a celebration of women in mathematics · a celebration of women in mathematics 32 n otices of the...

A Celebration ofWomen inMathematics

32 NOTICES OF THE AMS VOLUME 42, NUMBER 1

In March 1994 I organized a two-day conference atthe Massachusetts Institute of Technology in whichnine distinguished women mathematicians gavecolloquium lectures about their current mathe-matical research. There are many exciting mathe-

matical meetings where the speakers are all men: thiswas a meeting where exciting mathematics was pre-sented that was unusual only in that all the speakerswere women. From the comments that I heard after themeeting, the conference was clearly very successful. Theaudience contained a cross-section of the mathemati-cal community ranging from MIT Institute Professorsto undergraduates. It was pleasing to hear enthusiasmon all sides for the mathematics that was discussed. Ihave been told that the meeting was particularly in-spiring for many young women in the audience who arein the early stages of their mathematical careers.

The conference was funded by MIT and the VisitingProfessorsip for Women Program of the National Sci-ence Foundation. The Mathematics Department at MITwas a most supportive host for the conference and Ithank the many people in the department who con-tributed to the success of the conference, particularyDavid Benney and Haynes Miller and the excellent ad-ministrative staff of the department. To our pleasure,the conference was opened by Mark Wrighton, the MITProvost. Mildred Dresselhaus, a distinguished InstituteProfessor, introduced the second day with entertainingand illuminating remarks about her career at MIT. Shemade the very pertinent observation that for a woman

to be really helpful to the cause of women in sci-ence, it is important that she work from a baseof solid scientific achievement. The talks at themeeting were excellent illustrations of suchachievements.

This article contains short summaries of themathematics presented by the speakers:

Joan S. Birman, Studying Surfaces in Knot Com-plements

Ingrid Daubechies, Wavelets and Applications

Dusa McDuff, The Geometry of Symplectic Energy

Jill P. Mesirov, Mathematical Theory in ParallelAlgorithms

Cathleen S. Morawetz, The Wave Equation Re-visited

Jean E. Taylor, Surface Motion Due to Surface En-ergy Reduction

Chuu-Lian Terng, Soliton Equations and Differ-ential Geometry

Karen K. Uhlenbeck, Moduli Spaces and Adia-batic Limits

—Susan FriedlanderUniversity of Illinois-Chicago

(e-mail address: [email protected])

wim.qxp 3/16/99 3:06 PM Page 32

Studying Surfacesin KnotComplementsJoan S. Birman

My talk at the Celebra-tion of Women in Mathe-matics was about appli-cations of the theory ofbraids to the study ofknots in 3-space.

A knot K is a topolog-ical circle which issmoothly embedded in 3-space R3. Its knot type isits equivalence classunder isotopies of thepair (R3, K). Our interest

is in the knot problem, i.e.,the problem of howto decide whether two knots in 3-space determinethe same knot type. My current research (jointwith William Menasco) is directed toward solv-ing that problem algorithmically. The ingredientsneeded for our proposed algorithm are (i) a con-cept of a ‘simplest’ representative K0 of a knot,chosen so that each knot K has at least one andat most finitely many distinct simplest repre-sentatives; (ii) a complexity function c(K) whichmeasures how far K is from the same K0; (iii) amethod for reducing c(K) ; and (iv) an under-standing of how the finitely many simplest rep-resentatives are related to one another. Braidstructures provide the tools for attacking theseproblems.

Choose polar coordinates (z, r , θ) in R3. LetA be the z -axis. Then R3 −A is fibered by half-planes Hθ = {(z, r , θ) ∈ R3/θ = constant}. Ourknot K is represented as a closed braid with re-spect to the z -axis A if Kmeets every Hθ trans-versally, i.e. K is nowhere tangent to an Hθ . Forexample, if K is the ‘unknot’, an especially sim-ple representative would be the unit circleU0 = {(0,1, θ) ∈ R3; 0 ≤ θ < 2π} in the (r , θ)plane. This is our simplest representative, andit is unique up to isotopy in the complement ofthe braid axis.

Every smooth representative of every knottype bounds an orientable surface which is em-

JANUARY 1995 NOTICES OF THE AMS 33

bedded in 3-space, and so also an orientable sur-face S of minimum genus. For example, everyrepresentative U of the unknot bounds a disc D,our standard representative bounding the unitdisc D0 in the plane z = 0. This disc is foliatedradially by the arcs {D0 ∩Hθ,0 ≤ θ < 2π} .These arcs also define a flow on D0 as onepushes the Hθ ’s forward through the fibrationof R3 −A. See Figure 1a. What happens to thisfoliation and the associated flow when we replaceU0 with a nonstandard closed-braid represen-tative U of the unknot, say one which passes ntimes around the axis A before closing up to acircle? Clearly U will still bound a disc, say D,which will in general admit a singular foliation,the leaves of the foliation being the componentsof {D∩Hθ,0 ≤ θ < 2π}. It can be shown thatthe disc D can be assumed to have a particularlynice foliation: the leaves are all arcs, the singu-larities (if any) are all saddle points, and near eachpoint of A∩D the foliation is radial. Figure 1bshows the foliation in a neighborhood of twopoints in A∩D . It is immediately clear that ifthis foliation is to be extended to all of D, thensingularities must occur.

The foliations which we study can be shownto decompose the surface S into a union of tiles,each of which is a 2-cell which contains exactlyone singularity of the foliation. In particular, Sis a union of three types of foliated tiles, depictedin Figures 2a, 2b, and 2c, each of which has acanonical embedding in 3-space relative to ourcylindrical coordinates. Two of the tile types in-tersect K nontrivially, but the third is entirely inthe interior of S. These tiles are all depicted asthey appear when viewed on the positive side of

Supported in part by NSF grant DMS 91-06584.Joan Birman is professor of mathematics at ColumbiaUniversity, New York, New York. Her e-mail address [email protected].

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W.eW&g?/X?7He*@g?N1?@?eN@eW2@?e@?@?e?@e7R'?e@?@?e?@e@?f@?@?eJ@L?3T.?e@?@?e@@@?V+Y?e@?3Lhe?J5?V/he?.Y?

Figure 1

wim.qxp 3/16/99 3:06 PM Page 33


S. The black dots show the points where Apierces S. There is a well-defined sense of theflow about each pierce-point, clockwise or coun-terclockwise. The orientation of K is consistentwith the sense of the flow around the vertices.Notice that counterclockwise flows are preferred,reflecting the fact that the linking number of Kwith A is n > 0.

Returning to our example and pulling D back(with its foliation) to its preimage D under theembedding we are able to replace D, which isdifficult to visualize because of its complicatedembedding in 3-space, by the foliated modeldisc D. A simple example is given in Figure 1c.The disc is pierced 3 times by the braid axis andis foliated by two tiles of the type illustrated inFigure 2a. We leave it as an exercise for thereader to reconstruct the embedded disc and theclosed braid representative of its boundary fromthe foliated model disc of Figure 1c.

The kind of data which we can see in the fo-liation includes: (i) properties of the graph andsubgraphs formed by the singular leaves; (ii) thenumber and types of tiles which meet at each ver-tex; (iii) the sense of the flow near the vertices(clockwise or counterclockwise); (iv) the sign ofthe outward-drawn normal to S near a singularityof the foliation. It turns out that this data (andother combinatorial data) tells us how to mod-ify K so as to reduce c(K) . Similar methods alsoapply to the study of other surfaces which havea special position relative to that of K, e.g., closedincompressible tori in the complement of K.

This work is still in progress. At this writingwe have learned a great deal about the closedbraid representatives of knots and have an al-gorithm in a very special case: knots of braidindex at most 3, which are represented by aclosed 3-braid. We have partial results for closedbraid representatives of the unlink and other spe-cial cases.

Wavelets andApplicationsIngrid Daubechies

The relatively new de-velopment of wavelettheory in the last tenyears or so represents asynthesis of ideas frommathematics, physics,engineering, and com-puter science, some withroots going back to thestart of the century. Thegoal of the talk was to introduce wavelets, to ex-plain both their mathe-matical and algorithmi-

cal aspects, and to illustrate these by severalapplications. In 1 dimension, wavelets are typi-cally defined as the dilates and translates of onefunction,

a,b(x) = |a|−1/2(x− b

a

),

where is often required to have some smooth-ness and some decay, say | (x)|≤C(1+ |x|)−1−ε,|ˆ (ξ)| ≤ C(1 + |ξ|)−1−ε. Moreover, should sat-isfy

∫(x)dx = 0. It then turns out that any

square integrable function f (and, in fact, manyother functions as well) can be “decomposed”into such wavelets,

f (x) = C∫∞−∞

∫∞−∞〈f , a,b〉 a,b(x)a−2dadb,

a fact long known to harmonic analysts asCalderón’s formula and to quantum physicistsas the resolution of the identity for the ax + b-group. The interest of such a decomposition isthat it views f as a superposition of “buildingblocks” a,b , each of which is well localized in“time and frequency”; a typical example is

(x) = (1− x2)e−x2/2, where | a,b(x)| is mostlyconcentrated in the region |x− b| ≤ 2|a| , and|ˆ a,b(ξ)| mostly in |a|/4 ≤ |ξ| ≤ 4|a|. By cut-ting the domain of integration in a, b into dif-ferent parts, one effectively carves f into dif-ferent pieces, which may have very differentproperties. This is done in (e.g.) Littlewood-Paleytype arguments.

There are also wavelet families where the di-lation parameter a and the translation parame-ter b are assigned discrete values only. One

Ingrid Daubechies is a professor in the Program for Ap-plied and Computational Mathematics and the Math-ematics Department at Princeton University, Prince-ton, New Jersey. Her e-mail address is [email protected].

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?W.?W26Xg/X ?W.??W26X?@@e?/X? W.eW26Xf?/X? ??7H?.MB1gN1 ?7H??.MB1?N@L??N1? 7He.MB1f?N1? ??@fJ5?W-Xe?@ ?@f?J5??@)Xe@? @?fJ5?W2@e@? ??@e?W.Y?*>1e?@ [email protected]??@V1e@? @?e?W.Y?7R'e@? [email protected]??S@@e?@ ?@e?W.Ye?@?@e@? @?eW.Y??@f@? ??@e7YO@?7>@L??@ ?@e?7YO@?J@W5e@? @?e7YO@?3T.e@? ??@e@@@@?@0R/??@ ?@e?@@@@?@@0Ye@? @?e@@@@?V+Ye@? ??3L?heJ5 ?3L?he?J5? 3Lhe?J5? ??V/?he.Y ?V/?he?.Y? V/he?.Y? ?

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Figure 2

wim.qxp 3/16/99 3:06 PM 4


widely used choice is a = 2−j , b = 2−jk, withj, k ∈ Z. It is then also possible to choose insuch a way that the j,k = 2−j ,2−jk constitutean orthonormal basis for L2(R). Instead of theintegral representation above, we now have

f =∑j,k〈f , j,k〉 j,k,

which can again be understood in L2-sense, orin many other functional spaces. Harmonic analy-sis methods can be used to show that such j,kprovide unconditional bases for a variety of Ba-nach spaces, such as Lp(R),1 < p <∞, the Hardyspace H1(R) and its dual BMO, the Sobolevspaces Ws (R), the Besov spaces Bqp,s (R) , theHölder spaces Cs (R), etc. That is, for each ofthese spaces, there exists a criterion that de-cides whether f lies in the space or not, solelyby looking at the behavior of the absolute val-ues |〈f , j,k〉| of the wavelet coefficients. Thisunconditionality of wavelet bases for many func-tional spaces makes them a special and power-ful tool.

Wavelet bases are also linked with fast algo-rithms. Every wavelet base is derived from amultiresolution analysis (provided has somedecay and some smoothness). This means thatthere exists an auxiliary function φ, the scalingfunction, such that

φ(x) =∑ncnφ(2x− n), (x) =

∑ndnφ(2x− n);

moreover, ∫φ(x)dx = 1. Because of this latter

property, φ can be used to write a resolution ofthe identity, i.e.,

f (x) = limJ→∞

2J∫f (y)φ(2J (y − x))dy

for continuous f. In particular, the integrals2J∫f (y)φ(2Jy − k)dy can be approximated by

the sampled values f (2−Jk). (In practice, thisapproximation may have to be refined.) By therecurrence relations above, one can then, bymeans of a fast algorithm, compute for suc-cessive j < J the wavelet coefficients〈f , j,k〉 = 2j/2

∫f (y) (2jx− k)dx and the scal-

ing coefficients 〈f ,φj,k〉. In fact, algorithms ofthis type were already known to electrical engi-neers as subband filtering (with exact recon-struction).

A combination of these fast algorithms onone hand, and a thorough and deep under-standing of the mathematical properties of theunderlying functional analytic tool on the otherhand, has led to a variety of applications. A fewexamples discussed in the talk were compressionof large, dense matrices (Beylkin, Coifman, andRokhlin); theorems showing that wavelets giveasymptotically optimal denoising and estima-tion techniques for many different types of noisy

signals (Donoho—this application exploits thatwavelets are an unconditional basis in manyspaces); image compression and manipulationfrom edge information encoded in wavelet co-efficients (Mallat).

References[1] J. Benedetto, and M. Frazier, eds. Wavelets: Math-

ematics and Applications. CRC Press, 1993.[2] G. Beylkin, R. Coifman, and V. Rokhlin, Fast

wavelet transforms and numerical algorithms, Comm.Pure Appl. Math. 44 (1991), 141–183.

[3] D. Donoho, Unconditional bases are optimalbases for data compression and for statistical estima-tion, Applied and Computational Harmonic Analysis,1 (1992) , 100–115.

[4] C. Chui, ed., Wavelets: A tutorial in theory andapplications, Academic Press, 1992.

[5] I. Daubechies, Ten lectures on wavelets. CBMS-NSF Regional Conf. Ser. in Appl. Math., vol. 61, SIAM,Philadelphia, PA, 1992.

[6] ———, ed., Different perspectives on wavelets,Proc. Sympos. Appl. Math., vol. 46, Amer. Math. Soc.,Providence, RI, 1993.

[7] D. Donoho,Unconditional bases are optimal basesfor data compression and for statistical estimation,Appl. Comput. Harmonic Anal., 1 (1992), 100–115.

[8] S. Mallat and S. Zhong, Characterization of sig-nals from multiscale edges, IEEE Trans. PAMI 14 (1992).

[9] Y. Meyer,Ondelettes ét opérateurs, I: Ondelettes;II: Opérateurs de Calderón-Zygmund; III: Opérateursmultilinéaires, Hermann, Paris, 1990; English transl.Cambridge Univ. Press.

[10] ———, Wavelets: Algorithms & applications. SIAM,Philadelphia, PA, 1993.

[11] M. B. Ruskai et al., eds., Wavelets: Theory andApplication, Jones and Bartlett, Boston, 1992.

The Geometry ofSymplectic EnergyDusa McDuff

The talk was divided into three parts.

The first part was an elementary introductionto symplectic geometry—the geometry intro-duced on a smooth 2n-manifold M by a closed2-form ωwhich is nondegenerate (that is, its topexterior power ωn is a volume form). The basicexample of a symplectic manifold is Euclideanspace R2n with its standard form

ω0 = dx1 ∧ dx2 + · · · + dx2n−1dx2n.

Dusa McDuff, F.R.S., is professor of mathematics at theState University of New York at Stony Brook. Her e-mailaddress is [email protected].

wim.qxp 3/16/99 3:06 PM Page 35


One important concept is that of the sym-plectic gradient XH of a function H : M → R.This vector field is characterized by the equation

ι(XH )ω =ω(XH, ·) = dH,

and, in contrast to the gradient with respect toa Riemannian metric, points tangent to the levelsurfaces of H. Moreover, the flow which it gen-erates (called the Hamiltonian flow) preserves thesymplectic form ω. Thus, if M is the unit 2-sphere S2 ⊂ R3 with its usual area form, and ifH is the height function (given by the third co-ordinate function), the symplectic gradient XHis tangent to the horizontal circles H = const andgenerates a rotation about the north-south axis.Here, one can let the Hamiltonian depend ontime t ∈ [0,1]. One then gets a family XtH of vec-tor fields which integrates up to a familyφt,0 ≤ t ≤ 1, of symplectomorphisms. The mapφ1 is called the time-1 map of Ht . Essentially,any symplectomorphism (i.e., diffeomorphismwhich preserves ω) is the time 1-map of sucha flow.

Symplectomorphisms have important geo-metric properties, which are just beginning to beunderstood. One striking result is Gromov’s non-squeezing theorem (1985), which states that ifthere is a symplectomorphism which embeds a2n-ball of radius r into a cylinder of radius R,then r ≤ R. Here, a cylinder is a product of a 2-dimensional disc of radius R with another sym-plectic manifold (M,ω) of dimension 2n− 2.Gromov proved this result when M is Euclideanspace R2n−2 with its usual symplectic form, andit has been proved for all manifolds M byLalonde–McDuff.

This leads to the notion of a capacity. We de-fine the capacity c(B) of a ball of radius r to beπr2, and, for an arbitrary subset U in a sym-plectic 2n-manifold, define c(U ) to be the supre-mum of the capacities of the symplectic 2n-balls which embed symplectically in U. Thus, thenonsqueezing theorem implies that the capac-ity of the cylinder B2(r )× R2n−2 is πr2.

The second part introduced Hofer’s idea of theenergy of a symplectomorphism φ. We measurethe size of a Hamiltonian function H by

TotvarH = maxx∈M

H(x)−minx∈M

H(x),

and define the length of the path φt,0 ≤ t ≤ 1,generated by Ht to be

L(φt ) =∫ 1

0TotvarHt dt.

The energy e(φ) is then defined to be the infi-mum of the lengths of the paths φt which havetime-1 map equal to φ. Note that, although oneuses the first derivative of H in order to gener-ate the flow, the size of H depends only on its

values, not its derivative. Therefore, it is a pri-ori not clear that the energy of a nontrivial mapis always positive. That this is the case followsfrom Hofer’s energy-capacity inequality, whichstates that if U is an open set which is disjoinedby φ, i.e.,

φ(U )∩U =∅,then e(φ) ≥ c(U ) . As an example, consider thefunction H = y on R2 with ω0 = dx∧ dy . Theflow generated by H is translation in the x di-rection:

(x, y) 7→ (x + t, y),

and it is easy to check that the energy requiredto disjoin the rectangle [0,1]× [0, a] is preciselythe area a.

The third part of the talk outlined a geomet-ric proof of the energy-capacity inequality, whichis due to Lalonde–McDuff. The idea is to use aflow φt whose time-1 map disjoins a ball in Mof radius r to construct an embedded ball of ra-dius r in a cylinder B2(R)×M where

πR2 = L (φt ) +πr2/2 + ε.

The nonsqueezing theorem then implies that

πr2 ≤ L (φt ) +πr2/2 + ε,

and hence, taking the infimum over all suchpaths φt that

c(U ) = supπr2 ≤ 2e(φ).

A refinement of this argument gives (a slightlymodified version of) the full energy-capacity in-equality.

ReferenceF. Lalonde and D. McDuff, The geometry of symplecticenergy, Ann. of Math. (to appear).

MathematicalTheory in ParallelAlgorithmsJill P. Mesirov

Suppose we would like to understand a physi-cal phenomenon, for example, how air flowspast an airplane wing, or how a protein folds. Wemight perform real experiments and get datafrom measurements of actual phenomena in the

Jill Mesirov is Director of Research at Thinking Ma-chines Corporation in Cambridge, Massachusetts. Here-mail address is [email protected].

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laboratory, e.g., via awind tunnel. However, itwould be difficult infer-ring the behavior in un-known cases from thisdata. On the other hand,we might also try to builda mathematical model,perhaps adding somesimplifying assumptions,which characterizes thephysical phenomenon.Using the model we canthen do simulations. His-

torically, these calculations were done by hand,but now we use computers to do them. We cantest the model’s predictions by comparing theresults of simulations against real experimentaldata obtained for stereotypic examples. If the pre-dictions match the experimental data, then wegain confidence in our models and may use themto predict behavior in cases where it is unknown.

The mathematical model/simulation para-digm described above is used to help design thecars we drive, the medicines we take, and the air-planes in which we fly. This approach often rep-resents a more cost-effective way of doing some“experiments” via computer simulation. Its suc-cess has created an ever-increasing demand forhigher-performance computers to allow for realtime, interactive design, and graphics. Mathe-matics plays an important role in this process,both in modeling physical phenomena, and in de-signing efficient algorithms for high-perfor-mance parallel computers.

For the purpose of our discussion, a parallelcomputer consists of a collection of processors,each with its own local memory, connected by anetwork. A processor can only operate on datain its own local memory. If it has to use data thatis located in another processor’s memory, thenthat data must be sent over the network. Thistype of action is referred to as “communica-tions”. When designing algorithms for such dis-tributed- memory parallel computers, we seek tominimize the time spent in communications.This makes the algorithms more efficient inusing the processing power of the machine. Inthis lecture we described an interesting interplaybetween mathematics and the design of effi-cient parallel algorithms in the context of N-body algorithms for both fluid flow and molec-ular dynamics applications. We also showedsome videos of our simulations which the labo-ratories of the future may produce in real time.

The two cases we considered require differ-ent communication patterns. For the fluid flowproblem, we showed how rotated and translatedGray codes can be used to construct timewiseedge independent Hamiltonian paths on hyper-

cubes, yielding an optimal global communicationpattern for machines with this kind of network.In molecular dynamics codes for protein dy-namics, the force law is a local one. Thus, one isled to solving a local, i.e., on subcubes of side2a + 1, gossiping problem in multidimensionalgrids. We showed that by restricting to the casewhere the data distribution paths are transla-tionally and rotationally symmetric, the problemcorresponds to finding Hamiltonian paths inorbit graphs. Solutions were found for certain di-mensions and side lengths by finding solutionsin the case a = 1, and then giving a method ofextending those solutions to the cases a > 1.

Bibliography

[1] W. Taylor, J. Mesirov, and B. Boghosian, Optimizedlocal data distribution on lattice graphs (to appear).

[2] J-Ph. Brunet, A. Edelman, and J. Mesirov, Hy-percube algorithms for direct N-body solvers, SIAM J.Sci. Statist. Comput. 14 (1993), 1143–1158.

[3] J. Sethian, J-Ph. Brunet, A. Greenberg, and J.Mesirov, Two-dimensional, viscous, incompressible flowin complex geometries, J. Comput. Phys. 101 (1992),185–206.

[4] P. Tamayo, J. Mesirov, and B. Boghosian, Paral-lel approaches to short range molecular dynamics sim-ulations, Proceedings of Supercomputing ‘91, IEEEComputer Society Press, 462–470.

The Wave EquationRevisitedCathleen S. Morawetz

It is a surprising fact thatthere is still much tolearn about the waveequation. Beginning withrepresenting the solutionof the initial value prob-lem in terms of a funda-mental solution, it is aneasy step to see that in3D+T Huyghens’s princi-ple holds, i.e., if the sur-face of the light conefrom a point backwardsin time does not cut thesupport of the initial val-

ues, then the solution at that point is identicallyzero. This will be true even if there is a pertur-bation of the wave equation confined to the in-side of the cone.

Cathleen Morawetz is professor emeritus at the CourantInstitute of Mathematical Sciences, New York Univer-sity. Her e-mail address is [email protected].

Jill P. Mesirov

Cathleen Morawetz

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This has led to many investigations, includ-ing scattering theory for perturbations, and withHuyghens’s principle gives neater results in-cluding the exponential decay of the local energyif the wave equation is perturbed locally in cer-tain ways, e.g., the presence of a star-shaped re-flecting obstacle. See (LPM) and (M).

More recent questions are about finding“transparency conditions”. These are conditionson a boundary which guarantee that the solutioncould be continued outside the boundary as a so-lution of the wave equation. See (EM), (BT), and(KM) for approximate transparency conditions.H. Warchall (W) has shown under more generalbut analytic conditions that if, and we think hereonly of the wave equation, the initial data is sup-ported in a ball of radius b, then the solution in-side that ball for t > t1 > 0 depends only on thedata inside the ball at t = t1.

In fact, for the simple case of the wave equa-tion the transparency condition can be writtendown explicitly by giving the outward charac-teristic derivative in terms of an integral over theball of support at the same time. Conversely, ifthe solution satisfies this integral condition,then it can be continued outside as a solution ofthe wave equation. Cauchy integrals are involved,and thus the solution must have three deriva-tives.

Finally, this circle of ideas leads one to a wayof avoiding transparency conditions but stillsolving in a finite domain by repeated use ofHuyghens’s principle.

Bibliography[BT] A. Bayliss and E. Turkel, Boundary conditions

for exterior acoustic problems, Report No. 79-7, ICASE,NASA Langley Research Center, 1979.

[KM] G. A. Kriegsmann and C. S. Morawetz, Nu-merical solutions of exterior problems with the reducedwave equation, J. Comput. Phys. 28, (1978), 181–197.

[LPM] P. D. Lax, C. S. Morawetz, and R. S. Phillips,The exponential decay of solutions of the wave equa-tion in the exterior of a star-shaped obstacle, Comm.Pure Appl. Math. 16 (1963), 477–486.

[M] C. S. Morawetz, Exponential decay of solutionsof the wave equation, Comm. Pure Appl. Math. 19(1966), 439–444.

[ME] B. Engquist and A. Majda, Absorbing boundaryconditions for the numerical simulation of waves, Math.Comp. 31 (1977), 629–652.

[W] H. A. Warchall, Wave propagation at computa-tional domain boundaries (to appear).

Surface Motion Dueto Surface EnergyReductionJean E. Taylor

Suppose one has an ir-regular blob of some-thing, without any exter-nal forces such as gravityacting on it. Because it isnot a round ball, it doesnot have the smallest sur-face area for its volume.If it evolves so as to re-duce its surface area,how does it evolve? Whatdifferent evolution lawsare reasonable, and how

do the shapes differ under these different typesof motion? Can you tell, by some not-too-deli-cate features of the shapes, which law is, in fact,governing the motion?

For a crystalline material, such as a singlecrystal of ice, the shape of least surface energyfor a given volume is not a round ball, but some-thing else, like a hexagonal prism. How does asnowflake evolve towards this?

The claim is that physically interesting motionproblems can be formulated in a variational set-ting, as gradient flows under various naturalinner products. This enables existence proofs anda unifying framework for different motion lawsand different surface free energy functions. Italso underlies the whole crystalline approach,which is interesting both as a model in its ownright and as a possible means of approximationfor other surface free energy functions, andwhich has been implemented by computer pro-grams in many cases.

Surface Energy and Weighted MeanCurvatureWhy does mean curvature H arise for variationalproblems involving surface area? Because it is therate of decrease of surface area with volumeunder deformations. More precisely, the firstvariation of area of a smooth oriented surfaceS is the bounded linear operator on continuousvector fields g defined by

δS(g) =ddt

∫x∈htS

1dH 2x,

Jean Taylor is professor of mathematics at RutgersUniversity in New Brunswick, NJ. Her e-mail addressis [email protected].

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flow, in that the inner product space keepschanging, as S itself keeps changing.

One can use this fastest decrease property toconstruct a variational approach to motion byweighted mean curvature, by selecting a timestep ∆t and minimizing

∫h#S γ + 1

2t h • h startingwith S0 to get St, then with St to get S2t, etc. Con-vergence of these piecewise constant flows to acontinuous flow as ∆t goes to zero has beenshown [ATW] (with an approximately equal vol-ume integral replacing the L2 inner product). Theresulting flow is the PDE solution to motion byweighted mean curvature where that exists, andis motion by crystalline curvature for curvesand crystalline γ [AT].

Are there other reasonable inner productsthan the L2 inner product? Yes. In particular, mo-tion by the negative Laplacian of weighted meancurvature arises if one does gradient flow in theH−1 inner product. Mullins (who also intro-duced motion by curvature) derived that motionby the negative Laplacian of curvature modelssurface diffusion: H is a potential, D∇H givesthe flux of atoms down the gradient of the po-tential, and −D∆H is then the accumulation ofatoms, which translates into a velocity with theappropriate constants. For a survey of this field,see [CT].

This type of motion is difficult to model, how-ever. There is no maximal principle, and no level-set (viscosity solution) formulation. However, acrystalline formulation of this motion for curveshas been devised and implemented in a com-puter program [CRCT].

We have also derived from physical principlesa whole family of motion laws [CT]. At one ex-treme, where attachment kinetics are rapid andsurface diffusion is slow, there is motion by−∆κγ . At the other, where attachment kineticsare slow and surface diffusion is rapid, there ismotion by κγ − κav where κav is an overall av-erage; this is L2 gradient flow restricted to de-formations with integral 0. In between is a wholefamily of motion laws, which turn out to be flowby the inner product, which is the appropriatelinear combination of the L2 and H−1 innerproducts!

Comparing ShapesWe return to the original question of whetherthere are gross features of shapes moving undervarious laws that can be used to distinguish onelaw from another. We have made videos of com-puter simulations of several initial shapes mov-ing under each of the two extreme-case motionlaws discussed above. These videos can beviewed using Mosaic [M]. We comment on the fol-lowing observable features:

(1) In motion by κγ − κav, zigzag portions ofthe curve translate as a whole, with the same ve-


where htS = {y : y = x + tg(x) for some x ∈ S} .1

One can also compute that with n(x) denotingthe oriented normal at x and Hn the mean cur-vature vector,

δS(g) =∫x∈S

g · (−Hn(x))dH 2x.

For solids, with crystal lattice orientationsfixed in space, the surface free energy per unitarea is a function γ from the unit sphere to thereal numbers, and it is convenient to extend thedomain of γ to all vectors by γ(p) = |p|γ(p/|p|).If γ is C2 and convex and S smooth,

δγS(g) =ddt

∫x∈htS

γ(n(x))dH 2x

=∫x∈S

g · (−κγn)dH 2x

where κγ = ∂2γ∂p2

1κ1 + ∂2γ

∂p22κ2 . Here κi is the ith

principal curvature and pi the correspondingprincipal direction. We call κγ the weightedmean curvature. Note that if ∆E =

∫htS γ −

∫S γ

and ∆V denotes the signed volume from S tohtS, then for g with small support aroundx0, Et ≈ δS(g) ≈ −κγ (x0)Vt , so κγ (x0) ≈ − E

V .Given any γ there is associated the Wulff

shape

Wγ = {x : x · p ≤ γ(p)∀p}.

If Wγ is polyhedral, γ is crystalline, and the firstvariation is neither bounded nor linear. Instead,the crystalline weighted mean curvature (ab-breviated crystalline curvature) is nonlocal: oneuses the rate of decrease of surface free energywith volume, with the deformations moving en-tire segments. Edges and corners must be energy-minimizing to avoid infinite contributions tothe analog of first variation. For polygonal curves,the formula reduces to something quite simple:κγ(segment i)= −σiΛi/`i, where `i is the lengthof the segment, Λi is the length of the segmentof the Wulff shape boundary with the same nor-mal direction, and σi is 1 if the curve makes leftturns at each end (when traveling in the direc-tion of its orientation) is −1 if both ends makeright turns, and is 0 otherwise [T].

Inner Products and Gradient FlowsObserve that the action of the linear operator δSon a function g is given by the L2 inner prod-uct of g with −Hn. Thus, flow by mean curva-ture is gradient flow for surface area in the L2

inner product. It is an unusual type of gradient

1There are better definitions that allow for singulari-ties in S[A].

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locity, whereas in surface diffusion, there is aproximity effect, and each segment can have adifferent velocity.

(2) Curves that are convex (or, more generally,which have κγ nonpositive) remain so undermotion by κγ − κav, whereas long skinny por-tions tend to develop bulbs at the end under mo-tion by surface diffusion.

(3) Topology changes can happen under eithertype of motion, but tend to be more frequent anddramatic under motion by surface diffusion. Inparticular, a simple closed curve with κγ non-positive everywhere can break into two con-nected components under motion by surfacediffusion.

SummaryThe importance of regarding flows as being gra-dient flows under different inner products hasseveral advantages. First, it shows there is noth-ing magic about “decreasing energy fastest”—each decreases energy fastest in its own metric.Second, it provides a unifying approach, puttingdifferent surface energies, different motion laws,and even sharp versus diffuse interface ap-proaches under one umbrella [TC]. Third, it al-lows variational techniques to be used for mo-tion problems. Often both elliptic and highlynonelliptic problems can be handled in the sameproofs. And, in particular, in the crystalline con-text the gradient flow approach provides a goodcriterion for when a facet must bestepped [CRCT].

There are also advantages to considering thecrystalline approach. Some physical materialsare faceted. One can see what is special aboutarea, by comparing area-reducing motions tocorresponding surface-energy reducing motions.It is flexible; in both theory and computation, onecan alter the motion law without affecting thebasic underlying structure. It is computable: ituses a natural parametrization, by the Gaussmap. It is a good way to measure curvature nu-merically, and one can fairly easily detect andmake topological changes. One can use it for ap-proximations (convergence has been proved insome cases [G][S]). And it has a nice theory in itsown right, which is sometimes similar to that forarea-related problems and sometimes different.

A longer version of this lecture summary isexpected to appear [T2].

References[A] W. K. Allard, On the first variation of a varifold,

Ann. of Math. 95 (1972), 417–491.[AT] F. J. Almgren and J. E. Taylor, Flat flow is mo-

tion by crystalline curvature for curves with crystallineenergies, J. Diff. Geom. (to appear).

[ATW] F. J. Almgren, J. E. Taylor, and L. Wang, Cur-vature driven flows: A variational approach, SIAM J.Control Optim. 31 (1992), 387–437.

[CT] J. W. Cahn and J. E. Taylor, Surface motion bysurface diffusion, Acta Metal. Mater. 42 (1994),1045–1063.

[CRCT] W. C. Carter, A. Roosen, J. W. Cahn, and J. E.Taylor, Shape evolution by surface diffusion and sur-face attachment limited kinetics on completely facetedsurfaces, preprint.

[G] P. Girao, preprint, and P. Girao and R. Kohn,preprint.

[M] In Mosaic, Open URL to http://jeeves.nist.gov and follow the prompts.

[S] M. Soner, private communication.[T] J. E. Taylor, Mean curvature and weighted mean

curvature, Acta Metal. Mater. 40 (1992), 1475–1485.[T2] J. E. Taylor, Surface motion due to surface en-

ergy reduction, in elliptic and parabolic methods ingeometry (to appear).

[TC] J. E. Taylor and J. W. Cahn, Linking anisotropicsharp and diffuse surface motion laws via gradientflows, J. Stat. Phys. (1994) (to appear).

Soliton Equationsand DifferentialGeometryChuu-Lian Terng

In this talk, I explainedsome symplectic geo-metric properties of har-monic maps from theLorentze space R1,1 to asymmetric space: the ex-istence of a sequence ofcompatible sympleticstructures that make theharmonic map equationHamiltonian, a hierarchyof commuting flows, andan action of the affine

Kac-Moody group on the space of harmonicmaps.

It is well known that most finite dimensional,completely integrable, Hamiltonian systems canbe obtained by applying the Adler-Kostant-Symes(AKS) theorem to some Lie algebra G equippedwith an ad-invariant, nondegenerate bilinearform, and a decomposition G =K +N. The sym-plectic manifold is some coadjoint N -orbitM ⊂K⊥ 'N∗, and the equation is the Hamil-tonian equation of the restriction of some Ad(G)-

Chuu-Lian Terng is professor of mathematics at North-eastern University, Boston, Massachusetts. Her e-mailaddress is [email protected].

wim.qxp 3/16/99 3:06 PM Page 40


invariant function f on G to M. Applying the AKStheorem to the loop algebra of an affine Kac-Moody algebra, we obtain many well-known soli-ton equations and equations from differentialgeometry. In fact, let G/K be a symmetric space,G =K +P the Cartan decomposition, A a max-imal abelian subaglebra in P, P1 =A⊥ ∩P ,a ∈A a fixed regular element, and b ∈A. Weobtain a sequence of symplectic structures{ωr | r ∈ Z} on the space of Schwartz functionsM = S(R,P1), functionals {Fb,n : M→R|n ≥ 1},and evolution equations, the n-th flow associatedto G/Kwith respect to a, b, ut = [Qb,n(u), a] withn = −1 or n ≥ 2 such that(1) all ωr are compatible. That is, {, }r + µ{, }s

is again a Poisson structure on M for anyµ ∈ R and r , s ∈ Z, where {, }r is the Poissonstructure corresponding to ωr ,

(2) the Hamiltonian equation for Fb,n with re-spect to ωr is ut = [Qb,n+r+1(u), a],

(3) Fb,n are commuting Hamiltonians with re-spect to ωr for all n, r .

These flows contain many interesting equa-tions. For example, the equation for harmonicmaps from R1,1 to G/K is the −1-flow associ-ated to G/K, the nonlinear Schrödinger equationis the third flow associated to SU (2), the Modi-fied KdV equation is the fourth flow associatedto SU (2)/SO(2), and the Beals-Coifman evolutionequations are the n-th flows associated to acompact Lie group G with n ≥ 2.

There is also a system of partial differentialequations of n variables naturally arising fromthe second flow associated to a rank n sym-metric space. For example, the natural PDE sys-tem associated to

SO(2n + 1)/SO(n)× SO(n + 1),SO(2n)/SO(n)× SO(n)

and SO(2n,1)/SO(n)× SO(n,1)

is the equation for isometric immersions fromNn(c) into N2n(c) with flat normal bundle andindependent curvature normals with c = 1,0,−1respectively, where Nn(c) is an n-dimensionalspace form with sectional curvature c.

For these sequence of flows we also obtainanalogues of K. Uhlenbeck’s loop group actionon the space of harmonic maps from R1,1 intoLie group and the classical Bäcklund transfor-mations for the Sine-Gorden equation.

Moduli Spaces andAdiabatic LimitsKaren K. Uhlenbeck

The last ten years of dif-ferential geometry havebeen dominated by thestudy of very specialgeometric partial differ-ential equations. Theequations I have in mindinclude the minimal andconstant mean curvatureequations, the Yamabeequation, the Kähler-Ein-stein equation, harmonicmaps, and Yang-Millswith all its variations.

The origins of the equations vary tremendously,as they come from both very classical geometryand modern high-energy physics, but the equa-tions themselves are clearly very special. One ofthe interesting and useful features of most of theequations is the “large”, by which we mean multi-dimensional, solution spaces which are regu-larly found for these equations in special cir-cumstances.

By and large the origin of the equations ofgeometry is intimately connected with varia-tional principles. The classical equations areusually posed as minimization problems. It is ahard and fast rule of physics that interestingequations are the “equations of motion” forsome integral with a classical Lagrangian. Inthese cases, the Euler-Lagrange equation, whichis generated by the variational principle, is nearlyalways second order. However, for technical rea-sons, the second variation is self-adjoint, andhence has zero index if it can be represented asa Fredholm operator between Banach spaces.This means that formally speaking, the dimen-sion of a critical set, or the solution space of ageometric equation, tends to be zero dimen-sional.

There are a multitude of origins for the phe-nomena of “nonaccidental” solution manifoldsof large dimension. By “nonaccidental” we meanthat a small perturbation which does not changethe nature of the problem will not destroy them.The most common cause is the existence of acontinuous family of symmetries which leavesthe equation invariant. This is closely allied tothe phenomena of “integrals” which is important

Karen Uhlenbeck is Regents’ Chair in Mathematics atthe University of Texas at Austin. Her e-mail address [email protected].

wim.qxp 3/16/99 3:06 PM Page 41


for mean curvature surfaces. A second philo-sophically separate cause is the reduction of theoriginal variational problem, with its second-order geometric equation, to a first-order equa-tion for a topological minima. This mechanismreduces the second-order harmonic map equa-tion from a Riemann surface into an almost com-plex symplectic manifold to a pseudo-holomor-phic curve equation, and Yang-Mills to self-dualYang-Mills. The index of the first-order equa-tion obtained in this fashion is topologically de-termined and typically not necessarily zero. Afinal and not-so-well-known origin for multi-di-mensional parameter spaces is the introductionof point singularities, which changes the index.The Yamabe equation, which has zero index ona compact manifold, has index k on a manifoldwith k punctures.

An optimum strategy for progress includesthe search for new equations, as well as techni-cal progress on the well-studied equations. An-other angle is to search for new uses of these al-ready-studied moduli spaces of solutions togeometric partial differential equations. We de-scribe how a reduction technique can result inthe moduli space of solutions to one problem asthe target manifold for more geometry. Creditfor this idea, at least in its modern form in anapplication to monopoles, is due to the physi-cist Nick Manton. Here is a simplified version.

Consider the question of locating long, slow,closed orbits of the classical mechanics problemof a point mass moving in Rn with potential en-ergy V (x) and kinetic energy 12 |x|2. To make theproblem interesting we must have

M = {x ∈ Rn : dV (x) = 0}a nontrivial manifold, and technically speakingwe assume that M is a nondegenerate criticalmanifold in the sense of Bott. The action for theproblem is

J(x) =∫ T

0

(12 |x|2 − V (x)

)dt

with x(0) = x(T ). In our search for long, slow,closed orbits we renormalize t = Tτ to get

J(x) =∫ 1

0

12

T−2

∣∣∣∣ dxdτ∣∣∣∣2− V (x)

)T dτ.

Let ε = T−2. In renormalizing time τ, the Euler-Lagrange equations are

(*) ε( ddτ

)2

x(τ) + dV(x(τ)

)= 0.

As T →∞, ε → 0 and we hypothesize that thereis a smooth family of solutions x(ε, τ) which foreach ε solves (*), and which has a limit as ε → 0.We call this orbit at ε = 0 and adiabatic limit. Thefollowing theorem is cute, if nonobvious, and theproof can be done with advanced calculus.

Theorem. A necessary condition for x(τ) to bean adiabatic limit for the original mechanicsproblem is that x is a closed geodesic on M.

By replacing Rn by a function manifold andV : Rn → R by one of the geometric integrals ofthe type with a moduli space of critical points,one finds adiabatic approximations for hyper-bolic equations. It is also possible to replace tby static parameters to obtain “adiabatic” lim-its of static problems. Proving that the adiabaticlimit has some physical or geometric reality ismuch more difficult.

References[1] M. Atiyah and N. Hitchen, Geometry and dy-

namics of magnetic monopoles, Princeton Univ. Press,1988.

[2] S. Dostoglou and D. Salamon, Self-dual instan-tons and holomorphic curves, Ann. of Math., Universityof Warwick, preprint, 1992.

[3] ———, Cauchy-Riemann operators, self-duality,and the spectral flow, Proceedings of the First EuropeanCongress of Mathematics, Paris, 1992, preprint, Uni-versity of Warwick, 1992.

[4] R. A. Leese, Low-energy scattering of solitons inthe CP1 model, Nuclear Phys. B 344 (1990), 33.

[5] Leese, Peyrard, and Zakrzewski, Soliton scatter-ing in some relativistic models in (2 + 1) dimensions,Nonlinearity 3 (1990), 387.

[6] Leese, Peyrard and Zakrzewski, Soliton stabilityin the O(3) sigma model in (2 + 1) dimensions, Non-linearity 3 (1990), 773.

[7] N. Manton, A remark on the scattering of BPSmonopoles, Phys. Lett. B 110 (1982), 54–56.

[8] ——— , Monopole interactions at long range, Phys.Lett. B 154 (1985).

[9] R. Montgomery and L. Bates, Closed geodesics onthe stable two-monopoles, Comm. Math. Phys. 118(1988), 635–640.

[10] D. Stuart, Dynamics of abelian Higgs vorticesin the near Bogomolny range, Comm. Math. Phys.

[11] ———, The geodesic approximation for the Yang-Mills-Higgs Equation, preprint, University of California,Davis, 1993.

[12] R.S. Ward, Slowly-moving lumps in the CP1

model in (2 + 1) dimensions, Phys. Lett. 185 (1985),424–428.

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