the coda has begun,We decide which pathBest takes us to the end;A grand crescendo to a blustery climax,A quiet fade into the stillness of the night.That momentWhen past and future coalesceInto a predetermined point,Invented and revised;The silence never stops.

Call for ProposalsAll proposals can be submitted to the Director or Deputy Directoror any member of the Scientific Advisory Committee with a copyto proposals@msri.org. For detailed information, please see thewebsite msri.org.

Thematic ProgramsThe Scientific Advisory Committee (SAC) of the Institute meetsin January, May, and November each year to consider letters ofintent, pre-proposals, and proposals for programs. The deadlines tosubmit proposals of any kind for review by the SAC are March 15,October 15, and December 15. Successful proposals are usuallydeveloped from the pre-proposal in a collaborative process betweenthe proposers, the Directorate, and the SAC, and may be consideredat more than one meeting of the SAC before selection. For completedetails, see tinyurl.com/msri-progprop.

Hot Topics WorkshopsEach year MSRI runs a week-long workshop on some area of in-tense mathematical activity chosen the previous fall. Proposalsshould be received by March 15, October 15, or December 15 forreview at the upcoming SAC meeting. See tinyurl.com/msri-htw.

Summer Graduate SchoolsEvery summer MSRI organizes several 2-week long summer gradu-ate workshops, most of which are held at MSRI. Proposals must besubmitted by March 15, October 15 or December 15 for reviewat the upcoming SAC meeting. See tinyurl.com/msri-sgs.

Pacific Journal ofMathematicsFounded in 1951, The Pacific Journal

of Mathematics has published mathe-matics research for more than 60 years.PJM is run by mathematicians from thePacific Rim and aims to publish high-quality articles in all branches of math-ematics, at low cost to libraries and

individuals. PJM publishes 12 issues per year. Please con-sider submitting articles to the Pacific Journal of Mathe-matics. The process is easy and responses are timely. Seemsp.org/publications/journals/#pjm.

12

Puzzles ColumnElwyn Berlekamp and Joe P. Buhler

... ...

Modified from © Sergey Ilin | Dreamstime.com

0. How many meanings can you ascribe to the following wordsequence?

a pretty light brown doll house

1. An arbitrary square is removed from a 2

n by 2

n chessboard.Prove that the remaining squares can be exactly tiled by right tro-minoes. (A right tromino is the 3-square tile obtained by removingone square from a 2 by 2 chessboard.)

Comment: This year’s ITA (Information Theory and Applications)conference near UC San Diego included a special session in mem-ory of Solomon W. Golomb, which had a “puzzle” component. Thefirst five problems here (0–4) came up in that session, and problems0 and 1 are due to Sol.

2. The number 545 has the interesting property that if you eraseany of the digits and write in a new digit in that place (the new digitmay be a leading zero or the same as the old digit), then you neverget a multiple of 11. Find two more such integers.

Comment: We heard this from Richard Stong, and it was originallydue to Mike Boshernitzan.

3. Given a triangle T with vertices A, B, C, construct external30/30/120 triangles on the sides of T with the 120� angles being atthe new vertices X, Y, Z. Prove that XYZ is an equilateral triangle.

4. A group of n people plan to paint the outside of a fence sur-rounding a large circular field using the following curious process.Each painter takes a bucket of paint to a random point on the cir-cumference and, on a signal, paints towards their furthest neighbor,stopping when they reach a painted surface. What is the expectedfraction of the fence that will be left unpainted at the end of thisprocess?

Comment: We heard this from Paul Cuff, who remembers hearingit many years ago at one of Tom Cover’s seminars at Stanford. Thereader may enjoy figuring out the fraction of unpainted space ifthe painters paint towards their nearest neighbor, and deciding be-forehand by pure thought whether this should be more or less thanone-half.

5. The king has invited N quarrelsome knights to dinner. Manypairs of them have deep-seated mutual hostilities that are so badthat if seated next to each other bloodshed will likely ensue. Canyou always find a safe seating arrangement around a circular tableif each knight is hostile to less than (or at most?) half of the otherknights? An arrangement of the N knights is of course safe if nopair of adjacent knights are hostile.

Comment: We heard this problem from Zvezda Stankova.

Call for MembershipMSRI invites membership applications for the 2018–2019 academicyear in these positions:

Research Professors by October 1, 2017Research Members by December 1, 2017Postdoctoral Fellows by December 1, 2017

In the academic year 2018–2019, the research programs are:

Hamiltonian Systems, from Topology to Applications throughAnalysisAug 13–Dec 14, 2018Organized by Rafael de la Llave, Albert Fathi (Lead), VadimKaloshin, Robert Littlejohn, Philip Morrison, Tere M. Seara, SergeTabachnikov, Amie Wilkinson

Derived Algebraic GeometryJan 22–May 24, 2019Organized by Julie Bergner, Bhargav Bhatt (Lead), Dennis Gaits-gory, David Nadler, Nikita Rozenblyum, Peter Scholze, BertrandToën, Gabriele Vezzosi

Birational Geometry and Moduli SpacesJan 22–May 24, 2019

Organized by Antonella Grassi, Christopher Hacon (Lead), SándorKovács, Mircea Mustat,a, Martin Olsson

MSRI uses MathJobs to process applications for its positions. In-terested candidates must apply online at mathjobs.org after August1, 2017. For more information about any of the programs, pleasesee msri.org/scientific/programs.

Clay Senior ScholarshipsThe Clay Mathematics Institute (www.claymath.org) has announcedthe 2017–2018 recipients of its Senior Scholar awards. The awardsprovide support for established mathematicians to play a leadingrole in a topical program at an institute or university away fromtheir home institution. Here are the Clay Senior Scholars who willwork at MSRI in 2017–2018:

Geometric Functional Analysis and Applications (Fall 2017)William B. Johnson, Texas A&M UniversityGeometric and Topological Combinatorics (Fall 2017)Francisco Santos, Universidad de CantabriaGroup Representation Theory and Applications (Spring 2018)Gunter Malle, Universität KaiserslauternEnumerative Geometry Beyond Numbers (Spring 2018)Hiraku Nakajima, Kyoto University

13

2016 Annual ReportWe gratefully acknowledge the supporters of MSRI whose generosity allows us to fulfill MSRI’s mission to advance andcommunicate the fundamental knowledge in mathematics and the mathematical sciences; to develop human capital for thegrowth and use of such knowledge; and to cultivate in the larger society awareness and appreciation of the beauty, power,and importance of mathematical ideas and ways of understanding the world.

This report acknowledges grants and gifts received from January 1 – December 31, 2016. In preparation of this re-port, we have tried to avoid errors and omissions. If any are found, please accept our apologies, and report them todevelopment@msri.org. If your name was not listed as you prefer, please let us know so we can correct our records. Ifyour gift was received after December 31, 2016, your name will appear in the 2017 Annual Report. For more informationon our giving program, please visit www.msri.org.

Government SponsorsNational Science FoundationNational Security Agency

Current Foundation &Corporate SponsorsAmazonAnonymousArkay FoundationCarnegie Corporation of

New YorkCavali FoundationElwyn & Jennifer Berlekamp

FoundationFiredoll FoundationFrance Berkeley FundGoldman Sachs GivesGordon and Betty Moore

FoundationIBM Matching Grants Program*John Templeton FoundationLangsam Family

Foundation, Inc.Microsoft CorporationMicrosoft Matching Gifts

Program*Pacific Journal of MathematicsResearch Corporation for

Science AdvancementS. S. Chern Foundation for Math-ematical ResearchSanford J. Grossman

Charitable TrustSchmidt PhilanthropiesSimons FoundationThe Clowes FundTorchlight Investors, LLCVISA, Inc.Vision Maker Media

Hilbert SocietyWe recognize our most generousand loyal supporters whoseleadership and commitmentensures MSRI continues tothrive as one of the world’sleading mathematics researchcenters.

AnonymousEdward D. BakerElwyn & Jennifer Berlekamp**Susan & Paul ChernMay & Paul ChuWilliam & Margaret HearstMark KleimanMarilyn & Jim SimonsRoger StrauchAndrew Viterbi

Museion SocietyThe Museion Society, namedafter Musaeum, the Hall of theMuses in ancient Alexandria,recognizes our leadershipdonors in annual andendowment giving.

EuclidIan Agol**Elwyn and Jennifer BerlekampPaul & Susan ChernMay & Paul ChuNaren & Vinita GuptaWilliam & Margaret HearstMargaret Holen & David

CoulsonRonald Kahn and Julia Rowe**Mark P. Kleiman**Henry & Marsha LauferAnand Periasamy & Garima

Kapoor

Eric E. SchmidtFaye and Sandor StrausAshok and Gita Vaish

HypatiaAnonymousDan HeflinJohn MoussourisSorin and Marie-Anne PopescuAlun Wyn-Jones

PlatoMichael BershadskyIrina BogachevaRobert L. Bryant &

Réymundo A. GarciaJennifer Chayes & Christian

BorgsCedomir CrnkovicThomas R. DavisAlexander LesinDiviya MagaroHoward & Joan ResnikoffRonald L. RivestDario Villani and Lily ShapiroJade P. VinsonBob WisePeter N. Yianilos

Archimedes SocietyThe Archimedes Societyrecognizes MSRI’s annualsupporters.

RamanujanGreg AndorferJishnu and Indrani BhattacharjeeAlice Chang & Paul C. YangJim HerrnsteinRichard V. KadisonWilliam E. LangColin Rust and Jeannie TsengJustin Walker

NoetherAnonymousKen BaronPeter Bates**Mikhail BrodskyRuth CharneyJun S. Liu and Wei ZhangYakov EliashbergPaul FongDan FreedDavid GoldschmidtRobert & Shelley Ann GuralnickAlfred W. & Virginia D. HalesBrendan E. Hassett and

Eileen K. ChengJohn M. HosackDonald and Jill KnuthSylvia LevinsonAlbert & Dorothy MardenHoward A. MasurDusa McDuff & John MilnorMichael MorganMichael F. SingerJohn SmillieRichard & Christine TaylorKaren Uhlenbeck &

Bob WilliamsAlexander VaschilloHung-Hsi WuPeter and Gloria Yu

FermatAnonymous (5)Elliot AronsonSheldon AxlerHélène Barcelo & Steven

KaliszewskiChristine BreinerDanalee & Joe P. BuhlerTom & Louise Burns

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Richard C. ChurchillBruce CohenCharles W. CurtisAnthony D’AristotileJames DonaldsonPeter DurenE. Graham EvansMark Feighn & Edith StarrGisela FränkenAnita B. FriedmanJohn FriedmanAkito FutakiTheodore W. GamelinDaniel & Ryoko GoldstonPhillip GriffithRobin HartshorneJoel HassWilliam JohnsonDonald KornLynda & Robert KorsanJim LewisRafe MazzeoJon McCammondKathleen & Robert MegginsonAndrew OggWalter OhlemutzFreydoon ShahidiAaron N. Siegel & Olya GurevichJames T. SmithWilbur L. SmithHugo & Elizabeth SonnenscheinJohn StroikChuu-Lian Terng & Richard

PalaisMichael UriasWolmer V. VasconcelosJohn M. VoightCharles WeibelRoger & Sylvia WiegandScott A. WolpertCarol S. WoodJulius & Joan Zelmanowitz

Fibonacci

Anonymous (5)Eugene R. AlwardMichael T. AndersonEthan J. BerkoveRajiv Bhateja

Peter and Nancy BickelKirsten BohlAnna Marie BohmannDavid BressoudLawrence G. BrownRichard A. Champion, Jr.Jean Chan (In memory of

Dr. Peter Stanek)Ronan ConlonAnnalisa CrannellSteven & Margaret CrockettChris CunninghamSteven & Hema CutkoskyJohn DalbecHailong DaoDouglas DunhamPatrick B. EberleinPeter EggenbergerSteven P. EllisMichael FalkJeremiah FarrellRichard FinnElia FioravantiFrank M. FisherHarolyn L. GardnerDeene GoodlawCameron M. GordonDavid HarbaterNancy B. HingstonStanley S. IsaacsSheldon KatzKiran S. KedlayaHenry KimGaro KiremidjianEllen E. KirkmanMaria Klawe & Nicholas

PippengerDaniel H. KlingRob KusnerProf. Michel LapidusRichard K. and Joyce C. LashofJoseph MaherAnna L. MazzucatoClinton G. McCroryRobert McOwenJanet Mertz and Jonathan KanePolly MooreCarlos Julio MorenoJulie A. Newdoll

Yong Geun OhRichard & Susan OlshenAlice PetersThe Pisani FamilyMalempati M. RaoBruce ReznickWinston RichardsMarc A. RieffelJonathan RosenbergWilliam RosenbergDavid RowlandDonald E. SarasonMarkus SchmidmeierLeonard ScottTimo Seppalainen**Mohamed W. SesayBonnie SherwoodRegis A. SmithLouis & Elsbeth SolomonRalf SpatzierLee StanleyHarold StarkJames D. StasheffTroy L. StoryEarl J. TaftJames TantonPham H. TiepAlain ValetteBill VedermanDan Virgil VoiculescuStanley Wagon & Joan

HutchinsonTheodore WakarBrian WhiteSusan S. WildstromErling Wold

Gauss SocietyThe Gauss Society recognizesfriends who include MSRI intheir will or have made otherstipulations for a gift from theirestate, ensuring that MSRIremains strong in the future.

Edward D. BakerRobert L. Bryant &

Réymundo A. GarciaGary CornellDavid & Monika Eisenbud

Gisela FränkenRobert W. HackneyCraig Huneke & Edith ClowesWilliam E. LangDouglas LindDusa McDuff & John MilnorJoseph NeisendorferMarilyn & Jim SimonsHugo & Elizabeth SonnenscheinJim SotirosJulius & Joan Zelmanowitz

Book & Journal Giftsto the MSRI Library

Private DonorsAnonymousHélène BarceloElwyn BerlekampDavid EisenbudBevy HansenTsit-Yuen LamKenneth RibetYanir Rubinstein

PublishersAmerican Mathematical

SocietyBasic BooksBelgian Mathematical SocietyClay Mathematics InstituteCRC PressElsevierEuropean Mathematical SocietyFields InstituteHiroshima UniversityInstitut de Mathématiques

de JussieuInstitut de Mathématiques

de ToulouseInstitute for Advanced StudyJapan AcademyKyoto UniversityKyushu UniversityMathematical Society of JapanPrinceton University PressSpringerTohoku UniversityUniversity of Tokyo

*Provides matching funds for gifts by employees or directors; **Includes gifts to the MSRI Endowment Fund. Gifts to the endowment arecarefully and prudently invested to generate a steady stream of income.

To learn more about ways to support MSRI with a donation or a planned gift, please visit our website (www.msri.org/web/msri/support-msri), or contact the Director of Development, development@msri.org, 510-643-6056.

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MSRIMathematical Sciences Research Institute

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MSRI Staff & Consultant RosterAdd @msri.org to email addresses. All area codes are (510) unless otherwise noted.

Scientific and EducationDavid Eisenbud, Director, 642-8226, directorHélène Barcelo, Deputy Director, 643-6040, deputy.directorDiana White, National Association of Math Circles Director, 303-315-1720, namc.director

AdministrativeKirsten Bohl, Project Lead, National Math Festival, 499-5181, kbohlArthur Bossé, Operations Manager, 643-8321, abossePatricia Brody, Housing Advisor, 643-6468, pbrodyAaron Hale, IT Manager, 643-6069, ahaleMark Howard, Facilities and Administrative Coordinator, 642-0144, mhowardTracy Huang, Assistant for Scientific Activities, 643-6467, thuangClaude Ibrahimoff, International Scholar Advisor & Executive Assistant, 643-6019, cibrahimoffLisa Jacobs, Director’s Assistant and Board Liaison, 642-8226, lisajChristine Marshall, Program Manager, 642-0555, chrisJinky Rizalyn Mayodong, Staff Accountant, 642-9798, rizalynJennifer Murawski, Communications and Event Coordinator, 642-0771, jmurawskiMegan Nguyen, Grants Specialist, 643-6855, meganSandra Peterson, Development Assistant, 642-0448, spetersonLinda Riewe, Librarian, 643-1716, lindaJacari Scott, Scholar Services Coordinator, 642-0143, jscottJames Sotiros, Interim Director of Development, 643-6056, jsotirosSanjani Varkey, Family Services Consultant, sanjaniStefanie Yurus, Controller, 642-9238, syurus