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

Fall 2018

Hamiltonian Systems, from Topology to Applicationsthrough Analysis

Philip J. Morrison and Sergei Tabachnikov

OverviewThe special historic form of dynamical systems credited to Hamil-ton encompasses a vast array of fundamental and applied research.It is a basic form for physical law that has engendered developmentin many areas of mathematics, including analysis of ordinary andpartial differential equations, topology, and geometry. Our jumboprogram at MSRI has brought together a broad spectrum of math-ematicians and scientists with research spanning emphasis on theapplied to the rigorous.

It is hard to choose a starting point in the long history of Hamil-tonian dynamics and its concomitant variational principles. Onecan go as far back as ancient Greece (Euclid, Heron), onward tothe inspirational work of Fermat’s principle of geometric optics(17th century), and up to the voluminous works of Lagrange (18thcentury). Fermat stated that the path taken by light going fromone point to another in some medium is the path that minimizes(or, more accurately, extremizes) the travel time. This implies thelaw of optical reflection (the angle of incidence equals the angle ofreflection) and Snell’s law of refraction.

William Rowan Hamilton (19th century) studied the propagation ofthe phase in optical systems guided by Fermat’s principle and real-ized that one could generalize it and adapt it to particle mechanics.Here is a very brief description of Hamiltonian mechanics.

(continued on page 4)

Multiple trajectories for a billiard inside an ellipse: one of thepopular Hamiltonian systems is the “billiard problem.”

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Come and Celebrate . . .The Summer Research for Women program

created and first hosted in 2017 (page 8).

Our recent Celebration of Mind (page 11). !#Mafia, hats, and more in the Puzzles, as Elwynand Joe welcome a new contributor (page 15)!

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1

The View from MSRIHélène Barcelo, Acting Director

I’m excited (and I must admit, somewhat daunted!) to be takingthe role of Acting Director this year while Director David Eisen-bud is taking a much-deserved (and very active!) sabbatical. In therole of Assistant Director is the very capable and affable MichaelSinger, Professor Emeritus at North Carolina State University, andDeputy Director at MSRI from August 2001– July 2002.

This is shaping up to be a very busy year for us, as we are prepar-ing for the NSF recompetition for funding for the institute for theperiod 2020–25, formalizing our latest strategic vision, greatlyexpanding our fundraising efforts, and gearing up for the NationalMath Festival to be held in Washington, DC, this coming May.These and some other of our current projects are described below.

Survey of PostdocsMSRI has undertaken a survey of the 204 postdoctoral fellows thatwe have hosted since 2009. While the analysis of the responsesis not finished, so far we are encouraged by the data. Particularlystriking was the 90% of respondents who rated their overall expe-rience at MSRI as “excellent” or “very good.” We will continueto share the full results once complete.

Diversity at MSRIAs we prepare for the NSF recompetition and reflect on the in-stitute’s recent programs, we are very pleased to report that thework of so many past and present members of our advisory andplanning committees is coming to fruition. For example, overthe last decades the percentage of women participating in ourlong program has increased by nearly 5%, and in the 2017–18programs, 25% of the members, and fully 32% of the ResearchProfessors self-identified as women. In addition, our SummerResearch for Women in Mathematics program is accepting ap-plications for the third year — Assistant Director Michael Singerdiscusses the development of this newer initiative on page 8.

National Math FestivalOur biggest public outreach program, the National Math Festival,will return to Washington, DC, for a third time on May 4, 2019.This year’s festival includes not only presentations by mathemati-cians and activities for young children through adults, but alsodance performances, athletic competitions, and more. We encour-age any of you to join us for this celebration of math in manyforms.

(continued on next page)

David Eisenbud, Director(On sabbatical, 2018–19)

As readers will be aware, I’ve stepped back from many of theleadership duties at MSRI, which are being carried on very suc-cessfully this year by Acting Director Hélène Barcelo. This allowsme to devote more time to some other initiatives. And it alsomeans that in this Emissary, you get two views from MSRI: bothmy view (below), and Hélène’s, next door in the left column!

NSF ProposalThe NSF has decided to hold an Institutes recompetition everyfive years. Though our “core” NSF grant is now only a little morethan half our total funding, it is still the lifeblood of our researchenterprise, and we take the proposal process extremely seriously.The new proposal will have input from many people, but I’vetaken primary responsibility for writing it. I welcome any goodideas for things MSRI should propose or change! If you havesome, write them down, and send me an email (de@msri.org).

Fundraising and Strategic VisionAnnie Averitt is our new head of development, and I’m work-ing very closely with her on other projects to support MSRI

Annie Averitt

as well. Our endowment nowstands at around $26 million, butneeds to be much larger to ensurethat MSRI can continue to servethe math community for the gener-ations to come. Crucial for raisingmoney is to be able to say clearlywhat you want to do with it, andwe are thus engaged in a strate-gic planning exercise. Again, withmuch input from the trustees andfrom Hélène and Michael Singer,

I’ve taken primary responsibility for drafting the vision.

Film About Maryam MirzakhaniOne project for which we’ve already raised funds: a film aboutthe life and work of the wonderful mathematician Maryam Mirza-khani, who made great contributions to the study of the dynamicsand geometry of Riemann surfaces. The filming will take placein the U.S., Europe, and Iran. George Csicsery is the director,and we hope for a premiere at the Joint Mathematics Meetings inJanuary 2020.

The View from MSRI 2AMS/MSRI Briefing 3HS Program (cont’d) 4SRWM Program 8

2019 NMF 9Vered Rom-Kedar 10Upcoming Workshops 10Celebration of Mind 11Named Positions 11

Connections for Women 12Fellows and Medalists 12MSRI-UP at SACNAS 12New Faces at MSRI 12Call for Membership 13

Call for Proposals 13Jacques Féjoz 13Fall Postdocs 14Puzzles Column 15

Contents

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The View (continued from previous page)

Mirzakhani Research ProfessorshipsLast, but certainly not least, we wish to share some additional newsabout another project under development in honor of Maryam Mirza-khani: the creation of a five million dollar endowment to supportMirzakhani Research Professorships in MSRI’s scientific programs,with the enthusiastic support of her husband, Jan Vondrák.

Here is an excerpt from a piece that David and I wrote for theNovember AMS Notices on Maryam’s impact at MSRI:

Maryam was a key member of the MSRI Scientific AdvisoryCommittee from 2012–2016, the committee charged withselecting the scientific programs and their members. Duringpart of that time she was very ill with what turned out tobe her terminal cancer, but she never missed a meeting —even when she couldn’t travel from Stanford to Berkeleyshe attended by video, between harsh medical treatments —and she faithfully did the substantial committee homework.It is fair to say that she was an inspiration to the entirecommittee.

She played other important roles at MSRI, too. She wasone of the main organizers of the semester-long programon Teichmüller theory and Kleinian groups in 2007, and aresearch professor for the spring of 2015. She was even amember of the complementary program when her husband,Jan Vondrák, came to MSRI for a computational program inthe spring of 2005.

Finally, in the spring before her death, Maryam had agreedto be nominated as a trustee of MSRI — although she warnedDavid Eisenbud, MSRI’s director, in a long and deeply sadmeeting, that she might well not live to take office.

Professor Mirzakhani’s scientific work will endure and remain im-portant, and through Chairs such as the one we propose, we canpreserve the memory of her humble spirit of exploration, and herindomitable quest for knowledge.

The Mirzakhani Endowment

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To honor Maryam Mirzakhani’s legacy, MSRI has proposeda five million dollar endowment to support four MirzakhaniProfessors each year in programs at the institute. The photoabove shows Maryam with her fellow Fields medalists from2014 (Artur Avila at left and Martin Hairer at back), aswell as with her daughter Anahita and Manjul Bhargava(far right), at the ICM 2014 in Seoul. You can learn moreabout Maryam’s impact at MSRI in the View column onthis page.

If you wish to learn more or to contribute to the endow-ment in Maryam’s honor, you can visit msri.org/mirzakhanior contact Annie Averitt at development@msri.org or(510) 643-6056.

AMS/MSRI Congressional BriefingIn May 2018, the American Mathematical Society and MSRI hosted ajoint congressional briefing on Capitol Hill featuring MacArthur Fellowand MIT computer scientist Erik Demaine. Demaine’s talk on “OrigamiMeets Math, Science, and Engineering” shared surprising applicationsin manufacturing, robotics, animation, biology, medicine, nanotech, andspace technology that have grown from new fundamental research incomputational origami. Speakers at the Congressional briefing includedRepresentative Jerry McNerney of California, MSRI Director David Eisen-bud, AMS Associate Executive Director Karen Saxe, and AMS PresidentKen Ribet (UC Berkeley).

The next Congressional briefing takes place in December 2018, featuringRodolfo H. Torres (University of Kansas), whose research includes collab-orations with biologists, engineers, and economists. You can learn moreand view short films about past events: msri.org/congress.

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Hamiltonian Systems, from Topology to Applications throughAnalysis(continued from page 1)

A mechanical system is described by its configuration space, mathe-matically a smooth manifold Q whose points, q2Q, are understoodas positions. The system is described by a Lagrangian functionL(q,q) : TQ! R, depending on the position q and the velocity q.One considers the action functional

S[q(t)] =

Zt1

t0

L(q,q) dt, �q(t0) = �q(t1) = 0.

The time evolution of the mechanical system is described as a vari-ational principle that the action S be extremal. This implies theEuler–Lagrange equations,

@L

@q=

d

dt

✓@L

@q

◆.

Going from the tangent bundle TQ to the cotangent bundleT⇤Q, one introduces momenta p = @L/@q and the Hamilto-nian function T⇤Q ! R, given by the Legendre transformH(q,p) = pq-L(q,q). In the phase space T⇤Q, the motion isdescribed by Hamilton’s first order differential equations,

p=-@H

@qand q=

@H

@p.

The Hamiltonian form is not limited to finite-dimensional systems —indeed, action functionals are fundamental to the development of20th century field theories in physics. A case in point is �4 fieldtheory that has the action functional

S[�] =

Zt1

t0

dt

Zd3xL(�,@�) ,

with the Lagrangian density

L=1

2(@t�)

2-1

2|r�|2- 1

2m2�2-

g

4!�4 .

The Legendre transform of this system gives the associated Hamil-tonian form of the partial differential equations,

@t⇡=-�H

��and @t�=

�H

�⇡,

where �H/�� denotes the functional derivative and the Hamiltonianfunctional is

H=

Zd3x

✓1

2⇡2+

1

2|r�|2+ 1

2m2�2+

g

4!�4

◆,

where m and g are parameters of the theory. By direct calculationone obtains a nonlinear wave equation as a Hamiltonian system.

There are many ramifications and generalizations, practical and aes-thetic, of both the finite and infinite forms of Hamiltonian systems.We describe some of those addressed by our MSRI program below.

Dynamics on Symplectic Manifolds

The theoretical underpinning of Hamiltonian mechanics is sym-plectic geometry. (The term symplectic was adopted by H. Weylto avoid any connotation of complex numbers that his previously-used term, “complex group,” had suffered. It is simply the Greekadjective corresponding to the word “complex.”) The cotangentbundle T⇤Q of a smooth manifold carries a canonical symplecticstructure != dp^dq, a closed non-degenerate differential 2-form.Other manifolds, not necessarily cotangent bundles, may also carrya symplectic structure. Hamilton’s equations of motion describe theHamiltonian vector field (or the symplectic gradient) XH definedby the formula !(XH, ·) =-dH.

An important example of a Hamiltonian system, and one of theresearch topics in this program, is the motion of a charged particlein a magnetic field. Mathematically, a magnetic field is representedby a closed differential 2-form � on a Riemannian manifold Q.One considers the twisted symplectic structure !+⇡⇤(�) on thecotangent bundle T⇤Q, where ⇡ : T⇤Q!Q is the projection. Themotion of a charge is described as the Hamiltonian vector field ofthe energy function ||p||2/2 (the norm is taken using the metric onQ) with respect to the twisted symplectic structure. One of the prob-lems of contemporary interest is the existence of periodic motionsof the charge.

Iteration of the Standard Nontwist map [del-Castillo-Negreteet al., Physica D 91, 1 (1996)] that shows invariant tori (contin-uous curves) embedded in a chaotic sea of orbits. (Figurecourtesy of George Miloshevich.)

Other theoretical backgrounds of Hamiltonian dynamics include thecalculus of variations, Morse theory, and Floer theory. Floer theory,in particular, is designed for the study of symplectic dynamics ofArnold’s conjecture concerning fixed points of Hamiltonian diffeo-morphisms (flows of time-dependent Hamiltonian vector fields). Inthe simplest case, this conjecture states that an area and center ofmass preserving diffeomorphism of a torus has at least three, andgenerically at least four, distinct fixed points (proved in the 1980sby Conley and Zehnder).

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Integrability and Chaos

Hamiltonian systems exhibit a wide variety of dynamical behav-ior, from very regular (completely integrable) to chaotic. In acompletely integrable system (for example, Kepler’s problem incelestial mechanics), the motion is typically confined to tori thathave half the dimension of the phase space, and the motion on theseinvariant tori is described by constant vector fields.

While completely integrable systems are interesting, they are ex-tremely rare. The Kolmogorov–Arnold–Moser (KAM) theory stud-ies small perturbations of integrable systems; its fundamental resultis that some invariant tori persist under small perturbations.

If the dimension of the phase space is greater than two, the in-variant tori do not separate the space, and a phase trajectory mayescape to infinity. This and related phenomena are known as Arnolddiffusion. Arnold diffusion is one of the focal points of this topical

Normally hyperbolic invariant cylinder in a Froeschlé map. Thistype of object is relevant to the mechanism of Arnold diffusion;for example, moving around a 2-torus in a four-dimensionalmanifold. (Figure courtesy of Alex Haro.)

semester, and many experts in this field are participants of the pro-gram. KAM theory and the theory of Arnold diffusion have manyapplications, in particular, in celestial mechanics (for example, “Isthe solar system stable?”)

Flows on Poisson Manifolds

Poisson manifolds are a generalization of symplectic manifolds,where a manifold M has instead of a symplectic 2-form a Poissonbracket { , } : C1(M)⇥C1(M)! C1(M) for arbitrary functionsf,g,h 2 C1(M) satisfying

1. Bilinearity: {f+�g,h}= {f,h}+�{g,h} � 2 R

2. Skew symmetry: {f,g}=-{g,f}

3. Jacobi identity: {{f,g},h}+ {{g,h}, f}+ {{h,f},g}= 0

4. Leibniz Rule: {fg,h}= g{f,h}+ f{g,h}

Here, the Poisson bracket {f,g} = J(df^dg), written in terms ofthe Poisson bivector J, generates the vector field Xf = {f, ·} whoseintegral curves are the trajectories of the Hamiltonian dynamics ofinterest. Because the bivector does not have the usual form, systemsof this type are sometimes called noncanonical Hamiltonian systems.Unlike conventional Poisson brackets, the noncanonical Poissonbrackets of Poisson manifolds are degenerate with special invariantsknow as Casimir invariants C 2 C1(M) that satisfy {C,f}= 0 forall f 2 C1(M). Because of this degeneracy flows are constrainedto submanifolds that are in fact generically symplectic manifolds.

Cartoon of Poisson manifold with its foliation by symplecticleaves.

The study of Poisson manifolds is important both because of intrin-sic mathematical interest and because infinite-dimensional versionsof such flows generated by noncanonical Poisson brackets describemany physical systems. These flows are systems of partial differen-tial equations that have a Hamiltonian form given by

@t�= J(�)�H

��,

where � denotes the set of dependent field variables and J is a Pois-son operator that is a generalization of the Poisson tensor J of theHamiltonian bivector for finite systems.

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Mathematical BilliardsOne of the popular Hamiltonian systems is a mathematical billiardthat describes the motion of a mass-point in a domain, subject tospecular reflections off the boundary. Many mechanical systemswith elastic collisions — that is, collisions in which the energy andmomentum are preserved — are described as billiard systems.

On a basic level, billiards can be viewed as a study of non-smoothHamiltonian vector fields, one motivated by a physically relevant

Trajectories for a billiard inside an ellipse, broken out from thecover image. (Figure courtesy of Vadim Kaloshin.)

setup. The study of billiards was put forward by Birkhoff, whoobserved (in 1927) that “... in this problem the formal side, usuallyso formidable in dynamics, almost completely disappears, and onlythe interesting qualitative questions need to be considered.”

The billiard system can be considered as a continuous-time Hamil-tonian system with discontinuities corresponding to the reflections.It can be also considered as a discrete-time system, that is, a trans-formation acting on the oriented lines, thought of as segments of abilliard trajectory. The space of oriented lines (rays of light) carriesa symplectic structure, and the billiard ball map is a symplectictransformation.

Just like general Hamiltonian systems, planar billiards exhibit a fullspectrum of dynamical behaviors, from completely integrable tochaotic. A completely integrable example is the billiard inside anellipse: the interior of the billiard table is foliated by caustics whichare confocal conics.

A notoriously hard problem in this area is known as Birkhoff’sconjecture: If a neighborhood of the boundary of a billiard tableis foliated by caustics, then it is an ellipse. There was a very sub-stantial progress made toward the proof of this conjecture and itsvariation (for example, algebraic integrability, where the conservedquantity is a polynomial in momentum). A number of the mainplayers in this field participate in the program.

ApplicationsGiven the history of its development, it comes as no surprise thatHamiltonian dynamics plays an important role in modern theoreticalphysics and an array of applications, a few of which we note.

The Hamiltonian dynamics of bodies under the influence of gravity,celestial mechanics, is of course of basic importance for understand-ing the dynamics of the solar system and beyond (planets and stars).However, it also is needed for practical satellite navigation, forcommunication and space exploration, and the tracking of possiblydeleterious astroids.

The Hamiltonian dynamics of gravitating bodies is a special case ofso-called natural Hamiltonian systems where H(q,p) = T(q,p)+V(q) with T(q,p) the kinetic energy (mathematically, a Rieman-nian metric on the fiber of T⇤Q) and V(q) the potential energy.Examples of natural Hamiltonian systems are spring systems, pen-dula, particles in potential wells, and the N-body problem withqi 2 R3, i= 1, . . . ,N, where

H(q,p) =NX

i=1

||pi||2

2mi+

NX

i,j=1

cij

|qi-qj|

and cij represents the interaction. In the context of celestial me-chanics, the interaction represents gravitational attraction, but theN-body setup can also describe the electrostatic interaction of re-pelling electrons, electrons and ions (protons) that attract each other,and the collection of both that occurs in plasmas.

If charged particles experience the full electromagnetic interactionthen the Hamiltonian is given by

H(q,p) =||p-eA(q,t)||2

2m+e�(q,t) ,

6

Fermilab National Accelerator Laboratory near Chicago, IL hosts a large particle accelerator with radius of approximately 1km(photo) in which particles moving very close to the speed of light need to be stored for many hours. Understanding how thiscan be done is a complicated problem in Hamiltonian Dynamics, which in its essence boils down to correctly estimatingminute deviations from integrability, which are shown in the figure (inset) after magnification of 1 million.

Ferm

ilab

NA

SA/J

ohns

Hop

kins

APL

/Ste

veG

ribbe

n

The Tevatron — a large, 1 km-radius particle accelerator at Fermilabnear Chicago — accelerated particles to near the speed of light andstored them for many hours. Understanding the particle orbits in such asystem is a complicated Hamiltonian dynamics problem of estimatingminute deviations from integrability. (Inset courtesy of Martin Berz.)

The Parker Solar Probe, already in orbit around thesun, will characterize plasma near the sun (within10 solar radii) and investigate magnetic fields, ener-getic particles, and the creation of the solar windthat impacts the Earth’s environment and affectsspace weather.

where the magnetic field B = r ⇥ A and the electric fieldE=-r�-@tA. The dynamics under (relativistic versions of)such Hamiltonians is of great importance for the design of particleaccelerators such as Fermilab where one must corral and accelerateparticles with sufficient luminosity to probe the nature of elemen-tary particles. In addition accelerator technology is essential for themedical physics of radiotherapy, radiology, nuclear medicine, andoncology.

The Hamiltonian dynamics of charged particles is of fundamentalimportance for understanding naturally occurring plasmas such asthat near the sun that will be explored with the Parker Solar probe,which will investigate the origin of the solar wind that impactsthe earth’s magnetosphere. In addition the Hamiltonian dynamicsof charged particles is essential for understanding the laboratoryplasmas created in large machines such as the ITER Tokamak thatis being built in France. Particle dynamics in magnetic fields isparticularly important for the design of such devices that use astrong magnetic field to confine plasma within a toroidal regionwith the goal of producing controlled thermonuclear fusion to gen-erate power.

In addition to the finite-dimensional systems described above, thereare infinite-dimensional systems — field theories — that describecollective motion. An important example is the Vlasov–Maxwellsystem, a kinetic theory that takes into account the fact that charges,in addition to producing electric fields, have motions, currents, thatproduce magnetic fields and electromagnetic wavelike motions. Areduced form of this system describes average features of largescale stellar dynamics.

Other field theories include Euler’s equations of fluid mechanics;shallow water theory and the quasi-geostrophic equations, impor-tant equations of geophysical fluid dynamics that describe aspectsof atmospheric and ocean dynamics; magnetohydrodynamics andtwo-fluid theory, fluid theories of plasma physics that include themagnetic and electric fields as dynamical variables; and sundryadditional equations that describe various aspects of media treatedas a continuum.

All of these infinite-dimensional systems are Hamiltonian systems,and they all possess the form of flows on infinite-dimensional Pois-son manifolds as described above. Indeed, the discovery of theirHamiltonian form provided a major impetus for the study of Poissonmanifolds begun in the 1980s.

Structure-Preserving Computation

Given that Hamiltonian systems are defined in terms of a differential2-form, it comes as no surprise that various structures are preservedby the dynamics. In fact, the 2-form itself is preserved in the sense£XH

!= 0, where £XHis the Lie derivative with the Hamiltonian

vector field XH. One understands that Hamiltonian dynamics is aone-parameter temporal map of the phase space manifold to itselfby a canonical transformation (symplectomorphism) that preservessymplectic area defined by the 2-form.

Numerical algorithms that preserve this structure are known assymplectic integrators. In addition to the exact conservation of asymplectic area, which can be wedged together to make a notionof volume preservation, symplectic integrators prevent the Hamil-tonian (energy) from deviating significantly from its theoreticallyconstant value. Thus they provide improved performance for long-time computation.

Recently, the variational form of Hamiltonian dynamics has beenexploited to obtain variational integrators based on discretizing thevariational principle, giving rise to desirable preservation of geomet-ric structure. Poisson integrators are numerical algorithms that haveexact conservation of the symplectic leaves of a Poisson manifold,as well as being symplectic on symplectic leaves.

Currently, one major challenge is to extract from Hamiltonian partialdifferential equations finite-dimensional (semi-discrete) systems ofordinary differential equations that inherit their parent Hamiltonianform and then to implement the resulting system with a symplec-tic or Poisson integrator. This is particularly difficult for infinite-dimensional systems that have noncanonical Poisson brackets.

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MSRI Creates Summer Research for Women inMathematics Program

Michael Singer, Assistant Director

‘‘Our week at MSRI enabled us to develop a small project into something substantial. We are working slowlyfor exactly the same reasons that [the time at MSRI] helped us: we are overflowing with commitments athome (eight children across three authors!), and the isolation permitted the focus needed to advance.’’— SRWM participant, Summer 2017

The ChallengeResearch in mathematics requires concentration and time. It is oftengreatly enhanced by the opportunity to collaborate in a small group:motivation is increased by the social interaction, ideas bounce be-tween the collaborators, and team spirit plays a very positive role.

Women have not typically been encouraged to go into mathematicalresearch. Even after women enter the profession, a number of so-cietal forces tend to put them in less research-active environments.This, possibly together with other factors, has led to a waste of talent.A number of conferences for women in particular areas (Women inTopology, Women in Combinatorics, Women in Numerical Methodsfor PDEs, . . . ) have sprung up to ameliorate this situation.

In a typical conference for women, for example at the Banff Interna-tional Research Station (BIRS), a group of women meet for a weekand divide into small research teams, mixing established researchersand younger PhDs, to focus on particular research problems. How-ever, a week is too short: projects remain unfinished when the teamsreturn to their professional and personal responsibilities and theinteraction is hard to maintain.

To improve the odds of completion and enhance the quality of theproducts of such collaborations, as well as others begun in dif-ferent ways, MSRI initiated the Summer Research for Women inMathematics (SRWM) program. Groups of two to six women withestablished projects can apply for funding to spend a minimumof two weeks together at MSRI, living and working together andmaking use of the institute’s excellent resources for collaborativework in a period when the weather is cool — this is the Bay Area,after all — and MSRI has spare office capacity.

A Great Idea, Quickly Acted OnThe idea for such a program originated in May 2017 when BIRShosted a week-long workshop, Algebraic Combinatorixx 2, whichbrought together 42 women who worked, in small groups, on eightdifferent projects in algebraic combinatorics. The workshop was anastounding success: some of the participants had not even knowneach other prior to the workshop, but they formed cohesive groupsand made serious progress toward solving the problems that wereset forth by the team leaders. The energy and excitement of beingamong peers was palpable throughout the week.

As the end of the week approached, Georgia Benkart (Trustee ofMSRI) and Hélène Barcelo (MSRI’s Deputy Director and currentActing Director), who were among the participants, realized thatto preserve the energy built and the work accomplished through-out the week, these groups must be given the opportunity to meet

Summer 2018 participants Candice Price, Suzanne Sindi,Shelby Wilson, Nina Fefferman, and Nakeya Williams. On thecover: Summer 2018 participant Zajj Daugherty.

again. MSRI acted quickly and opened its doors to the participantsduring the following summer to so that they could pursue (and,when possible, complete) the projects started at BIRS.

Immediate SuccessFour groups (a total of 13 researchers) applied and spent a week inBerkeley during the summer of 2017. One group has had a paperpublished, one has submitted a paper, and the remaining two groupshave nearly completed manuscripts and hope to submit papers inthe next few months.

It is remarkable that so many of the participants who attended boththe BIRS workshop and the week at MSRI were able to produce apublication from their work. Several mentioned that knowing theywould reunite at MSRI helped them maintain contact following theBIRS workshop. This added to the energy to push their projectsalong, and ensured that they would make the most out of the addedtime together.

Keeping the Program GoingThe SRWM program continued in the summer of 2018 with sup-port from the National Security Agency (NSA), supplemented byMSRI funds. MSRI was able to expand its reach beyond womenfrom the BIRS workshops, and we received applications from 22groups (totaling 81 individual applicants). MSRI was able to accept21 researchers forming six groups of two to five researchers each,covering topics in Lie algebras, Riemannian geometry, Hecke al-gebras, PDEs, and evolutionary biology. All six groups continue

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to work together, and many expect to be able to submit papers forpublication within the next few months.

An important factor in the success of this project was MSRI’s abil-ity to offer family support funds to the many women with children.Because of this, they were able to either bring their children toBerkeley accompanied by a family member or leave them at homewith trusted childcare. Being away from family during the summeris not easy, and without adequate financial support, it would nothave been possible for many of them to participate in these workinggroups at MSRI.

In exit surveys, participants noted the benefits of distraction-free,in-person collaboration; the value of candid discussions amongpeers regarding family and work/life balance; and their unexpectedsense of relief and comfort in sharing partial or incomplete ideasand asking questions in a women-only setting.

The program will continue in Summer 2019 with an increased num-ber of participants, thanks to the generous support of the Lyda HillFoundation, the NSA, Microsoft Research, and the National ScienceFoundation. To learn more, visit msri.org/srw2019.

Save the Date! — National Math Festival: May 4, 2019

Welcome to the Third NMF!We welcome you to join us at the National Math Festival this com-ing May 4th in Washington, DC! This free, all-ages event takesplace at the Washington Convention Center and features a day oflectures, demos, art, films, performances, puzzles, games, children’sbook readings, and more to bring out the unexpected in mathemat-ics for everyone. Over 20,000 visitors joined the 2015 and 2017Festivals. The 2019 National Math Festival is organized by MSRIin cooperation with the Institute for Advanced Study (IAS) and theNational Museum of Mathematics (MoMath).

HighlightsThe Alfred P. Sloan Foundation Film Room will be home to manyshort, creative math films throughout the Festival, as well as oppor-tunities for Q&A with several 2019 Festival presenters and winningfilmmakers from the National Science Foundation-sponsored “WeAre Mathematics” contest.

The Make or Take Spiral hosts dozens of national and local mathorganizations offering hands-on activities and take-home resourcesfor math educators and visitors of all ages.

Explore Mathical Book Prize award-winning titles with live authorreadings for ages 2-18!

You can find more information at nationalmathfestival.org, includ-ing resources to share the invitation with your family, friends, andcommunity. A full schedule of all 2019 Festival events is online atnationalmathfestival.org/events.

Featured PresentersBARKIN/SELISSEN PROJECT • Holly Krieger (University of

Cambridge) • Mark Mitton • Lillian Pierce (Duke University) •Annie Raymond (University of Massachusetts Amherst) •

Emily Riehl (Johns Hopkins University) • Marcus du Sautoy(University of Oxford) • Nancy Scherich (University ofCalifornia, Santa Barbara) • Francis Su (Harvey MuddCollege) • James Tanton (Mathematical Association ofAmerica) • Amelia Taylor (Zymergen) • Joseph Teran

(University of California, Los Angeles) • John Urschel(Massachusetts Institute of Technology) • Suzanne L. Weekes

(Worcester Polytechnic Institute) • Avi Wigderson (IAS) •Mary Lou Zeeman (Bowdoin College)

Major Activity PresentersNational Museum of Mathematics (MoMath) • Julia

Robinson Mathematics Festival • MIND Research Institute •Natural Math and Math On-A-Stick • The Bridges Organization •

The Young People’s Project

We thank our 2019 Festival Sponsors, including the Simons Foun-dation, the Alfred P. Sloan Foundation, Eric and Wendy Schmidt, theNational Science Foundation, the Gordon and Betty Moore Founda-tion, the Kavli Foundation, the American Mathematical Society, theCharles and Lisa Simonyi Fund for Arts and Sciences, and NorthropGrumman.

9

Focus on the Scientist: Vered Rom-KedarVered Rom-Kedar is a Chern Professor in the MSRI programHamiltonian Systems, from Topology to Applications through

Vered Rom-Kedar

Analysis. She has made outstandingcontributions in the field of dynam-ical systems and their applications.

Vered was born in Haifa, Israel. Hermother, Yael Rom, was the first fe-male pilot trained and certified bythe Israeli Air Force.

By the age of 19, Vered had al-ready obtained her first universitydegree from the Technion–IsraelInstitute of Technology located inHaifa. Then, after completing twoyears of military service in the IDF,

Vered moved to Cal Tech where she obtained a Ph.D. in appliedmathematics. After a postdoctoral fellowship at the University ofChicago, she settled at the Weizmann Institute of Science.

Since 2009 she has been the Estrin Family Chair of ComputerScience and Applied Mathematics at the Weizmann Institute.She served as the head of the Computer Science and AppliedMathematics Department from 2014–16.

In spite of the Weizmann Institute being principally a researchestablishment, Vered is actively involved in the development ofmathematical education at various levels. In addition to super-vising M.Sc. and Ph.D. students, she served as the head of theMoross Research School of Mathematics and Computer Science

from 2010–13. Since 2013 she has been a committee member forthe development of national high school curricula in mathematics.

She is also a member of the editorial board for the SIAM Journalon Applied Dynamical Systems and a member of the editorial ad-visory board of Chaos: An Interdisciplinary Journal of NonlinearScience.

Vered is well known for her important contributions to the prob-lems of transport in chaotic systems, creation of islands of stabil-ity, and mathematical features of Fermi acceleration. In addition,she has applied her knowledge of dynamical systems to the mod-eling of the immune response in humans.

Her current research interests are centered around Hamiltoniansystems, including both the theoretical aspects, such as near-integrable and near-billiard dynamics in multi-dimensional sys-tems, and applications of the theory to fluid mixing, non-linearoptics and resonant waves. She also works on mathematical mod-eling of the innate immune system with medical implications.

On her department website, Vered writes about herself: “In life,like in science, [I like] to do (too?) many different things (moth-ering three kids, painting, reading, water sports, biking, skiing,scuba-diving, and others), and thus suffer from the very goodproblem of lack of time to take a breath (which is yet anotherthing I like to do from time to time).” In addition, she is anexcellent painter and a person highly respected and appreciatedby her colleagues and friends.

— Vasily Gelfreich

Forthcoming WorkshopsJan 28–30, 2019: Connections for Women:Derived Algebraic Geometry, BirationalGeometry, and Moduli Spaces

Jan 31–Feb 8, 2019: IntroductoryWorkshop: Derived Algebraic Geometryand Birational Geometry and ModuliSpaces

Mar 6–8, 2019: CIME 2019:Mathematical Modeling in K-16:Community and Cultural Context

Mar 25–29, 2019: Derived AlgebraicGeometry and its Applications

Apr 8–12, 2019: Hot Topics: RecentProgress in Langlands Program

May 6–10, 2019: Recent Progress inModuli Theory

Jun 10–12, 2019: Improving thePreparation of Graduate Students to TeachUndergraduate Mathematics

Jun 15–Jul 28, 2019: MSRI-UP 2019:Combinatorics and Discrete Mathematics

Aug 15–16, 2019: Connections for Women:Holomorphic Differentials in Mathematicsand Physics

Aug 19–23, 2019: Introductory Workshop:Holomorphic Differentials in Mathematicsand Physics

Aug 29–30, 2019: Connections for Women:Microlocal Analysis

Sep 3–6, 2019: Introductory Workshop:Microlocal Analysis

Summer Graduate SchoolsJun 3–14, 2019: Commutative Algebraand its Interaction with Algebraic Geometry

Jun 10–21, 2019: Random and ArithmeticStructures in Topology

Jun 24–Jul 5, 2019: RepresentationStability

Jul 1–13, 2019: Séminaire deMathématiques Supérieures 2019: CurrentTrends in Symplectic Topology

Jul 1–12, 2019: Geometric Group Theory

Jul 8–19, 2019: Polynomial Method

Jul 22–Aug 2, 2019: Recent Topics onWell-Posedness and Stability ofIncompressible Fluid and Related Topics

Jul 29–Aug 9, 2019: Toric Varieties

Jul 29–Aug 9, 2019: Mathematics ofMachine Learning (Microsoft)

Jul 29–Aug 9, 2019: H-Principle

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.

10

Celebration of Mind 2018: Math, Magic, Games, Puzzles, & MoreOn October 21, 2018, MSRI hosted over 300 visitors at the in-stitute’s annual Celebration of Mind festival. Developed by theGathering 4 Gardner Foundation, each year events are held aroundthe world on or around Martin Gardner’s birthday (October 21) sothat people can meet and share in the legacy of this polymath.

This year’s event featured stage presentations and activities by CarloSéquin (UC Berkeley), Henry Segerman (Oklahoma State Univer-sity), Brady Haran (Numberphile), magician Mark Mitton, andspeedcuber Sydney Weaver. There was also hands-on fun with theCalifornia Math Festival, the Julia Robinson Mathematics Festival,and the Elwyn and Jennifer Berlekamp Foundation. In one activity,dozens of attendees worked together to build a three-story-tall dualhelix sculpture with MoMath co-founder Glen Whitney. MSRIstaff also hosted a Zometool geometric bubble blowing station andMathical Book Prize reading nook. Mark your calendar to join usnext October!

Left column: Gretchen Muller (at right) of the California MathFestival shares activities with families, while three young par-ticipants take a break with Mathical Book Prize winning titlesin the Hearst Library. Center: Considering a move at the JuliaRobinson Mathematics Festival in our commons room. Rightcolumn: UC Berkeley math students and festival visitors as-semble a three-story dual helix with MoMath co-founder GlenWhitney, and the sculpture hanging in the atrium.

Named Positions, Fall 2018MSRI is grateful for the generous support that comes fromendowments and annual gifts that support faculty and postdocmembers of its programs each semester.

Chern, Eisenbud, and Simons ProfessorsGonzalo Contreras, Centro de Investigación en MatemáticasWilfrid Gangbo, University of California, Los AngelesJeffrey Lagarias, University of MichiganJames Meiss, University of Colorado BoulderRichard Montgomery, University of California, Santa CruzVered Rom-Kedar, The Weizmann InstituteAntonio Siconolfi, Sapienza – Università di RomaZensho Yoshida, University of Tokyo

Named Postdoctoral FellowsHuneke: Rosa Vargas, Universidad Nacional Autónoma de

MéxicoStrauch: Gabriel de Oliveira Martins, University of California,

Santa CruzUhlenbeck: Thomas McConville, MITViterbi: Joshua Burby, Courant Institute of Mathematical

Sciences

11

More News from MSRIConnections for Women, Fall 2018

Lisa

Jaco

bs

Connections for Women (CfW) is an ongoing workshop series thatserves as an introduction to each semester’s programmatic topics. InAugust, Nancy Hingston (The College of New Jersey) gave a CfWworkshop on loop products and self-intersections in the SimonsAuditorium for this semester’s Hamiltonian Systems program.

MSRI Connections Abound in RecentFields Medalists and AWM FellowsIn August, the International Mathematical Union announced thisyear’s Fields Medalists at the International Congress of Mathe-maticians in Rio de Janeiro, Brazil. MSRI extends our sincerecongratulations to the winners, all of whom have been connected tothe Institute: three of the four — Alessio Figalli (ETH Zurich), Pe-ter Scholze (Universität Bonn), and Akshay Venkatesh (Institutefor Advanced Study) — have been in residence for recent programsat MSRI, while the work of the fourth winner, Caucher Birkar(University of Cambridge), was featured in a Hot Topics workshopon canonical models of algebraic varieties.

The Association for Women in Mathematics (AWM) has namedtheir second class of AWM Fellows, whose commitment to sup-porting and growing women across the mathematical sciences ispraised by their students and colleagues. Amongst the 2019 Fellowsare MSRI Acting Director Hélène Barcelo, MSRI Trustees MariaKlawe (Harvey Mudd College) and Dusa McDuff (Barnard Col-lege), and Committee of Academic Sponsors Chair Judy Walker(University of Nebraska). Fourteen of this year’s 23 fellows (and22 of 32 fellows in the 2018 inaugural class) have been involved ingovernance and/or scientific programs at MSRI. For a complete list,visit tinyurl.com/AWMfellows.

MSRI-UP at SACNAS

Paul

Sam

uelI

gnac

io

MSRI-UP participants Esteban Escobar (Cal Poly Pomona; pic-tured) and Heyley Gatewood (Stetson University) won awards forposter presentations at this year’s SACNAS national meeting. Theposters presented results from their group research projects in the2018 “Mathematics of Data Science” Research Experience for Un-dergraduates, funded by the National Science Foundation and theAlfred P. Sloan Foundation.

New Faces at MSRI

MSRI welcomed the following new staff members duringthe summer and fall of 2018. Pictured at top (left to right):Bertram Ladner, Facilities and Food & Beverage Coordina-tor; Kimberly Manning, Executive Assistant to the DeputyDirector; Nate Orage, Technical Support. At bottom (leftto right): Demaur Owens, Systems Administrator; andBenjamin Parker, Receptionist. Annie Averitt, Director ofAdvancement and External Relations, is introduced in the Viewfrom MSRI on page 2.

12

Call for MembershipMSRI invites membership applications for the 2019–2020 academicyear in these positions:

Research Members by December 1, 2018Postdoctoral Fellows by December 1, 2018

In the academic year 2019–2020, the research programs are:

Holomorphic Differentials in Mathematics and PhysicsAug 12–Dec 13, 2019Organized by Jayadev Athreya, Steven Bradlow, Sergei Gukov,Andrew Neitzke, Anna Wienhard, Anton Zorich

Microlocal AnalysisAug 12–Dec 13, 2019Organized by Pierre Albin, Nalini Anantharaman, Kiril Datchev,Raluca Felea, Colin Guillarmou, Andras Vasy

Quantum SymmetriesJan 21–May 29, 2020Organized by Vaughan Jones, Scott Morrison, Victor Ostrik, EmilyPeters, Eric Rowell, Noah Snyder, Chelsea Walton

Higher Categories and CategorificationJan 21–May 29, 2020Organized by David Ayala, Clark Barwick, David Nadler, EmilyRiehl, Marcy Robertson, Peter Teichner, Dominic Verity

MSRI uses MathJobs to process applications for its positions. Inter-ested candidates must apply online at mathjobs.org after August 1,2018. For more information about any of the programs, please seemsri.org/scientific/programs.

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 1,October 1, and December 1. 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 ofintense mathematical activity chosen the previous fall. Proposalsshould be received by March 1, October 1, or December 1 forreview at the upcoming SAC meeting. See tinyurl.com/msri-htw.

Summer Graduate SchoolsEvery summer MSRI organizes several two-week-long summergraduate workshops, most of which are held at MSRI. Proposalsmust be submitted by March 1, October 1, or December 1 forreview at the upcoming SAC meeting. See tinyurl.com/msri-sgs.

Focus on the Scientist:Jacques FéjozJacques Féjoz is a mathematician of depth and impres-sive power. He has made several important contribu-tions to stability and instability in celestial mechanics

Jacques Féjoz

and Hamiltonian dynamics. Letme start with his remarkableresult on the stability of aplanetary system.

Stability versus instability inthe solar system is one of themost important problems indynamics. It is natural to studydynamics of the solar systemusing Newtonian equations.The most important result aboutstability of the solar system saysthat for a planetary system withsmall enough masses of planets,there is a positive measure of initial conditions leading toquasiperiodic motions, densely filling invariant tori in thephase space. The first important step toward proving thisresult was made by V. Arnold in the 1960s. In the 1990s,M. Herman made another important step, and finally in theearly 2000s, Jacques completed the proof.

Another one of Jacques’ fascinating results is proof ofthe existence of reparametrized quasiperiodic solutionsaccumulating to a double collision. This is fairly surprisingsince, heuristically, collisions lead to sensitive dependenceon initial conditions.

Analysis of instabilities in the solar system is full of puzzlesand open questions. One place in the solar system whereinstabilities are quite illuminating is the asteroid belt. It turnsout that between the orbits of Jupiter and Mars, there are overone million astroids of diameter greater than 1 km. When onelooks at the distributions of asteroids with respect to orbitalperiod, there are gaps in the distribution. These so-calledKirkwood gaps correspond to rational relations between theperiods of the asteroids and Jupiter. The most notorious onesare 1:3, 2:5, and 3:7.

One possible explanation of the 1:3 Kirkwood gap wasproposed by J. Wisdom and, independently, by A. Neish-tadt. Jacques, jointly with Guadia, Kaloshin, and Roldan,proposed another explanation and proved the existenceof unstable motions in the regime of small Jupiter eccentricity.

Jacques is versatile. He does mathematics, plays oboe in theUniversité PSL orchestra, and also plays water polo, all ata high level. Jacques explains that doing mathematics andplaying oboe have at least one thing in common: peoplegenerally don’t know how hard it is.

— Vadim Kaloshin

13

Named Postdocs — Fall 2018Huneke

Rosa Vargas

Rosa Maria Vargas Magaña isthe Huneke Postdoctoral Fellowin the Hamiltonian Systems, fromTopology to Applications throughAnalysis program. In 2017 she re-ceived her Ph.D. in MathematicalSciences from the Universidad Na-cional Autónoma de México work-ing on surface water waves underthe supervision of Panayotis Panay-otaros. Her principal research inter-ests are the derivation of simplifiedmodels for the study of physical phenomena that lead to nonlin-ear partial differential equations with rich mathematical struc-ture. In particular, she introduced a Whitham–Boussinesq-typemodel that describes the effects caused by large order depth vari-ations on surface water waves when the full linear dispersionis included in a weakly nonlinear approximation and exhibits acombination of nonlinear and variable depth effects involvingnon-local operators.

The Huneke postdoctoral fellowship is funded by a generous en-dowment from Professor Craig Huneke, who is internationallyrecognized for his work in commutative algebra and algebraicgeometry.

Strauch

Gabe Martins

Gabriel Martins is this semester’sStrauch fellow in the HamiltonianSystems, from Topology to Appli-cations through Analysis program.He grew up in São Paolo and Riode Janeiro, Brazil. He got his bach-elor’s and master’s degrees in theUniversidade Federal do Rio deJaneiro where he wrote a master’sthesis on the basics of Morse The-ory and Symplectic Topology ad-vised by Umberto Hryniewicz. Hereceived his Ph.D. from UC Santa Cruz in the spring of 2018,where he worked with Richard Montgomery. Gabriel stud-ies the dynamics of a charged particle under the influenceof a strong magnetic field, specifically in a container whosefield tends to infinity at the boundary. He proved that ifthe field blows up fast enough, then the charged particle istrapped in the interior for all time. Gabriel’s methods alsoshow promise for providing quantitative leakage results forcontainers with bounded magnetic fields — such as have beenproposed as a way to achieve sustained nuclear fusion. Gabriel’s

current work also involves exploring the quantum/classical dis-crepancy between his work for classical particles and a quantumtrapping theorem proven by Colin de Verdière and Truc, as wellas investigating trapping when using the non-Abelian general-ization of magnetic fields, known as Yang–Mills fields.

The Strauch postdoctoral fellowship is funded by a generousannual gift from Roger Strauch, Chairman of The Roda Group.He is a member of the Engineering Dean’s College AdvisoryBoards of UC Berkeley and Cornell University, and is alsocurrently the chair of MSRI’s Board of Trustees.

Viterbi

Joshua Burby

Joshua Burby is the Viterbi Post-doctoral Fellow in this semester’sHamiltonian Systems, from Topol-ogy to Applications through Anal-ysis program. After receiving hisPh.D. in 2016 from Princeton Uni-versity’s Plasma Physics Labora-tory, under the supervision of HongQin, he obtained a US DOE FusionEnergy Sciences Postdoctoral Re-search Fellowship with residenceat the Courant Institute of Mathe-matical Sciences in New York. After the Viterbi Fellowship, hewill take up residence at the Los Alamos National Laboratoryas a Feynman Postdoctoral Fellow. He was also awarded, butdeclined, a Humboldt Research Fellowship. Josh has broadinterest in the geometric structure of physical law, but partic-ularly the Hamiltonian and variational structure of dynamicalsystems that describe plasmas. Given the multi-scale com-plexity of these systems, he brings geometrical techniques offast-slow dynamical systems theory to obtain faithful reducedmodels that maintain structure, both in removing fast scalesto obtain reduced partial differential equations, and further toobtain structure-preserving discretizations for numerics.

The Viterbi postdoctoral fellowship is funded by a generous en-dowment from Dr. Andrew Viterbi, well known as the co-inventorof Code Division Multiple Access (CDMA) based digital cel-lular technology and the Viterbi decoding algorithm, used inmany digital communication systems.

Correction!

In the Spring issue, the brief bio of Karen Uhlenbeck that ap-peared with her fellowship stated that she was the first womanmathematician named to the National Academy of Sciences in1986. This is not the case: Julia Robinson, in 1976, becamethe first woman to be elected to the mathematical section of theNational Academy of Sciences. We apologize for the error.

14

Puzzles ColumnElwyn Berlekamp, Joe P. Buhler, and Tanya Khovanova

This puzzle column has been running for almost exactly 18 years.There have been guest editors along the way, but Elwyn and Joe,

and MSRI, would like to take this oppor-tunity to welcome Tanya Khovanova

as a new long-term composer and edi-tor. Tanya is at MIT, where she is HeadMentor for RSI (the Research ScienceInstitute), and for MIT PRIMES. Tanyais well known in the recreational math-ematics community for her blog, heronline videos, and her clever problems.Two of her papers were chosen for theBest Writing on Mathematics volumesin 2014 and 2016. Tanya received her

Ph.D. in 1988 from the Moscow State University (under IsraelGelfand) and worked as a systems engineer, applied mathematician,and lecturer before coming to MIT. Her early research was in repre-sentation theory and related topics, and she is now also interestedin combinatorics and recreational mathematics.

1. Amy and Bob play a game. Initially, there are 20 cookies on ared plate and 14 on a blue plate. A legal move consists either ofeating two cookies on one plate, or moving one cookie from the redplate to the blue plate. The last player to move wins, and of courseboth Amy and Bob want to win. Amy makes the first move, andthen they alternate moves. Who will win?

Comment: This problem appeared on the 2014 Bay Area Mathemat-ical Olympiad.

2. Five people, A, B, C, D, E, sit around a table playing “Mafia.”The players are secretly assigned roles: two are in the mafia, one isa detective, and other two are innocent (that is, neither a detectivenor in the mafia). The mafiosi always lie, and the other playersalways tell the truth. The mafiosi know each other, and the detectiveknows who they are. The mafia members do not know who thedetective is, and the two innocents know nothing about the others’roles. All players know the rules, and the number of players giveneach role. The following four statements are made in order.

A: I know who B is.B: I know who the detective is.C: I know who B is.D: I know who E is.

Who is who?

Comment: The game is well known (for example, it has a Wikipediapage), and this puzzle can be found on the QWERTY YouTubechannel. It also appeared on Tanya’s blog.

3. Let A= {1,2,3,4,5} and let P be the set of all nonempty subsetsof A. A function f from P to A is a “selector” function if f(B) is inB, and f(B[C) is either equal to f(B) or f(C). How many selectorfunctions are there?

Comment: We saw this on Stan Wagon’s Problem of the Week blog(stanwagon.com/potw/), and it apparently originally appeared in aTurkish national competition.

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4. Let C be a circle of radius r> 1 in the plane. You are allowed toaugment the set C by repeatedly performing the following operation:Take two points in C that are a distance 1 apart, and add the linesegment between them to C. Show that it is possible to performfinitely many such operations so that the center of the circle is inthe resulting set.

Comment: This problem is due to Gregory Galperin, who also asksthe much harder question of what a reasonable upper bound is, as afunction of r, on the number of augmentations that are required.

5. Assume that a correctly set clock has hour, minute, and secondhands of the same length, and that the motion of those hands iscontinuous. Prove that it is impossible for the tips of those handsto be at the vertices of an equilateral triangle. Suppose that a clockbuilder chooses a wrong gear and the result is a clock which hascorrect hour and second hands, but the minute hand makes only11 revolutions in a 12-hour period. Show that, despite the clock’sflaws, it does have the virtue that the hand tips do occasionally lieat the vertices of an equilateral triangle. (Assume that at noon andmidnight the hands all point straight up, as on a normal clock.)

6. Although hats problems have been something of a staple in ourpuzzle columns, a quite novel one appeared in an article by Tanyain the Mathematical Intelligencer in 2014. We follow this with avariant that we heard from Larry Carter and Stan Wagon; Wagon,Rob Pratt, Michael Wiener, and Piotr Zielinski have a paper on thearXiv that discusses further generalizations of these questions.

(a) There are n prisoners and n+ 1 hats. Each hat has its owndistinctive color. The prisoners are put into a line by their friendlywarden, who randomly places hats on each prisoner (note that onehat is left over). The prisoners “face forward” in line which meansthat each prisoner can see all of the hats in front of them. In par-ticular, the prisoner in the back of the line sees all but two of thehats: the one on her own head, and the leftover hat. The prisoners(who know the rules, all of the hat colors, and have been alloweda strategy session beforehand) must guess their own hat color, inorder starting from the back of the line. Guesses are heard by allprisoners. If all guesses are correct, the prisoners are freed. Whatstrategy should the prisoners agree on in their strategy session?

(b) Suppose that there are three prisoners and five hats — that is, thewarden will place three hats, and there will be two extra hats. Therules are as above. Find an optimal strategy!

15

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Non-ProfitOrganizationUS Postage

PAIDBERKELEY, CAPermit No. 459

Autumnal Greetings from MSRI!. . . and from the postdocs in this fall’s jumbo program on Hamiltonian Systems, who enjoyed carving some jumbo

pumpkins at a Halloween social in October.

Join us at the 2019 JMM in BaltimoreMathematical Institutes Open House

Wednesday, January 16 Marriott Inner Harbor5:30 – 8:00 pm Grand Ballroom East West ABC

1st Floor

MSRI Reception for Current and Future DonorsThursday, January 17 Hilton Baltimore

6:30 – 8:00pm Carroll A3rd Floor

For more information, contact development@msri.org.

MSRI DirectorateAdd @msri.org to email addresses. All area codes are (510) unless otherwise noted.

David Eisenbud, Director (on sabbatical 2018–19), 642-8226, deHélène Barcelo, Acting Director, 643-6040, directorMichael Singer, Assistant Director, 2018-19, 643-6070, msingerAnnie Averitt, Director of Advancement and External Relations, 643-6056, aaveritt

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