www.esf.org

METHODS OF INTEGRABLE SYSTEMS, GEOMETRY,APPLIED MATHEMATICS (MISGAM)

Standing Committee for Physical and Engineering Sciences (PESC)

RESEARCH NETWORKING PROGRAMME

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The European Science Foundation (ESF) was established in 1974 to create a common European platform forcross-border cooperation in all aspects of scientific research.

With its emphasis on a multidisciplinary and pan-European approach, the Foundation provides the leadershipnecessary to open new frontiers in European science.

Its activities include providing science policy advice (Science Strategy); stimulating co-operation betweenresearchers and organisations to explore new directions (Science Synergy); and the administration of externallyfunded programmes (Science Management). These take place in the following areas: Physical and EngineeringSciences; Medical Sciences; Life, Earth and Environmental Sciences; Humanities; Social Sciences; Polar;Marine; Space; Radio Astronomy Frequencies; Nuclear Physics.

Headquartered in Strasbourg with offices in Brussels, the ESF’s membership comprises 75 national fundingagencies, research performing agencies and academies from 30 European nations.

The Foundation’s independence allows the ESF to objectively represent the priorities of all these members.

Cover picture : Helicoid with two handles; minimal surface constructedusing the methods of the theory of integrable systems.A.Bobenko, U.Pinkall and M.Schmies

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Integrable systems and random matricesnowadays arise in various problems of pure andapplied mathematics and statistics.Integrable systems with finite degrees of freedomhave been the object of intense investigation bymathematicians and physicists for many centuriesbecause of their applications in mechanics andgeometry. The trajectories of integrable mechanicalsystems display a regular behaviour as opposed toa chaotic one.

The discovery of integrable behaviour in classicaland quantum physical systems with infinitely manydegrees of freedom is one of the most excitingdevelopments of Mathematical Physics in thesecond half of the 20th century. Such systems arisein fluid dynamics, classical and quantum fieldtheory, statistical physics, optical communications.Algebraic and differential geometry play aprominent role in the mathematical foundations ofthe theory of integrability.

The theory of random matrices originates from thestudy of energy levels of heavy nuclei. It proved tobe efficient also in statistics, combinatorics and instudying growth processes. Recent developmentsin random matrix theory are intimately connectedwith the theory of integrable systems. The aim of the Methods of Integrable Systems,Geometry, Applied Mathematics (MISGAM)Programme is to join the efforts of Europeanresearchers actively involved in different aspects ofthe theory of integrable systems and randommatrices. The mathematical techniques developedhave to be transformed into working tools forApplied Mathematics. A significant boost to thisdomain of research in Europe will certainlycontribute to the spread of these new ideas in alarge number of related fields.

MISGAM unifies the European efforts on thisexciting interdisciplinary project creating a fruitfultraining ground for young researchers.

The running period of the ESF MISGAM ResearchNetworking Programme is for five years from July2004 – June 2009.

Introduction

The figures shownumerical simulations ofthe evolution andinteraction of singularsolutions of the EPDiffequation from DD Holmand MF Staley. EPDiff isshort for Euler-Poincareequation on thediffeomorphisms. EPDiffmay be derived as ashallow water equation,which applies to thepropagation andinteraction of nonlinearinternal waves.The surface effects ofthese internal waves areobserved in Nature, forexample, on the Oceansurface, such as at theStrait of Gibraltar shownin the figure above. Inthe figures, two straight segmentsare initialized movingrightward. The one behind hastwice the speed of theone ahead, and the twosegments are offset inthe vertical direction. The segments eachbreak into curved stripsof a certain width and these undergoovertaking collisions. The segments retaintheir integrity as theyexpand until theovertaking collisionoccurs. Upon overtakinga slower segment, thefaster segment transfersits momentum to theslower one ahead and aremarkable"reconnection" or"melding" of the segments mayoccur.

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Integrable systems have been studied since the 18thcentury by the giants of geometry and mechanics,like Euler, Hamilton, Jacobi and Lagrange, whorevealed the deep symplectic and algebro-geometricnature of these problems. Around 1895 Kortewegand de Vries (KdV) described shallow water waves bymeans of a celebrated nonlinear differential equationwhich is shown to have soliton solutions and hasmany of the features similar to those of the 18thcentury integrable mechanical problems. In the late1960s it was discovered that the Schrödingerequation whose potential is given by a solution of theKdV equation has a spectrum that remains invariantin time. It means that the theory of KdV flows makespart of the spectral theory of the Schrödingerequation known to be of fundamental importance forQuantum Mechanics.

These discoveries stimulated an outburst of ideaswhich marked the beginning of the modern theory ofintegrable systems. This led to the discovery of theHamiltonian nature of the KdV equation and itsgeneralisations like the Kadomtsev-Petviashviliequation to algebro-geometric solutions and the Lie-theoretic description to the tau-functionrepresentation of solutions, just to name a few.

Geometry and integrable systems grew even closerafter the discovery that the celebrated KdV equationplays a fundamental role in the matrix and topologicalmodels of two-dimensional quantum gravity. Namely,the partition function of 2D gravity turns out tocoincide with the tau-function of a particular solutionto the KdV hierarchy. More recent investigations,stimulated and motivated by the theory of Gromov -Witten invariants of compact symplectic manifolds,suggest that the constitutive rules of a wide class ofintegrable partial differentials equations are encodedin certain topological characteristics of moduli spacesof algebraic curves and their mappings. It is achallenge to uncover and explain this novelrelationship between the theory of integrable systemsand geometry and to make these new ideas work inapplied mathematics.

Random matrix models play an important role in thefollowing intimately related problems.- The distribution of the eigenvalues of a random

matrix, having a certain symmetry condition toguarantee the reality of the spectrum, is given bymatrix integrals, which are related to solutions ofintegrable systems.

- The problem of the statistics of the length of thelongest increasing sequence in random permutationor random words (Ulam's problem) has recentlybeen reduced to questions on random matrices.

- Integrals over groups and symmetric spaces lead toa variety of interesting matrix models, which satisfynon-linear integrable differential equations. As astriking feature, they also satisfy Virasoro constraintsand the coefficients of the expansions arecombinatorial or topological quantities.

- The sample covariance matrix of a Gaussianpopulation is used to estimate the true covariancematrix of that population and the study of thestatistical distribution of the largest eigenvalues ofthe sample covariance matrix requires randommatrix technology and has led to useful results instatistics where the sizes of the sample and thepopulation are of the same order. This is activelybeing applied to medical statistics, financialmathematics and communication technology.

The four problems above and their time-perturbationsare all solutions to integrable equations or lattices.Matrix integrals point the way to new integrablesystems, and also to the formulation of newcombinatorial and probabilistic problems.

Further progress in integrable theory, as well aspossible applications to real physical problems, willstrongly depend on understanding the asymptoticintegrability of classical and quantum systemsdepending on a parameter. The Whitham averagingtechnique proved to be an efficient tool not only in theasymptotic theory of nonlinear waves but also inquantum field theory and in the theory of randommatrices. Geometric and analytic foundations of theaveraging methods rely upon the study of behaviourof trajectories of integrable systems in theneighbourhood of a finite dimensional invariantmanifold. The results of this study are used inapplying dispersive integrable limits to weakturbulence.

Scientific Background

Surface with constant mean curvatureand rotationally symmetric metric described

by a solution of a Painlevé equation.A.Bobenko and U.Eitner

Deformation theory of integrable systems and Frobenius manifolds

The modern theory of integrable systems beganwith the discovery that certain remarkable nonlinearevolutionary Partial Differential Equations (PDEs) areintegrable, such as the Korteweg - de Vries (KdV)equation, nonlinear Schrödinger equation, Todalattice, etc. Nevertheless the classification ofsystems of integrable PDEs remains an openproblem.

The perturbative approach to the classificationproblem proved to be successful in classifyingsystems of spatially one-dimensional integrableevolutionary PDEs admitting the hydrodynamiclimit. The moduli space of the PDEs is closelyrelated with the moduli space of semisimpleFrobenius manifolds. The families of PDEs classifiedin this way possess properties inspired by 2Dtopological field theory. The bihamiltonian structureof the integrable PDEs constructed in this way canbe considered as a deformation of classical W-algebras associated to simply-laced simple Liealgebras. We study the full symmetry algebra ofthese PDEs in order to produce deformations ofquantum W-algebras. We also plan to develop analternative construction of the families of integrablePDEs in terms of a suitable class of infinite-dimensional Lie algebras.

The geometry of integrable Hamiltonian PDEspossesses much of the richness of finitedimensional integrable systems, together with otherfeatures coming from classical analysis of partialdifferential equations. In this framework, asatisfactory Hamilton - Jacobi theory for integrableevolutionary PDEs, as an effective tool to actuallyfind solutions to these PDEs, is still underconstruction. The interest in such a theory hasgrown in relevance in connection with the so-calledsymplectic field theory. This theory suggests a newlink between Gromov - Witten invariants ofsymplectic manifolds and integrable hierarchies.The potential of these invariants can be calculatedas the value of the Hamilton - Jacobi generatingfunctional in certain points of the Lagrangiansubspace.

Integrable systems, singularity theory, and deformation theory of algebraic varieties

There are three completely different sources ofFrobenius manifolds:•Hierarchies of integrable partial differential

equations;•Quantum cohomology;•Singularity theory.

The main questions are: how much of the richstructure of the theory of integrable partialdifferential equations and of the theory of Gromov -Witten invariants can be constructed for theFrobenius manifolds in singularity theory? What isthe role of the tt* geometry in the theory ofintegrable partial differential equations as well as inthe theory of Gromov - Witten invariants? Moregenerally, the interplay between integrable systemsand generalisations of variations of Hodgestructures attached to families of non-commutativeobjects, such as A-infinity algebras or categories, isunder investigation. Furthermore a development ofa general approach to the problem of relationshipsbetween integrable systems with deformationtheory of algebraic varieties and singularities iscurrently in progress.

Quantisation and integrable systems

The comparison of semiclassical techniques withmethods based on the theory of Lie superalgebrasand with approaches based on deformationquantisation leads to two classes of problems.

The first one concerns quantum many particlesand/or spin systems. A current area of research isthe study of the deformed quantum Calogero -Moser (CM) problems in relation with Liesuperalgebras and symmetric superspaces.Preliminary results suggest that these issues arerelated to the solutions of the WDVV equations. Weare deepening this line of research to clarify to whatextent deformations of the quantum CM model canprovide new solutions to the WDVV associativityequations. The second class of problems focuseson the quantisation of integrable systems and alsocomprises Topological Field Theory (TFT). We aremainly studying the problems in TFT associated tothe quantisation of Poisson diffeomorphisms and ofmore general Poisson maps.

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Research Topics

6 MISGAM

Applications of integrable systems to geometryof surfaces and to its discrete generalisations

Differential equations describing interesting specialclasses of surfaces (as well as classicalparametrisations of general surfaces) areintegrable, and, conversely, many importantintegrable systems admit a differential-geometricinterpretation. In our project we use thisphenomenon to develop new geometric methodsof the integrability theory as well as to obtain newinteresting geometries and dynamical systems.The problem of an “intelligent discretisation” iscentral to our research. We study this problem inframes of discrete differential geometry, which aimsto develop discrete equivalents of thegeometric notions and methods of differentialgeometry. Current interest in this field derives notonly from its importance in pure mathematics butalso from its relevance for computer graphics. Animportant example are polyhedral surfacesapproximating smooth surfaces.

One may suggest many different reasonablediscretisations with the same smooth limit. Whichone is the best? From the theoretical point of viewthe best discretisation is the one which preservesall fundamental properties of the smooth theory.Such a discretisation clarifies the structures of thesmooth theory and possesses importantconnections to other fields of mathematics. On theother hand, for applications the crucial point is theapproximation: the best discretisation is supposedto possess distinguished convergence propertiesand should represent a smooth shape by a discreteshape with just few elements. Although thesetheoretical and applied criteria for the bestdiscretization are completely different, in manycases “theoretical” discretisations turn out topossess remarkable approximation properties andare very useful for applications. We are working onboth theoretical and applied aspects of discretedifferential geometry.

Recently the roots of integrable differentialgeometry were identified in the multidimensionalconsistency of discrete nets (the so-calledconsistency approach). This led to a new(geometric) understanding of integrability itself.First, we adhere to the viewpoint that the centralrole in this theory belongs to discrete integrablesystems. Further, and more important, this leads toa constructive and almost algorithmic definition ofintegrability of discrete equations as theirmultidimensional consistency. We focus on thedevelopment of the consistency approach forintegrable systems on (discrete) manifolds. Inparticular we classify discrete integrable equationsin dimensions two and three and address thequestion whether there are discrete integrableequations in more than three dimensions.

Research Topics

Discrete surface with constant negative Gaussian curvatureand two straight asymptotc lines. The corresponding smooth

surface is described by a Painlevé solution of the sine-Gordonequation. In the discrete case one has a discrete

Painlevé and a discrete sine-Gordon equation.T.Hoffmann and A.Bobenko

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Random matrices, random permutations,matrix models and applications

The study of matrix integrals and random matricesremains an extremely important domain ofresearch, which bridges several areas, probability,geometry, combinatorics and mathematicalphysics. This interaction has already proved to befruitful. Indeed, it has lead to the solution of thefamous Ulam's problem in combinatorics: what isthe distribution of the length of the longestincreasing subsequence in large randompermutations? Unitary matrix integrals play a big role. Similarly,integrals over symmetric spaces, likeGrassmannians, lead to a variety of importantmatrix models, which satisfy nonlinear integrabledifferential equations and Virasoro constraints.Furthermore, the computation in the large N limit ofa class of matrix integrals with polynomial potentialover complex N x N matrices is also both wellknown and surprisingly instructive and rich inapplications: the result beyond the leading large Nlimit contains information on the number of mapsdrawn on Riemann surfaces of a given topology.A relatively new development in the theory ofrandom matrices concerns normal matrix models.The new feature with respect to the Hermitianmatrix models is that the eigenvalues lie in thecomplex plane rather than on the real axis, whichgives rise to geometrically interesting situations. Westudy this model from a rigorous analytical point ofview, beyond the realm of formal power series usedin the existing literature, introducing suitableregularisations of the (ill-defined) integrals overmatrices. A related class of matrix integrals (in a certain sensethe holomorphic square roots of normal matrixintegrals) have recently made their appearance inthe string literature, and are expected to beimportant objects in supersymmetric gauge theoryand strings. Here a formal saddle point calculationsuggests that the eigenvalues are asymptoticallydistributed along certain curves. The determinationof these curves and the investigation of theirpossible relation with integrable models is anotherpart of this project. Again, a rigorous definition ofthe integrals over matrices is a prerequisite.

Research Topics

The pictures shows ten nonintersecting Brownian paths withtwo different starting points and one common end point.

At any intermediate time the positions of the paths aredistributed like the eigenvalues of a Gaussian unitary random

matrix with external source. As the number of paths increasesthe paths fill out a two-dimensional region whose boundary isindicated in black. This random model has a phase transition.

For small time there are two well-separated groups of pathsthat come together at a critical time and then continue as onegroup. At the critical time the boundary has a cusp singularity.

A. Kuijlaars

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Analytic and geometric approaches to asymptotic integrability

The weak dispersion expansion of integrable systemsis a problem of increasing importance. The so-calledmethod of hydrodynamic reductions discovered in thestudy of the dispersionless limits reveals a deeprelationship with topics in complex analysis. Theconnections between these ideas and the knowngeometrical theory of integrable hierarchies need to beexplored further. The use of Riemann - Hilbert problems to describesolutions of dispersionless limits of integrable systemsleads to an interesting twistor formulation ofdispersionless integrable systems. The twistorformalism will be used to investigate relevant sets ofadditional symmetries and to find new classes ofsolutions of these equations.Conformal maps find interesting applications toquadrature domains. These are special classes of two-dimensional domains parametrised by compactRiemann surfaces. It is proposed to explore furtherconnections between integrable systems theory andthese special classes of domains with emphasis onpotential applications to free surface flows in fluiddynamics.Other specific aspects of fluid dynamics constitute animportant part of the research project. We studymeasure-valued solutions for a family of evolutionarypartial differential equations in one, two and threespatial dimensions. Concerning the borderline between integrable andnon-integrable behaviour in evolutionary partialdifferential equations, semiclassical problems for non-integrable equations are somehow reduced tointegrable problems. Among our plans is theclarification of this issue. We also study numerically theproperties of the partial differential equations obtainedby a truncation of the perturbative expansions of theintegrable systems.

Research Topics

Soliton solutions to the Korteweg de Vries equation provide aninfinite extended solutions to the Kadomtsev Petviashvili (KP)

equation which are called line solitons. In the case of the socalled KP1 equation, these line solitons are unstable. The figure

shows the evolution of a perturbed line soliton of the KP1equation. Instabilities are developed in the form of peaks.

C. Klein, C. Sparber and P. Markowich

Multiperiodic solutionof the Korteweg de Vries equation

J. Frauendiener and C. Klein

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European Context

There are a great number of collaborations betweenEuropean researchers in geometry and integrablesystems, random matrices, and their applications.However, for the most part these have beenestablished on an individual basis and not as part ofany formal European network. The MISGAMprogramme aims at stimulating the exchange ofideas, ranging from theory to applications, and atimproving cross-fertilisation of researchers andstudents across boundaries. The topics involved arehighly related, and at the same time have an impacton many different areas of geometry, combinatorics,probability, statistics and physics. Nowadays,European scientists play an important and leadingrole in these fields, however they are scattered all overEurope. Many interesting questions, bothfundamental and applied, are ready to be tackled.With this in mind, we propose a very broad networkto provide ample opportunities of interaction andgrowth. The MISGAM programme is providing thenecessary impetus to improve communicationsbetween researchers, not only in different countries,but also in different areas of science. Yet the project issufficiently focused in order to serve as a fruitfultraining ground for young researchers.

The activities of the MISGAM programme are:

Short visits grants

A visitor exchange programme is active for visits of upto two weeks with an electronic open call through allthe year. Processing time for short visit grants isroughly three weeks. Applications can be submittedon the web-page.

Exchange grants

A visitor exchange programme is active for visits for aperiod between two weeks and six months withelectronic open call through all the year. Processingtime for exchange grants is roughly six weeks.

Workshops/Conferences

Each year, few thematic workshops or conferencesand brainstorm meetings (5-10 participants) will beorganised.

Applications should be submitted on the web page,http://www.esf.org/misgam. Proposal for workshopsand conferences will be processed once a yearduring the Steering Committee Meeting. Proposal fora brainstorm meeting should be submitted at leastfour months in advance of the event date.

Website

There are two websites where all the relevantinformation can be found: http://misgam.sissa.it andhttp://www.esf.org/misgam.

Activities

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ESF Research Networking Programmes are principally funded by the Foundation’s Member Organisations onan à la carte basis. MISGAM is supported by:

• Fonds zur Förderung der wissenschaftlichen Forschung (FWF)Austrian Science Fund, Austria

• Fonds National de la Recherche Scientifique (FNRS) National Fund for Scientific Research, Belgium

• Fonds voor Wetenschappelijk Onderzoek (FWO) Fund for Scientific Research – Flanders, Belgium

• Commissariat à l’Énergie Atomique/Direction des Sciences de la Matière (CEA) Institute for Basic Research of the Atomic Energy Commission, France

• Deutsche Forschungsgemeinschaft (DFG)German Research Foundation, Germany

• Scuola Internazionale Superiore di Studi Avanzati (SISSA), Italy

• Slovenská Akadémia ViedSlovak Academy of Sciences, Slovak Republic

• Comisión Interministerial de Ciencia y Tecnología (CICYT)Interministerial Committee on Science and Technology, Spain

• Vetenskapsrådet (VR)Swedish Research Council, Sweden

• Schweizerischer Nationalfonds (SNF)Swiss National Science Foundation, Switzerland

• Engineering and Physical Sciences Research Council (EPSRC), United Kingdom

Funding

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Professor Boris Dubrovin (Chair)International School for AdvancedStudies (SISSA)Via Beirut 2-434100 TriesteItalyTel: +39 040 3787 461Fax: +39 040 3787 528Email: dubrovin@sissa.it

Professor Alexander BobenkoInstitut für MathematikTechnische Universität BerlinStrasse des 17. Juni 13610623 BerlinGermanyTel: +49 30 314 24 655Fax: +49 30 314 79 282Email: bobenko@math.tu-berlin.de

Professor Giovanni FelderDepartment of Mathematics ETH ZurichHG G 46Rämistrasse 1018092 ZürichSwitzerlandTel: +41 44 632 3409Fax: +41 44 632 1085Email: giovanni.felder@math.ethz.ch

Professor Nigel HitchinMathematical InstituteUniversity of Oxford24-29 St GilesOxford OX1 3LBUnited KingdomTel: +44 1865 273515Fax: +44 1865 273588Email: hitchin@maths.ox.ac.uk

Professor Kurt JohanssonMatematiska institutionenKTHLindstedtsvägen 25100 44 StockholmSwedenTel: +46 8 790 61 82Email: kurtj@math.kth.se

Professor Arno KuijlaarsDepartment of MathematicsKatholieke Universiteit LeuvenCelestijnenlaan 200 B3001 Leuven (Heverlee)BelgiumTel: +32 16 32 70 52Fax: +32 16 32 79 98Email: arno.kuijlaars@wis.kuleuven.be

Professor Luis Martínez AlonsoDepto. De Física Teórica IIUniversidad Complutense de Madrid28040 MadridSpainTel: +34 91 394 4504Fax: +34 91 394 5197Email: luism@fis.ucm.es

Professor Peter MichorFakultät für MathematikUniversität WienNordberstrasse 151090 WienAustriaTel: +43 1 4277 50618Fax: +43 1 4277 50620Email: Peter.Michor@esi.ac.at

Professor Ladislav SamajInstitute of PhysicsSlovak Academy of SciencesDubravska cesta 9845 11 Bratislava 45Slovak RepublicTel: +421 2 59410 522Email: ladislav.samaj@savba.sk

Professor Pierre Van MoerbekeDépartement de MathématiqueUniversité de Louvain2, Chemin du Cyclotron1348 Louvain-la-NeuveBelgiumTel: +32 10 47 31 87Tel: +32 10 47 33 10 (sec)Fax: +32 10 47 25 30Email: vanmoerbeke@math.ucl.ac.be

Professor Jean-Bernard ZuberLPTHEUniversité Paris VIBoîte 1264 Place Jussieu75252 Paris cedex 5FranceTel: +33 1 44 272170Tel: +33 1 44274121 (sec)Fax: +33 1 44277393Email: zuber@lpthe.jussieu.fr

Advisory expert:

Professor Alexander VeselovDepartment of Mathematical SciencesLoughborough UniversityLoughboroughLeicestershire LE11 3TUUnited KingdomTel: +44 1509 222866Fax: +44 1509 223969Email: A.P.Veselov@lboro.ac.uk

Project Coordinator:

Dr. Tamara GravaInternational School for AdvancedStudies (SISSA)Via Beirut 2/4, 34100 TriesteItalyTel: +39 040 3787 445Fax: +39 040 3787 528Email: grava@sissa.it

Other participants:

Austria: A. Hochgerner (Wien)France: S. Barannikov, P. Di Francesco,B. Eynard, C. Viallet, P. Zinn-Justin (Paris),P. Vanhaecke (Poitiers). Germany: C. Hertling (Bonn), C. Klein (Leipzig),M. Praehofer, H. Spohn, Y. Suris (Munich).Greece: S. Kamvissis (Athens).Italy: M. Pedroni (Bergamo), G. Falqui, P. Lorenzoni (Milan), B. Fantechi, (Trieste),A. Abenda (Bologna), P. Santini,O. Ragnisco (Roma).Portugal: R. Loja Fernandes (Lisbon)Poland: A. Panasyuk (Warsaw),A. Doliwa (Olsztyn).Netherlands: J. van de Leur (Utrecht).Spain: M. Mañas (Madrid).Sweden: J. Hoppe (Stockholm).Switzerland: A. Cattaneo, T. Kappeler (Zürich).UK: E. Sklyanin (York), M. Dunajski (Cambridge),E. Ferapontov, R. Halburd (Loughborough), M. Mazzocco (Manchester), L. Mason, N. Woodhouse (Oxford), Y. Chen, D. Crowdy, J. Elgin, J. Gibbons, D.D. Holm (London).

ESF Liaison

Dr. Thibaut LeryScience

Ms. Catherine WernerAdministration

Physical and Engineering SciencesUnit (PESC)European Science Foundation1 quai Lezay-MarnésiaBP 9001567080 Strasbourg cedexFranceTel: +33 (0)3 88 76 71 28Fax: +33 (0)3 88 37 05 32Email: cwerner@esf.org

For the latest information on thisResearch Networking Programmeconsult the MISGAM website:www.esf.org/misgam

MISGAM Steering Committee

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www.esf.org