Spring 2016

EMISSARY

Mathematical Sciences Research Institutewww.msri.org

Willmore Bending Energy on the Space of Surfaces
Rob Kusner, Andrea Mondino, and Felix Schulze
The interplay between curvature and topology abounds at MSRI thisspring, with Differential Geometry the theme of our semester-long program. Since it would be impossible to touch upon everything happening here now, we focus on a topic close to our recent interests: how thecurvature-based Willmore bending energy can be used to turn a sphereinside out without creasing or tearing, explain the shapes of red blood cells,
estimate the mass of the cosmos, and more.

Bending Energy: From Robert Hooke to Tom
Willmore via Sophie Germain
Imagine a surface in space made from an ideal isotropic material that canbe effortlessly stretched, but resists being bent. The “bending version” of
Hooke’s Law for elasticity predicts that the energy density of the surface
is proportional to the sum of the squares of its principal curvatures, κ1 andκ2 , which can be rewritten κ21 + κ22 = (4H2 − 2K) in terms of the mean
curvature H = 12 (κ1 + κ2 ) and the Gauss curvature K = κ1 κ2 . The Gauss–
Bonnet formula,Z
K = 2π χ(Σ),Σ

relates the total Gauss curvature to the Euler number χ(Σ) of a closed surface Σ in R 3 . This formula — or the corresponding relation for a surface
with suitable boundary conditions — implies the bending energy is proportional toZ
W(Σ) = H2Σplus a topological constant. Although now named for T. J. Willmore, whoin the 1960s rekindled interest in W for closed surfaces, this bending energy was introduced by Sophie Germain to explain the sand patterns seen(continued on page 5)

At right: Two stereographic projections of Lawson’s minimal surfaces
ξg,1 of genus g = 3, 4, 5, 6, with principal curves drawn. These areconjectured to be the Willmore energy minimizers for each genus.
(Graphics by N. Schmitt, Universität Tübingen GeometrieWerkstatt.)

Contents
The View from MSRI
Math Circles
Sun-Yung Alice Chang
Public Understanding
DG Program (cont’d)

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3
3
4
5

Call for Proposals
Spring Postdocs
Tobias Colding
Named Positions
Staff News

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11

Clay Senior Scholars
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CME Group–MSRI Prize 12
Workshops
12
Call for Membership
12
Puzzles Column
131Mathematical Art
2015 Annual Report
Staff Roster

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The View from MSRIDavid Eisenbud, Director

David Eisenbud

I have the great pleasure of announcing some news that is particularly welcome in this era of federal budget belt-tightening: a number of long-term development efforts have come to fruition thisyear!

New Funding for Research
Science

Shiing-Shen Chern Endowment. First, MSRI will receive a new$5 million endowment, named in honor of MSRI’s founding Director: the Shiing-Shen Chern Endowment will be funded by Chern’s
son and daughter, Paul Chern and May Chu; and by one of Chern’scollaborators, Jim Simons, and his wife Marilyn Simons. The endowment will help MSRI attract more of the most distinguishedmathematicians to its scientific programs. These major figures inmathematics form the core of our programs, mentoring the postdoctoral fellows and other junior members, and attracting the participation of a large community of mathematicians. Starting soon,
when the endowment is complete, MSRI will name at least one
such “Chern Professor” each semester.

All this good news for the future does not eclipse our current scientific programs: the wonderful jumbo program on Differential Geometry now under way (see the cover article); the Critical Issues
in Math Education workshop about observing (in video) teachersin action; the prize that we gave, jointly with the CME Group, toDouglas Diamond for his work in mathematical economics (seepage 12); and, last but not least, this year’s Hot Topics workshop,
on Wall Crossing and Cluster Algebras, an extraordinary application of ideas from string theory to an expanding field that used toseem very distant from that subject.

Public Understanding of Math

Postdoc funding. The second piece of good news is that MSRI is
receiving new funding for its sought-after postdoctoral positions.
Coming to MSRI is an unmatched experience for postdocs in thefields of its semester programs, and for this reason the demand for
positions is very high — this year there were over 300 applications(a record) for the 2016–17 programs, despite the high degree of
specialization. However, coming to MSRI for a semester has represented more and more of a financial sacrifice: the stipends for
postdoctoral fellows (in sync with the NSF Postdoc stipends) havenot been raised for many years, despite the substantial increase in
the cost of living in the Bay Area.

Our outreach programs are buzzing as well. Building on the great
public enthusiasm for the National Math Festival that we ran jointlywith the Institute for Advanced Study in Washington, DC, lastApril, we’re planning a second festival for April 2017. The Simons Foundation and the Kavli Foundation have already pledged
their support, and we expect that a number of other foundations
and companies will soon join them.
The Mathical Book Prize, our award for outstanding math-related
books for kids of all ages, is now a partnership with the National
Council of Teachers of Mathematics and the National Council of
Teachers of English, vast organizations whose collaboration will
help us make the prize known and get the winning books into
the hands of children throughout the country, particularly in lowincome communities. The National Association of Math Circles
is pursuing similar goals. Numberphile, the very popular YouTube
video channel created by journalist Brady Haran with MSRI support, now has over 1.6 million subscribers, and a brand new subseries on “Tadashi’s Toys” (Tadashi Tokieda is an applied mathematician and a truly great popularizer). More details about all ofthese programs are in the update on page 4.

Now, an anonymous donor’s five-year pledge has enabled MSRI to
invite almost one third more postdocs and to increase the stipends
by 20–40%, making them competitive with excellent university positions. Causation is hard to attribute, but it is the case that 100%of the 34 postdocs to whom we made offers are planning to cometo MSRI next year; I doubt that many other institutions in mathematics can make this boast.

Simons support. In addition to all this, Jim and Marilyn Simons
have greatly increased their unrestricted support of MSRI; this will
give MSRI new flexibility and allow us to improve both the financial circumstances of visitors and other aspects of the environment,
Community Does It
such as computing resources and services.
In all these activities, our purpose is to support the mathematical
Support for women with families. Targeted donations will allow sciences community in both its research mission and its relation to
us to achieve another long-desired goal: increasing the possibil- the public, but in this we depend on community participation: weities for women with children to participate in our programs, by host, rather than perform. And of course we depend on commuproviding adequate financial support. Several years ago, MSRI es- nity support in many other ways, including financial — the donors
tablished the position of Family Services Coordinator with a gener- of large gifts are very concerned with knowing that they are reallyous gift from Roger Strauch to support all men and women coming supporting the things that are important to the mathematics comto MSRI with families; now, additional funding from trustee Mar- munity, and this is partly measured by the number of contributors
garet Holen and her husband David Coulson, as well as a second in our community. I want to thank our many supporters, listed in
gift from Vinita and Naren Gupta, will provide additional support, the Annual Report on page 14. Your gifts impact our work at MSRI
specifically for women, at levels corresponding to the costs of day- and help us with raising additional funds from private sources tocare or school.
keep growing our programs. Thank you!
2

Math Circles: On Screen, In Print
Math Circles have been having their moment in the sun lately, both in print and on thescreen. MSRI’s support program for Math Circles, the National Association of Math Circles (NAMC), was delighted to have some of its members interviewed for a feature article
in the March issue of The Atlantic Monthly, “The Math Revolution.” The article explores
why the number of American teens who excel at advanced math has surged, and Math
Circles naturally play a starring role! The public interest in Math Circles triggered by the
Atlantic article led to an Education Week interview with NAMC Director Diana White andAssociate Director Brandy Wiegers.
The new documentary film Navajo Math Circles is now traveling around the country forscreenings and is available for DVD purchase. Several of the students and teachers fromthe film attended the world premiere at the Joint Mathematics Meetings in January, and will
also be appearing at the Navajo Nation Museum in Arizona in April.
To read the Atlantic and Education Week articles, visit tinyurl.com/MathRevolution andtinyurl.com/EduWeek-MathCircles. To learn more about Navajo Math Circles and screenings near you, you can see www.navajomathcirclesfilm.com.

N AVAJO M ATH C IRCLES
A fi l m by G E O RG E C SI CS ERY

Focus on the Scientist: Sun-Yung Alice Chang
Sun-Yung Alice Chang, theEugene Higgins Professor of
Mathematics at Princeton University, has made outstanding
contributions both in geometry
and analysis.

bounded analytic functions on domains. Her deep analytical intuition has shaped her view in partial differential equations, establishing a solid bridge between analysis and geometry.
Her interest in geometry initially came from mathematical collaboration with her husband. They worked on the problem ofprescribing Gaussian curvature on the sphere, where a crucial ingredient is the study of extremals for a Sobolev-type inequality.
Alice also got involved in conformal geometry, which is the natural higher dimensional analogue for the geometry of surfaces,and spectral invariants in particular.

Alice was born in China in theancient capital city of Xi’an.
During the period of the Chinese Revolution, her family
moved to Hong Kong and thento Taiwan when she was two.
She grew up in Taiwan and atSun-Yung Alice Changtended National Taiwan University. Later she moved to the U.S. where she obtained her
Ph.D. in mathematics at UC Berkeley. There she met her husband, Paul Yang, who was a graduate student working on geometric problems.

One of Alice’s main achievements is the understanding of conformal invariants in higher dimensions. While Gauss curvatureis the canonical invariant for surfaces, Alice has made a deep
contribution to the understanding of fourth-order Q-curvature,
which is the natural replacement in four dimensions since it controls both the geometry and the topology of such a manifold. She
has also considered fully nonlinear versions of scalar curvaturein conformal geometry, which in particular yields classification
Alice held positions at the State University of New York at Buf- results for four-dimensional manifolds.
falo, the University of California, Los Angeles, and the University of Maryland before moving to Princeton University in In her own words: “I have always felt that mathematics is a lan1998. Her deep mathematical contributions have been widely guage like music. To learn it systematically, it is necessary torecognized, including a plenary talk at the 2002 International master small pieces and gradually add another piece and thenCongress of Mathematicians in Beijing as well as an Ameri- another. Sitting in a good mathematics lecture is like sitting in acan Mathematical Society Colloquium talk in 2004. In 1995, good opera. Everything comes together. They get to the heart ofshe received the Ruth Lyttle Satter Prize in Mathematics. The the problem and I enjoy it!”prize is awarded every two years to a woman who has made an Alice has succeeded in transmitting this passion for mathematics
outstanding contribution to mathematics research in the previ- and the appreciation of its beauty to her students (more than fifous five years. Alice is currently a member of the US National teen already). Being Alice’s student and collaborator has engenAcademy of Sciences.
dered my deep admiration and respect for her, both as a matheAlice’s first interests came from harmonic analysis, as shown matician and as a woman.
— María del Mar González Nogueras

in her Ph.D. work, where she studied the boundary behavior of3

Public Understanding of Mathematics
Mathical Book Prize

Tadashi’s Toys

In January, MSRI and the Children’s Book Council announced the2016 winners of the Mathical: Books for Kids from Tots to Teens
youth literature prize. The Mathical Book Prize cultivates a love of
mathematics in the everyday world. Each year’s winners and honor
books join a selective and ever-growing list of new and previously
published fiction and nonfiction titles for kids of all ages.

Brady Haran and Tadashi Tokieda

Filmmaker George Csicsery and journalist Brady Haran, creatorof the MSRI-supported YouTube channel Numberphile, have been
making a series of videos featuring Stanford professor Tadashi
Tokieda’s mathematical toys, which have been entertaining viewers of all ages. If you haven’t seen “Tadashi’s Toys” yet, you may
be pleasantly surprised: objects you can make in a matter of minutes at home, which illustrate principles of mathematics, will delight both children and seasoned mathematicians. To see videos
demonstrating Tadashi’s inventions, visit msri.org/toys.

This year’s award winners are 8: An Animal Alphabet, by Elisha
Cooper; Max’s Math, by Kate Banks with art by Boris Kulikov;
and graphic novel Secret Coders, by Gene Luen Yang and Mike
Holmes. Gene Luen Yang, co-creator of Secret Coders, has also
been named the fifth National Ambassador for Young People’s Literature by the Children’s Book Council, Every Child a Reader, andthe Library of Congress. For a full list of current and past winners,
visit mathicalbooks.org.
Not on the TestMSRI continues to partner with Berkeley City College for the“Not on the Test: The Pleasures and Uses of Mathematics” lecture
series. September 2015 featured Barbara Simons, Board Chair of Verified
Voting, speaking on “Internet Voting: Wishful Thinking?” October’s
talk on “The Math and Science ofPixar Films” by Danielle Feinberg,
Director of Photography at Pixar Animation Studios, was extremely well attended, with over 300 attendees! February and March featured talks by two National MathFestival alumni: Kimberly Bryant of the nonprofit Black Girls
Code shared her work to promote computer science education forgirls of color; and Mariel Vazquez of the University of California,
Davis, taught the audience about how using mathematics and computational tools is critical for her research in microbiology, as mostcells in our bodies contain over two meters of DNA, which tends toknot in confined spaces. You can find videos of many of the talks
at msri.org/e/not-on-the-test.

National Math Festival
At the USA Science and Engineering Festival in April, MSRI
announced that the National Math Festival
will return to the nation’s capital on Saturday, April 22, 2017! The festival will takeplace at the Walter E. Washington Convention Center in downtown Washington, DC.
Stay tuned for more announcements overthe coming months about featured speakers
and activities as the festival kicks into gear!
For more news about the 2017 Festival (aswell as about math events around the U.S.
and resources for math lovers of all ages),
visit our brand new website: www.nationalmathfestival.org.

Illustrative Mathematics
Math educators around the U.S. can participate in a virtual lecture series that runs through May 2016. Developed in partnershipwith Illustrative Mathematics, the “Speaking of Mathematics Education: Productive Conversations with Families” webinars
feature exclusive content from an education
forum held in conjunction with the 2015
National Math Festival. Presenters include
William McCallum (University of Arizona), Suzanne Wilson (University of Connecticut), Deborah Ball (University of Michigan),and Peggy Brookins (National Board for Professional TeachingStandards). Archived webinars are also available for viewing. For
more info, visit tinyurl.com/MathEducationWebinar.

Harmonic Series Music Concerts
MSRI has welcomed the public to enjoy our free Harmonic Series
music concerts throughout the year, and February’s performance by the Left Coast Chamber
Ensemble featured the music of Debussy, Schumann, and an original work by pianist Eric Zivian. Additional concerts in 2015–16 featured
The Honey-Doos and the Galax Quartet. News
about upcoming concerts in 2016–17 will be released in earlyfall.
4Differential Geometry —Willmore Bending Energy(continued from page 1)

on Chladni’s vibrating plates. Napoleon had set a prize in 1808 forthe best solution, and the only relevant submission — Germain’s —was judged incomplete. Poisson then submitted his own solution
and was awarded the prize; curiously, he also noted that the K partof the bending energy density is a “divergence term” that does notaffect the Euler–Lagrange equation for equilibrium — or the corresponding wave equation for vibration — foreshadowing later versions of the Gauss–Bonnet relation by many decades!
By the 1920s, W. Blaschke and his students investigated what theycalled conformal area whose density 14 (κ1 − κ2 )2 = H2 − K is not
only scale-invariant, but also unchanged
R by2 Möbius transformations of space. It follows thatW(Σ)=
Σ H is Möbius-invariantR
for closed surfaces, since Σ K is a topological invariant. Integrating the inequality H2 > K over an immersed sphere and using
Gauss–Bonnet immediately gives W > 4π. In fact — as Willmore
himself noted — for any closed Σ immersed in R 3 , one can instead use the better inequality H2 > K+ = max(K, 0) along with
One example of a minimax sphere eversion, shown around
H. Hopf’s observation that “oneR can lay Σ on a table in any di- the border of the image. (Courtesy of J. M. Sullivan.)rection” to conclude W(Σ) > Σ K+ > 4π, with equality only forround spheres.

Sphere Eversion and the Smale Conjecture(Now the Hatcher Theorem) via W-Flow

W-Critical Surfaces and Minimal Surfaces in
3-Sphere and 3-Space

It is a remarkable fact that a sphere S2 in R 3 can be everted —that is, turned inside out, without tearing or creasing. Existenceof a sphere eversion is a consequence of Smale’s theorem (provenshortly before he joined the UC Berkeley faculty in 1960) that any
two immersed spheres in R 3 are regularly homotopic — that is,
the space of immersed S2 in R 3 is path connected. For several
years it remained a mystery how to explicitly carry this out, until
A. Shapiro, B. Morin, and Bill Thurston (MSRI Director, 1992–97)
each invented their own sphere eversions.

The main sources of W-minimizing or W-critical surfaces are minimal (H ≡ 0) surfaces in R 3 and S3 . In the 1970s, J. Weiner investigated the stability of W-critical surfaces, rediscovering something known to Blaschke: stereographic images Σ ⊂ R 3 of minimal surfaces Σ̃ ⊂ S3 — in particular, the plethora constructed by
H. B. Lawson that decade — are always W-critical, with W(Σ) =area(Σ̃).
Robert Bryant (MSRI Director 2007–13, current AMS President) then proved a classification theorem for immersed W-criticalspheres in R 3 : using Hopf’s holomorphic differential techniques,as well as ideas going back to Darboux, Cartan, and Chern (founding Director of MSRI), he showed these must arise from “conformally compactifying” complete minimal surfaces with planar embedded ends; in particular, a round sphere (the conformal compactification of a flat plane) is the only embedded W-critical sphere in
R 3 , and further, the critical values of W on the space of immersed
spheres are quantized in multiples of 4π. Many examples of minimal spheres with planar embedded ends were soon constructed,
not only by Bryant, but also by C. K. Peng and the first author, as
well as higher genus examples by C. Costa and N. Schmitt.

The minimax sphere eversion using Willmore flow on the space of
immersed spheres was conceived by the first author while a gradstudent at UC Berkeley in the early 1980s. (This is described in detail at http://www.gang.umass.edu/∼kusner/other/minimax.pdf andis illustrated in the border of the image above.) Its animation
was initiated by the first author with K. Brakke, G. Francis, andJ. M. Sullivan during a 1992 MSRI workshop, and debuted at the1998 ICM in Berlin as a short film The Optiverse (which you canfind on YouTube). Sullivan also discusses sphere eversions further
at http://www.math.uiuc.edu/∼jms/Papers/isama/bw/.

Bryant’s result is related to an important inequality for W provenby P. Li and S.-T. Yau, and sharpened by the first author: if d“sheets” of a closed immersed surface Σ pass through a point of
R 3 , then W(Σ) > 4πd; equality occurs only if there is a complete minimal surface Σ0 in R 3 with d embedded planar ends — or“sheets at ∞” — with a Möbius transformation taking Σ0 to Σ (and∞, to the d-uple point on Σ).

The space of embedded spheres in R 3 deformation retracts to the subspace of round spheres, and thus iscontractible.
An equivalent formulation in terms of unknotted loops in R 3 hadjust been proved by A. Hatcher using combinatorial surgery methods, and in light of Bryant’s then-recent result that the only embedded W-critical sphere in R 3 is round, this approach seemed

The seminal motivation for the minimax sphere eversion was ananalytic proof — using a modified Willmore gradient flow — of the
Smale conjecture:5

promising. Unfortunately, the lack of a maximum principle is ahuge obstacle to using this fourth-order W-flow directly: besides
work of E. Kuwert and R. Schätzle with the strong assumptionthat W < 8π, little is known about how to preserve embeddedness under the flow; there are even examples of initial immersed
spheres which become singular in finite time, so — despite compelling computer animations and numerical simulations — moreremains to be done to put the minimax sphere eversion via W-flow
on a firm analytic foundation.

The Willmore Conjecture and the Willmore
Problem

Images A–E show observations of conformal deformation ofa genus-2 phospholipid vesicle (scale bar = 10 µm). The bottom series shows computed images of the deformation. (A–E from X. Michalet and D. Bensimon, Observation of stable
shapes and conformal diffusion in genus 2 vesicles, Science
269 (1995), 666–668. Reprinted with permission from AAAS.
Bottom series from Jingmin Chen’s thesis.)

One impetus for the MSRI research program in Differential Geometry this spring was the recent proof by F. C. Marques and A. Nevesof the Willmore conjecture:
The W-minimizing torus Σ1 ⊂ R 3 is a stereographic
image of the Clifford minimal torus in S3 with area
2π2 = W(Σ1 ).

W-critical surface, the stereographic image of another Lawson
minimal surface from S3 , also conjectured to be the W-minimizer.

In his 1980s lecture “Three Obdurate Conjectures in Differential
Geometry” at UC Berkeley, Bob Osserman (MSRI Deputy Director 1990–95, Special Projects Director 1995–2011) alluded to theunexpected difficulty and richness of this conjecture by calling it“the sleeper” among the three. As noted above, the round sphereachieves the absolute minimum W = 4π for closed surfaces, and
L. Simon later showed that for higher genus surfaces the absolute minimum is strictly greater. The punchline of the Marques–
Neves argument is F. Urbano’s index characterization of the minimal Clifford torus, but their main contribution is a refinement of
the Almgren–Pitts min-max methods in geometric measure theory, allowing them to show (for instance) that any minimal surface
in S3 — other than a great 2-sphere or Clifford torus — must havearea strictly greater than 2π2 .

The Marques–Neves solution to the Willmore conjecture uses a
clever degree argument depending on the orientability of the surface, and their tools from geometric measure theory rely on embeddedness and codimension-1 in an essential way. It is unclear if
their methods can be adapted to study surfaces in higher codimension. For example, the corresponding conjecture remains open for
tori in R n , as does the following even more basic conjecture of thefirst author:

Any closed surface Σ in n-space not diffeomorphic
to S2 has W(Σ) > 6π, with equality if and only if Σ
comes from stereographic projection V ⊂ R 4 of theVeronese minimal projective-plane in S4 .
The general Willmore problem is to find the W-minimizer of Note that V and its mirror image solve the Willmore problem for
fixed topology — intrinsic diffeomorphism type or regular homo- the two regular homotopy classes of embedded projective planes intopy class — among closed immersed surfaces in R 3 or R n . The R 4 , by work of Li and Yau.
invariance of W under the noncompact group of Möbius transformations makes this problem delicate. The existence of higher
Extending the Bending Energy to a Frame
genus embedded W-minimizing surfaces in R 3 was established inwork over the past three decades by Simon, and by M. Bauer and Energy
Kuwert, using ideas of the first author, who has also conjectured:
The Willmore bending energy W has been discussed above to
study the space of immersed surfaces in R n . A natural extension of
The Lawson surface ξg,1 is the lowest-area minimal
W is a new energy introduced by the second author and T. Rivière
surface of genus g in S3 , and it stereographicallyfor framed immersions of a surface into space: the frame energy is
3
projects to the W-minimizer of genus g in R .
essentially the Dirichlet energy for tangent frames on a surface Σ
Some low genus Lawson surfaces are shown earlier in the figure immersed in R n . The frame energy is bounded below by W(Σ),
on the cover.
and it is much more coercive than W: indeed, an upper bound on
the frame energy prevents the degeneration caused by the action of
For regular homotopy classes that do not contain embedded surMöbius transformations of space or the degeneration of the conforfaces, finding an immersed W-minimizing surface representative
mal class of the underlying surface Σ.
remains an open problem, with one exception: Bryant and thefirst author constructed explicit immersed real projective planes, T. Rivière and the second author have proven a Willmore-type inthe simplest of which realizes a W-minimizing Boy’s surface B equality for any codimension: the frame energy of any framedin R 3 with W(B) = 12π (this surface is well known to anyone torus Σ immersed in R n is at least 2π2 , with equality if and only
who has visited Oberwolfach). For immersed Klein bottles in R 3 if n > 4 and there is a homothety carrying Σ to the square Clifford
there are three regular homotopy classes, but only the class of B torus S1 × S1 ⊂ R 4 ⊂ R n endowed with the standard orthonorconnect sum with itself (or the mirror image) is known to contain a mal frame (∂θ , ∂ϕ ), up to a fixed rotation. The frame energy may
6consider the analogous Canham problem: minimizing W(Σ) under
the isoperimetric constraint v(Σ) = v alone.

also be used to find canonical representatives for regular homotopy classes: for instance, in each of the two regular homotopy
classes of immersed tori in R 3 , there exists a smooth minimizerof the frame energy. Recently, Rivière extended the frame energy
to immersions of spheres into R m and used it as a viscosity term
to investigate the first author’s min-max proposal for constructing
W-critical spheres in R 3 .

There is a “tautological” solution Σ to the Canham problem forgenus g > 0: Let Σg be any (unconstrained) W-minimizing surface of genus g in R 3 , and let vg denote the smallest value of theisoperimetric ratio v(µ(Σg )) for its image under any Möbius transformation µ. Note (by the isoperimetric inequality) that vg < 1,
and also v(µ(Σg )) tends to 1 as µ becomes more extreme, with thesurface µ(Σg ) limiting to a round sphere (as a “microscopic hanThe Canham–Helfrich Model and Conformaldle” of genus g “vanishes”). Thus for any v in the interval [vg , 1),
Diffusion for Vesicles
we simply take Σ = µ(Σg ) for some Möbius transformation µ with
A vesicle is a lipid bilayer enclosing a fluid region, typical of cell v(µ(Σg )) = v: since W is Möbius invariant, this Σ achieves themembranes or the endoplasmic reticulum within a cell. In vesi- minimum value W(Σ) = W(Σg ).
cle models proposed by P. Canham and W. Helfrich in the 1970s,
between these two lipid layers lies an embedded surface Σ ⊂ R 3
critical for the modified bending energy,Z
WH0 (Σ) =(H − H0 )2 ,Σsubject to constraints on the surface area A(Σ) and enclosed
volume V(Σ). Including the “spontaneous curvature” term H0amounts to varying the original bending energy W(Σ) while
R imposing a third constraint on the total mean curvature M(Σ) = Σ H,
which geometrically accounts for the difference in areas of the twovesicle layers.
The Canham–Helfrich problem is to minimize W(Σ) among embedded surfaces Σ ⊂ R 3 of a given topological type subject to thesethree constraints. The Möbius invariance of W means it suffices to Dupin cyclides solve the genus-1 Canham problem. (FromJ. Chen’s thesis.)
impose the two scale-invariant constraints
For example, when g = 1, the solution to the Willmore conjecturelets one explicitly compute v1 ≈ 0.13 to be the isoperimetric ratio
realized by the stereographic projection of the square Clifford torus
on the isoperimetric ratio and normalized mean curvature when as a surface of revolution in R 3 ; this gives a large interval [v1 , 1)
minimizing W(Σ). Recall that the isoperimetric inequality asserts for which the Canham problem is solved by Dupin cyclides. (Seethe figure above; the first author’s recent work with J. Chen andv(Σ) 6 1, with equality if and only if Σ is a round sphere.
T. Yu suggests that these solutions are also unique for this interval
For a generic surface Σ ⊂ R 3 , a 3-parameter family of Möbius of isoperimetric ratios.)
transformations (inversions in unit spheres centered at points noton Σ) changes its shape. Thus, by the implicit function theorem, For g = 0, we have seen the W-minimizer Σ0 is a round sphere;
for regular values of the two constraints v and m, one expects a since Möbius transformations preserve round spheres, v0 = 1 is
1-parameter family of solutions to the Canham–Helfrich problem. the only isoperimetric ratio where the preceding “tautological” apThe possibility of such shape-changing families was pointed out proach works. J. Schygulla overcame this difficulty by applying
in the first author’s early work with L. Hsu and J. M. Sullivan Möbius transformations to a perturbation of the inverted catenoid
computing W-minimizing surfaces; dubbed “conformal diffusion” instead: for each v in (0, 1], this gives a surface Σ with v(Σ) = v andby physicist U. Seifert, this phenomenon was experimentally con- W(Σ) < 8π; he then used Simon’s “ambient approach” to show
firmed for real — but rather rare — lipid vesicles of genus 2 in the there exists a genus-0 solution of the Canham problem for the enÉcole Normale Supérieure biophysics laboratory of D. Bensimon tire interval (0, 1]. The solution is not explicit, nor is uniquenessand X. Michalet soon afterward. (See the figure at the top of the known (except for the round sphere at v = 1), but it is believed to berotationally invariant, shaped like a red blood cell for intermediateprevious page.)values of v, and like a “gastrula” for v near 0.
v(Σ) = 36π

V(Σ)2
=v
A(Σ)3

m(Σ) =M(Σ)2=m
4πA(Σ)The Canham Problem: Minimizing Energy with In collaboration with L. Keller and Rivière, the second author used
Rivière’s “parametric approach” to extend the range of isoperimetan Isoperimetric Constraint
ric ratios for which the higher genus Canham problem is solvThe simpler Canham model for vesicles ignores the constraint on able to an open interval containing [vg , 1). While the shape ofM(Σ). Thus it is both natural and mathematically more tractable to an unconstrained W-minimizer Σg for g > 1 is only conjectural,7

some recent numerical work of Chen and Yu with the first author indicates the optimal Möbius transformation one applies to
Σg to estimate vg . They also have explored the shapes of Canhamminimizers with small values of the isoperimetric ratio: these haveW ≈ 8π and look like doubled spheres joined by small necks;
however, the distribution of necks is unclear. A simple yet important open question that should lead to a resolution of the higher
genus Canham problem is to decide whether the strict inequality
W(Σ) < 8π can be achieved by surfaces of genus g > 0 for anyv(Σ) near 0.

of a Riemannian 3-manifold whose metric is asymptotically
Schwarzschild. The effective mass of the “black hole” represented
by this asymptotic metric can be computed by a flux integral at infinity which defines mADM , the ADM mass for such an end. The
Hawking mass,mH (Σ) = (16π)−3/2 A(Σ)1/2 16π − 4W(Σ) ,David Gregory & Debbie Marshall, Wellcome Images | CC BY-NC-ND 4.0

which is a proxy for the mass enclosed by a surface Σ within aspatial slice
R of spacetime, also makes use of the bending energy
W(Σ) = Σ H2 . While mH is well-defined for all closed immersed
surfaces, the mass it measures is “too small” on most of them:
indeed, as observed earlier, if Σ is a closed surface in Euclidean
space, then W(Σ) > 4π — and so mH (Σ) 6 0 — with equality only
for round spheres.
It is interesting to compare these two notions of mass. Using R. Geroch’s observation that mH (Σ) should evolve monotonically under inverse mean curvature flow, G. Huisken and T. Ilmanen haveshown that mH (Σ) 6 mADM for any outward minimizing Σ inN, yielding a proof of the long-conjectured Riemannian Penrose
inequality.

A related concept is the center of mass for an isolated gravitational system modeled by an asymptotically Schwarzschild end N.
By proving there exists a unique foliation of the end by constantmean curvature surfaces, asymptotic to a foliation by concentric
spheres, G. Huisken and S.-T. Yau defined such a center. Their
Observed and computed red blood cells. At left, scanning foliation is more closely related to yet another notion, the isoperielectron micrograph of normal blood cells; at right, com- metric mass, and this raises the question of whether there exists
puted images showing normal (top image) and gastrula-like an analogous center related to the Hawking mass. Together withvesicle (bottom image). (Computed images from J. Chen’s T. Lamm and J. Metzger, the third author has proposed maximizing
thesis.)mH (Σ) — or equivalently minimizing W(Σ) — among surfaces Σwith given area A(Σ). They show there exists a family of constrained W-critical surfaces — in fact, local minimizers — that foThe Hawking Mass and the Riemannian
liates the asymptotic region of N such that the leaves are asympPenrose Inequality
totic to centered spheres; it remains an open question if such foliIn general relativity it is natural to model the asymptotic spa- ations define a unique asymptotic center, and whether their leaves
tial geometry of an isolated gravitational system by the end N are global minimizers for W.

Call for Proposals
All proposals can be submitted to the Director or Deputy Director or any member of the Scientific Advisory Committee with a copy toproposals@msri.org. For detailed information, please see the website www.msri.org.

Thematic Programs
The Scientific Advisory Committee (SAC) of the Institute meets in January, May, and November each year to consider letters of intent,
pre-proposals, and proposals for programs. The deadlines to submit proposals of any kind for review by the SAC are March 15,
October 15, and December 15. Successful proposals are usually developed from the pre-proposal in a collaborative process between
the proposers, the Directorate, and the SAC, and may be considered at more than one meeting of the SAC before selection. For complete
details, see http://tinyurl.com/msri-progprop.

Hot Topics Workshops
Each year MSRI runs a week-long workshop on some area of intense mathematical activity chosen the previous fall. Proposals should
be received by March 15, October 15, or December 15 for review at the upcoming SAC meeting. See http://tinyurl.com/msri-htw.

Summer Graduate Schools
Every summer MSRI organizes several 2-week long summer graduate workshops, most of which are held at MSRI. Proposals must be
submitted by March 15, October 15 or December 15 for review at the upcoming SAC meeting. See http://tinyurl.com/msri-sgs.
8

Named Postdocs in Differential Geometry — Spring 2016
Strauch Postdoc
Heather Macbeth is this spring’s Strauch Postdoctoral Fellow. Heather received her undergraduate degree from the University of Auckland, New Zealand, in 2009, and a Part III fromthe University of Cambridge in 2010. She was a graduate student at Princeton University under the supervision of Gang Tian, receiving her Ph.D. in June 2015. Since July 2015 she has
been a Moore Instructor at MIT. Heather’s research is in Kähler geometry and geometric analysis. She has studied the existence problem for Kähler–Einstein metrics on Fano manifolds,
extending the existence criteria in terms of alpha-invariants due to Tian. Her other projects
include work on the Yamabe invariant, conformal geometry, and projective geometry.
The Strauch Fellowship is funded by a generous annual gift from Roger Strauch, Chairman of
The Roda Group. He is a member of the Engineering Dean’s College Advisory Boards of UC
Berkeley and Cornell University, and is also currently the chair of MSRI’s Board of Trustees,on which he has served for more than 15 years.

Heather Macbeth

Viterbi Postdoc
Chengjian Yao is the Viterbi Postdoctoral Fellow in Differential Geometry. Chengjian received his
undergraduate degree from the University of Science and Technology of China in Hefei. In May
2015, he obtained his Ph.D. from Stony Brook University under the supervision of Xiuxiong Chen.
He is currently on leave from the Chercheur postdoctoral position at Université Libre de Bruxelles
in Belgium. Chengjian’s research has focused on the study of the existence and regularities of conical Kähler–Einstein metrics on compact Kähler manifolds. He was able to smoothly approximate
conical Kähler–Einstein metrics with singularities along a big class of simple normal crossing
divisors, using the classical Aubin–Yau continuity path. Together with his collaborators Spotti
and Sun, they affirmed the Yau–Tian–Donaldson conjecture in the case Q-Gorenstein smoothable
Fano-varieties. More recently, he has worked on a geometric flow, called definite triple flow, to
try to attack one of Donaldson’s questions which aims to give a symplectic characterization of a
hyper-Kähler 4-manifold.
Chengjian Yao

The Viterbi Endowed Postdoctoral Scholarship is funded by a generous endowment from Dr. Andrew Viterbi, well known as the co-inventor of Code Division Multiple Access based digital cellular technology and the Viterbi
decoding algorithm, used in many digital communication systems.

Huneke Postdoc
Andrea Mondino is the Huneke Endowed Postdoctoral Fellow in Differential Geometry. Andrea
received his bachelor’s degree from Torino University in 2006 and his master’s degree fromSISSA & Trieste University in 2008. In September 2011, he obtained his Ph.D. from SISSA(Trieste) under the supervision of Andrea Malchiodi. Starting in September 2016, he will be
an assistant professor at the University of Warwick. Andrea is broadly interested in differential
geometry/geometric analysis and its application to general relativity and the optimal shapes ofcell membranes. In his work on variational geometric problems, he has investigated the optimal
shapes that arise as minimizers — or more generally as stationary points — of energy functionals.
Andrea is also interested in non-smooth spaces that satisfy lower Ricci curvature bounds in the
optimal transportation sense. With collaborators, he proved results about the structure of such
spaces and about their geometric-analytic properties.
The Huneke postdoctoral fellowship is funded by a generous endowment from Professor Craig
Huneke, who is internationally recognized for his work in commutative algebra and algebraic
geometry.

9

Andrea Mondino

Focus on the Scientist: Tobias Colding
Tobias Colding, a Clay Senior Scholar and Eisenbud Professor
in this semester’s Differential Geometry program, is no stranger
to the pleasures of MSRI — healso held a Clay Senior Scholar
position here in 2011 as part of
the Quantitative Geometry program. He is also one of the organizers of the current Differential
Geometry program.
Tobias earned his Ph.D. fromthe University of Pennsylvania
in 1992 and completed his final year of graduate studies while
visiting MSRI with his advisor
Chris Croke. During his time as
Tobias Colding
a Courant Instructor at NYU, hereturned to MSRI as a postdoctoral fellow in 1993 as part ofthat year’s Differential Geometry program. He rose through theranks at NYU and was named a full Professor in 1999, a position he maintained until 2008. After holding visiting positions at MIT and Princeton, he moved permanently to MIT in
2005, where he is now the Cecil and Ida Green Distinguished
Professor.

imal surfaces.” In a series of four papers published in 2004,
the pair determined a compactness theory for sequences of minimal disks without presuming curvature or area bounds. This
result was immediately applied to resolve many open problems
including the longstanding conjecture on the uniqueness of thehelicoid.
Their work began with the seemingly basic — but highly technical — observation that every embedded minimal disk can belocally modeled by a minimal graph or a piece of the helicoid.
With the help of this local picture, they determined global structural results for embedded minimal surfaces with finite genus.
In particular, they resolved the Calabi–Yau conjecture for minimal surfaces with finite topology, proving that every complete,embedded example must be proper.

Again in collaboration with Minicozzi, Tobias recently developed infinite-dimensional Łojasiewicz inequalities for noncompact hypersurfaces with bounded entropy. The noncompactness
required new techniques because, unlike in Leon Simon’s celebrated work, they cannot prove these inequalities by reducing
from an infinite-dimensional space to the classical setting. As an
immediate application, they established the uniqueness of tangent flows at a generic singularity of the mean curvature flow, aTobias is undaunted by mathematical challenges since his enthu- result which itself has many further applications.
siasm and optimism propel him. While to many mathematicians,
Tobias is best known for his work on the structure of spaces with
Tobias is a member of the American Academy of Arts and Scilower Ricci curvature bounds, we focus on two of his most reences and the Royal Danish Academy of Sciences and Letters.
cent seminal contributions in the calculus of variations.
He was also an invited speaker to the International Congress of
In 2010, Tobias was awarded the Oswald Veblen prize, jointly Mathematicians in Berlin in 1998.
with William P. Minicozzi, II, for their “profound work on min— Christine Breiner

Named Positions for Spring 2016
MSRI is grateful for the generous support that comes from endowments, annual gifts, and awards that support faculty and postdoc
members of its programs each semester.
Eisenbud and Simons Professors

Natasa Sesum, Rutgers University
Michael Singer, University College London
Gang Tian, Princeton University
Jeff Viaclovsky, University of Wisconsin
Guofang Wei, University of California, Santa Barbara
Xiaohua Zhu, Peking University

Olivier Biquard, École Normale Supérieure
Simon Brendle, Stanford University
Sun-Yung Chang, Princeton University
Tobias Colding, Massachusetts Institute of Technology
Simon Donaldson, Imperial College, London
Akito Futaki, The University of Tokyo
Ursula Hamenstaedt, Universitäat Bonn
Tom Ilmanen, ETH Zurich
Vitali Kapovitch, University of Toronto
Zhiqin Lu, University of California, Irvine
Curtis McMullen, Harvard University
Richard Schoen, Stanford University
Felix Schulze, University College London

Named Postdoctoral Fellows
Huneke: Andrea Mondino, ETH Zurich
Strauch: Heather Macbeth, Massachusetts Institute of Technology
Viterbi: Chengjian Yao, Stony Brook University
Clay Senior Scholar
Tobias Colding, Massachusetts Institute of Technology

10Staff News: Recognizing Five New MSRI Faces . . .

Patricia Brody

Aaron Hale

Tracy Huang

Patricia Brody joined MSRI in December as the new Housing Advisor. Prior to coming to MSRI, Patricia ran a nonprofit foundationin San Francisco. She graduated from UCLA with a B.A. in political science and earned her J.D. there as well, and has been active inthe legal world. Her favorite pastimes include time with her family
and travel — and housing!

Jennifer Murawski

Britta Shillingsburg

looking forward to more travel in her future.

Jennifer Murawski joined MSRI in December as the Communications and Event Coordinator. She was previously an assistant director at the University of Pittsburgh’s Asian Studies Center, coordinating undergraduate academic programs and K-12 educational
outreach. She is an alumna of the Japan Exchange and TeachAaron Hale joined MSRI in January as IT Strategist and Technicaling Program and volunteers with US–Japan sister cities to promote
Lead. Aaron has worked as an IT contractor and consultant for four
youth exchange programs.
years. Prior to that, Aaron worked as a lab coordinator at Pennsylvania College of Technology. In his free time, Aaron likes to
Britta Shillingsburg joined MSRI in December as Event Planner
spend time with his wife and daughter, camping and hiking aroundand Marketing Specialist, based out of Washington, DC. She prethe Bay Area. His interests include particle physics, foundational
viously worked for the Direct Selling Association as an exhibit andproblems in quantum mechanics, and theoretical biology.
marketing coordinator. Britta volunteers with the DC Lean In ExTracy Huang joined MSRI in January as the Assistant for Scien- ecutive Board to plan professional development events for women,tific Activities. Tracy recently graduated from the University of and she teaches high school students about small business ownerSouthern California with a B.A. in East Asian languages and cul- ship for Pennsylvania Free Enterprise Week. Her hobbies includetures. She has studied in Beijing and Nanjing, China, and she is yoga, hiking, and running.

. . . and Honoring 32 Years of Service
In January, MSRI directors and staff gathered to say farewell to Jackie Blue on theoccasion of her retirement. Jackie was MSRI’s International Scholars and Housing
Advisor for the last 32 years, helping our visiting scholars and their families findaccommodation and obtain the proper visas, and providing judicious and friendly
advice. In honor of her many years of service, Jackie’s name will be added to thecommemorative wall in MSRI’s outdoor Berlekamp Garden.
Jackie Blue in the Berlekamp Garden

Clay Senior Scholarships

Geometric Group Theory (Fall 2016)
Karen Vogtmann, University of Warwick

The Clay Mathematics Institute (www.claymath.org) has announced the 2016–2017 recipients of its Senior Scholar awards.
The awards provide support for established mathematicians toplay a leading role in a topical program at an institute or university away from their home institution. Here are the Clay Senior
Scholars who will work at MSRI in 2016–2017:11

Harmonic Analysis (Spring 2017)
Alexander Volberg, Michigan State University
Analytic Number Theory (Spring 2017)
Manjul Bhargava, Princeton University
Rodney Heath-Brown, Oxford University

CME Group–MSRI Prize in Innovative Quantitative Applications
The tenth annual CME Group–MSRI Prize in Innovative Quantitative Applications was awarded to
Douglas W. Diamond, Merton H. Miller Distinguished Service Professor of Finance at the Universityof Chicago Booth School of Business, at a luncheon in Chicago in February.
Diamond’s research has focused on the study of financial intermediaries, financial crises, and liquidity. A panel discussion on “Non-Bank Runs and Financial Crises” was held in conjunction with theaward ceremony.
The annual CME Group–MSRI Prize is awarded to an individual or a group to recognize originality
and innovation in the use of mathematical, statistical, or computational methods for the study of the
behavior of markets, and more broadly of economics. You can read more about the CME Group–
MSRI Prize at http://tinyurl.com/cme-msri.
Douglas W. Diamond

Forthcoming Workshops

July 25 – August 5, 2016: Chip Firing and Tropical Curves, organized by Matthew Baker (Lead), David Jensen, Sam Payne

May 2 – 6, 2016: Geometric Flows in Riemannian and Complex
For more information about any of these workshops, as well as
Geometry, organized by Tobias Colding, John Lott (Lead), Natasa
a full list of all upcoming workshops and programs, please see
Sesum
www.msri.org/scientific.
June 11 – July 24, 2016: MSRI-UP 2016: Sandpile Groups, organized by Federico Ardila, Duane Cooper, Maria Mercedes Franco,
Herbert Medina, Suzanne Weekes (Lead)

Call for Membership

August 17 – 19, 2016: Connections for Women: Geometric Group MSRI invites membership applications for the 2017–2018 acaTheory, organized by Ruth Charney (Lead), Indira Chatterji, Mark demic year in these positions:Feighn, Talia Fernós
Research Professors
by October 1, 2016
August 22 – 26, 2016: Introductory Workshop: Geometric Group
Research Members
by December 1, 2016
Theory, organized by Martin Bridson, Benson Farb, Zlil Sela
(Lead), Karen Vogtmann
Postdoctoral Fellows by December 1, 2016
September 27 – 30, 2016: Groups Acting on CAT (0) Spaces, organized by Ian Agol, Pierre-Emmanuel Caprace, Koji Fujiwara, In the academic year 2017–2018, the research programs are:
Alessandra Iozzi, Michah Sageev (Lead)
Geometric Functional Analysis and Applications
October 25 – 28, 2016: Geometry of Mapping Class Groups and Aug 14 – Dec 15, 2017
Out(Fn ), organized by Yael Algom-Kfir, Mladen Bestvina (Lead), Organized by Franck Barthe, Marianna Csornyei, Boaz Klartag,
Alexander Koldobsky, Rafal Latala, Mark Rudelson
Richard Canary, Gilbert Levitt
December 6 – 9, 2016: Amenability, Coarse Embeddability and Geometric and Topological Combinatorics
Fixed Point Properties, organized by Goulnara Arzhantseva, Cor- Aug 14 – Dec 15, 2017nelia Drutu (Lead), Graham Niblo, Piotr Nowak
Organized by Jesus De Loera, Vic Reiner, Francisco Santos, Francis Su, Rekha Thomas, Günter M. Ziegler

Summer Graduate Schools

Group Representation Theory and Applications
May 30 – June 11, 2016: Séminaire de Mathématiques Supérieures
2016: Dynamics of Biological Systems (Canada), organized by Jan 16 – May 25, 2018
Organized by Robert Guralnick, Alexander (Sasha) Kleshchev,
Thomas Hillen, Mark Lewis, Yingfei Yi
Gunter Malle, Gabriel Navarro, Julia Pevtsova, Raphael Rouquier,June 13 – June 24, 2016: Harmonic Analysis and Elliptic Equa- Pham Tiep
tions on Real Euclidean Spaces and on Rough Sets, organized by
Enumerative Geometry Beyond Numbers
Steven Hofmann (Lead), Jose Maria Martell
Jan 16 – May 25, 2018
June 20 – July 1, 2016: Mixed Integer Nonlinear Programming:
Organized by Mina Aganagic, Denis Auroux, Jim Bryan, Andrei
Theory, Algorithms and Applications (Spain), organized by FranOkounkov, Balazs Szendroi
cisco Castro-Jiménez, Elena Fernández, Justo Puerto
MSRI uses MathJobs to process applications for its positions. InJuly 11 – 22, 2016: An Introduction to Character Theory and theterested candidates must apply online at www.mathjobs.org after
McKay Conjecture, organized by Robert Guralnick, Pham Tiep
August 1, 2016. For more information about any of the programs,
July 18 – 29, 2016: Electronic Structure Theory, organized by Lin please see www.msri.org/scientific/programs.
Lin (Lead), Jianfeng Lu, James Sethian12

Puzzles Column
Elwyn Berlekamp and Joe P. Buhler
1. As a respite from the current election discussions, it seems useful to recall and extend a problem (from many years ago) with thehope of elevating conversations about elections to a more rarefied
level.

5. Alice, Bob, and Catherine have positive integers on their foreheads. One of the numbers is the sum of the other two. (Each cansee the other two but not their own, no side communication is allowed, each knows that one number is the sum of the other two,they are all brilliant logicians, etc.).

You are a citizen of Oddonia, which has two political parties. The
parties are so closely matched that from your point of view anyone The following dialog ensues:else’s vote might as well be viewed as determined by the flip of afair coin.
Alice:
I don’t know my number.
Bob:I
don’t know my number.
Oddonia has three equally populous states, each with 2n+1 voters.
Catherine:I
don’t know my number.
Two systems for electing the President are being considered: (1) a
Alice
:
I
know my number, and it is 50.
popular vote, or (2) an electoral college with one electoral vote foreach state.

What are the other two numbers?
(a) Which system gives the most power to you? You should regard
one system as more powerful than another if the probability Comment: This problem apparently first appeared in a 2004 isthat your vote is decisive (that is, determines the election) is sue of Math Horizons; we ran into it in the new Springer book
One Hundred Prisoners and a Light Bulb by Ditmarsch andlarger.
Kool, which contains a number of such brainteasers as well as an(b) More generally, what can you say about comparing electoral overview of epistemic logic.
college systems assuming d or d 0 states, where d and d 0 are
6. Show that every sufficiently large integer is a sum of cubes ofdivisors of the total population size?
distinct positive integers.
2. Four people come to a river in the night. There is a rickety
narrow bridge, and it can only hold at most two of the people at atime. Moreover, it is very dark and there is no railing, so a flashlight must be used whenever the bridge is crossed. There is only
one flashlight.

Mathematical Art at MSRI

Person A can cross the bridge in one minute, B in two minutes, C infive minutes, and D in eight minutes. When two people cross thebridge together, they must of course move at the slower person’space. How quickly can they all get across the bridge?
Comment: This came up in conversation recently, and we learned
that the problem is “well-known,” at least in the sense that it has its
own wiki page.
3. For n > 1, consider an n × n chessboard and place identicalpieces at the centers of different squares.
(a) Show that no matter how 2n identical pieces are placed on theboard, one can always find four pieces among them that are thevertices of a parallelogram.
(b) Show that there is a way to place (2n − 1) identical chesspieces so that no four of them are the vertices of a parallelogram.
4. Find a positive integer N and a1 , a2 , . . . , aN , where each ai isequal to 1 or −1, such that
20162016 = a1 · 13 + a2 · 23 + a3 · 33 + · · · + aN · N3 ,or show that this is impossible.
Comment: The preceding two problems (3 and 4) were both on the2016 Bay Area Mathematical Olympiad, which had a number of
excellent problems. Problem 6 below was somehow inspired byproblem 4; although both succumb to a specific technique, there
seem to be a larger number of ways to approach problem 6.
13Henry Segerman (far left) brought an exhibit of 3D-printed“illustrated mathematics” to display at MSRI in February
2016. The display featured twelve pieces of his solo andcollaborative artwork, which can also be viewed online atwww.segerman.org.
Dr. Segerman is an assistant professor in the Departmentof Mathematics at Oklahoma State University, working
mostly in three-dimensional geometry and topology.

2015 Annual Report
We 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 forthe growth 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, 2015. In preparation of this report, 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. If
your gift was received after December 31, 2015, your name will appear in the 2016 Annual Report. For more informationon our giving program, please visit www.msri.org.

Government Sponsors Hilbert Society
National Science Foundation
National Security Agency

Current Foundation &Corporate Sponsors
Anonymous (2)AllianceBernstein Foundation
Fund*
Arkay Foundation
Clay Mathematics Institute
CME Group
Firedoll Foundation
Goldman Sachs Gives
Heising Simons Foundation
Howard Hughes Medical
Institute
IBM Matching Grants Program*
JAZEM Fund
The Kavli Foundation
Langsam Family
Foundation, Inc.
Maplesoft
Math for America
Meyer Sound Laboratories
Microsoft Research
Microsoft Matching Gifts
Program*
Mosse Foundation for
Education & the Arts
Pacific Journal of Mathematics
Research Corporation for
Science Advancement
D. E. Shaw & Co.
Simons Foundation
Torchlight Investors, LLC
Vision Maker Media

Helen & John Meyer
Roger A. Strauch
Faye & Sandor Straus
Gita & Ashok Vaish
Andrew Viterbi

Brendan E. Hassett & Eileen K.
ChengRichard V. Kadison
William E. Lang
Andrew Odlyzko

Hypatia

Noether
Anonymous
Deborah L. Ball
Ruth Charney
Thomas R. Davis
Dr. Charles Fefferman
Paul Fong
Eric & Emma Gimon
Solomon W. Golomb
John M. Hosack
Donald & Jill Knuth
Langsam Family Foundation
Albert & Dorothy Marden
Dusa McDuff & John Milnor
Michael Morgan
Hyungsook Kim & Bernd Sturmfels
Leon H. TatevossianRichard & Christine Taylor
Chuu-Lian Terng & Richard Palais
Roger & Sylvia Wiegand
Hung-Hsi Wu

We recognize our mostgenerous and loyal supporters
whose leadership andcommitment ensures MSRI
continues to thrive as one of
the world’s leading
mathematics research centers.

Anonymous
Edward D. Baker
Jennifer Chayes & Christian
Borgs
Sanford J. Grossman
Dan Heflin
Henry & Marsha Laufer
Sorin & Marie-Anne Popescu

Anonymous
Edward D. Baker
Elwyn & Jennifer Berlekamp
Susan & Paul Chern
May & Paul Chu
William & Margaret Hearst
Margaret Holen & David
CoulsonMarilyn & Jim Simons
Roger Strauch
Andrew Viterbi

PlatoMuseion Society

Anonymous
Michael Bershadsky
David desJardins & Nancy
Blachman
Irina Bogacheva
Robert L. Bryant &Réymundo A. Garcia
Cedomir Crnkovic
Diviya & Alexander Magaro
Howard & Joan Resnikoff
Ronald L. Rivest
Colin Rust & Jeannie Tseng
Governor Bob Wise
Peter N. Yianilos

The Museion Society, named
after Musaeum, the Hall of theMuses in ancient Alexandria,
recognizes our leadership
donors in annual andendowment giving.
EuclidElwyn & Jennifer Berlekamp
Paul & Susan Chern**
Miriam & David Donoho
Theodore W. Gamelin**
Vinita & Naren Gupta**
Joan & Irwin Jacobs
Margaret Holen & David
CoulsonRonald Kahn & Julia Rowe**
Mark P. Kleiman
Anna Patterson & Tom Costello
Anand Periasamy & Garima
KapoorArchimedes Society
The Archimedes Society
recognizes MSRI’s annual
supporters.

14

Ramanujan
Jishnu Bhattacharjee
S.-Y. Alice Chang & Paul C.
Yang

FermatAnonymous (4)Sheldon Axler
Hélène Barcelo & Steve
Kaliszewski
Ken Baron
Danalee & Joe P. Buhler
Tom & Louise Burns
Richard C. Churchill
Bruce Cohen
Anthony D’Aristotile
Peter DurenJohn A. Eagon
Gisela Fränken

Dan Freed
John Friedman
Daniel & Ryoko Goldston
Phillip Griffith
Alfred W. & Virginia D. Hales
Joel Hass
Alessandra Iozzi
Garo Kiremidjian
Lynda & Robert Korsan
Jim Lewis
Jon McCammondFabio Mercurio
Andrew Ogg
Walter Ohlemutz
Richard S. Palais
John F. & Marilyn M. Richards
Timothy Riley & Tara H. Riley
Donald E. Sarason
Freydoon Shahidi
Michael F. Singer
James T. Smith
Wilbur L. Smith
Hugo & Elizabeth
Sonnenschein
John Stroik
Eva Tardos
Michael Urias
Wolmer V. VasconcelosCharles Weibel
Carol S. Wood
Peter & Gloria Yu
Fibonacci
Anonymous (13)Michael T. Anderson
Jayadev AthreyaPeter & Nancy Bickel
Kirsten Bohl
Anna Marie Bohmann
Norman BooksteinDavid Bressoud
Mikhail Brodsky
David Brown
Lawrence G. Brown
Richard A. Champion, Jr.
Jean Chan (In memory of
Dr. Peter Stanek)
Kenneth Collins
Walter Craig & Deidre Haskell
Chris Cunningham
Steven & Hema Cutkosky
John Dalbec

Marc A. Rieffel
Rachel D. Roberts
Dr. Jonathan Rosenberg
William H. Rowan &Charlene M. Quan**Markus SchmidmeierFrank-Olaf Schreyer
Leonard Scott
Timo Seppalainen
Mohamed W. Sesay
Bonnie Sherwood
William Sit
Bruce M. Solomon
Louis & Elsbeth Solomon
Jim Sotiros
Ralf SpatzierThomas Spencer
Lee Stanley
Harold Stark
Troy L. Story
Earl J. Taft
Pham H. Tiep
David Van Vactor
Bill Vederman
Dan-Virgil Voiculescu
John M. Voight
Jonathan Wahl
Guofang Wei
Erling Wold
Michael Wolf
Alexander Wright

Hailong Dao
Martin Davis
Percy A. Deift
James Donaldson
Bruce Driver
Douglas Dunham
Patrick B. EberleinPeter Eggenberger
Richard Ehrenberger
Hans P. Engler
E. Graham Evans
Michael Falk
Jeremiah Farrell
Richard Finn
Frank M. Fisher
Anita B. Friedman
Harolyn L. Gardner
Michael A. Geline
William M. & Emily A.
Goldman
Cameron M. Gordon
Frances & Denny Gulick
Hayssam Hajar
Eloise & Kenn HamannDavid Harbater
Alexander Hart
Nancy B. Hingston
Elaine Holliman
Stanley S. Isaacs
Sheldon Katz
Rinat Kedem**
Kiran S. Kedlaya
Ellen E. Kirkman
Ralph M. Krause
Prof. Michel Lapidus
Richard K. & Joyce C. Lashof
Gye-Seon Lee
Clement Lutterodt
Yvan Martel
Anna L. Mazzucato
Robert McOwen
Janet Mertz & Jonathan Kane
Rev. Dr. Polly Moore
Carlos Julio Moreno
Walter Neumann
Yong-Geun Oh
Richard & Susan Olshen
Yuval & Deborah Peres
The Pisani Family
Malempati M. Rao
Winston Richards
Eleanor Rieffel

Gauss Society
The Gauss Society recognizes
friends who include MSRI in
their will or have made otherstipulations for a gift from theirestate, ensuring that MSRI
remains strong in the future.
Edward D. Baker
Robert L. Bryant &Réymundo A. Garcia
Gary Cornell
David & Monika Eisenbud
Gisela Fränken
Robert W. Hackney
Craig Huneke & Edith Clowes
William E. Lang
Douglas Lind
Dusa McDuff & John Milnor
Joseph Neisendorfer***

Marilyn & Jim Simons
Hugo & Elizabeth
Sonnenschein
Jim Sotiros
Julius & Joan Zelmanowitz

Book Donations to
the MSRI Library
Private Donors
Sheldon Axler
Hélène Barcelo
Elwyn Berlekamp
Mark Harbison
Lee Kaskutas (estate ofWilliam Arveson)
Alexander Kheyfits
Dmitry Kleinbock
Stephen Krantz
Tsit-Yuen Lam
Dave Witte Morris
Kenneth Ribet
Margaret & Christine Wertheim
Cédric Villani
Publishers
American MathematicalSociety
Belgian Mathematical Society
Bloomsbury Press
Cambridge University Press
CRC Press
Elsevier
European Mathematical Society
Farrar, Straus and Giroux
Fields Institute
Institute for Advanced Study
International Press
Japan Academy
Kyushu University
Mathematical Society of Japan
Oxford University Press
Princeton University Press
Research Institute forMathematical Sciences,
Kyoto University
Springer
Tohoku University
Mathematical Institute
WGBH
World Scientific* Matching funds for gifts by employees or directors; ** Includes gifts to the MSRI Endowment Fund. Gifts to the endowment are
carefully and prudently invested to generate a steady stream of income; *** New member of the Gauss Society in 2015.

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

15

MSRI
Mathematical Sciences Research Institute
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510.642.0143 • FAX 510.642.8609 • www.msri.org
ADDRESS SERVICE REQUESTEDMSRI Staff & Consultant Roster
Add @msri.org to email addresses. All area codes are (510) unless otherwise noted.
Scientific and Education
David Eisenbud, Director, 642-8226, director
Hélène Barcelo, Deputy Director, 643-6040, deputy.director
Diana White, National Association of Math Circles Director, 303-315-1720, namc.director
Administrative
Kirsten Bohl, Outreach Producer, 499-5181, kbohl
Arthur Bossé, Operations Manager, 643-8321, abosse
Patricia Brody, Housing Advisor, 643-6468, pbrody
Heike Friedman, Director of Development, 643-6056, hfriedman
Aaron Hale, IT Strategist and Technical Lead, 643-6069, ahale
Mark Howard, Facilities and Administrative Coordinator, 642-0144, mhoward
Tracy Huang, Assistant for Scientific Activities, 643-6467, thuang
Claude Ibrahimoff, International Scholar Advisor & Executive Assistant, 643-6019, cibrahimoff
Lisa Jacobs, Executive Coordinator and Board Liaison, 642-8226, lisaj
Christine Marshall, Program Manager, 642-0555, chris
Jinky Rizalyn Mayodong, Staff Accountant, 642-9798, rizalyn
Jennifer Murawski, Communications and Event Coordinator, 642-0771, jmurawski
Megan Nguyen, Program Analyst, 643-6855, megan
Linda Riewe, Librarian, 643-1716, linda
Jacari Scott, Scholar Services Coordinator, 642-0143, jscott
Britta Shillingsburg, Event Planner and Marketing Specialist, 202-420-8954, bshillingsburg
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