BULLETIN

INTERNATIONAL CENTER FOR MATHEMATICS

June 2006 20

Contents

Coming Events 1

CIM News 6

News from our Associates 7

Past Events - Scientific Reports 12

Feature Article: Poisson vs. Symplectic Geometry by Rui Loja Fernandes 15

Math in the Media 20

Interview: Marcelo Viana 25

Coming Events

June 28-30: Workshop “From Lie Algebras toQuantum Groups”

Organizers

Helena Albuquerque and Joana Teles (University ofCoimbra), Samuel Lopes (University of Porto).

Aims

This workshop is a Satellite Conference of the Interna-tional Congress of Mathematicians (ICM, Madrid, Au-

gust 22-30, 2006) and will bring together leading spe-cialists in the topics of Lie algebras, quantum groupsand related areas. It aims to present the latest de-velopments in these areas as well as to stimulate theinteraction between young researchers and establishedspecialists.

The event will be held at the Department of Mathe-matics of the University of Coimbra.

Keynote Speakers

Helena Albuquerque (Univ. of Coimbra, Portugal)

Georgia Benkart (Univ. of Wisconsin-Madison, USA)

Alberto Elduque (Univ. of Zaragoza, Spain)

George Lusztig (MIT, USA)

Shahn Majid (Univ. of London, UK)

Carlos Moreno (Univ. Complutense de Madrid, Spain)

Michael Semenov-Tian-Shansky (Univ. of Bourgogne,France)

For more information about the event, see

www.aim.estt.ipt.pt/~jmmp/CIM/Lie/index.htm

July 1: 9th European Workshop on Applicationsand Generalizations of Complex Analysis

Organizers

Amılcar Branquinho and J. Carvalho e Silva (Univ. ofCoimbra), Ana Foulquie and Maria Isabel Cacao (Univ.of Aveiro).

Aims

This workshop is satellite to the “12th European In-tensive Course on Complex Analysis and Applicationsto Partial Differential Equations” to be held in bothUniversities of Coimbra and Aveiro from June 26 toJuly 7, 2006. It is intended to give an opportunity fordiscussions between junior and senior researchers fromseveral European countries in various fields of mathe-matics related to Complex, Quaternionic and CliffordAnalysis, like Algebra, Geometry, Numerical Analysis,Differential Equations, and Special Functions.

The event will be held at the CIM premises at the As-tronomical Observatory of the University of Coimbra.

For more information about the event, see

www.mat.uc.pt/~ajplb/9th.htm

July 19-21: Workshop on Mathematics inChemistry

Organizers

Jean-Claude Zambrini, J. Pinto Paixao and F. BastosPereira (University of Lisbon).

Aims

To identify and discuss research problems in the areaof the chemical sciences whose development is stronglydependent on mathematical techniques. To foster thecollaboration between leading researchers in chemistryand mathematics.

Chemistry is an exact science since it relies on quantita-tive models that can be described and applied by usingthe mathematical language. For instance, the theoryof chemical bonding and molecular structure, rates andequilibria of chemical reactions, molecular thermody-namics, relationships involving energy, structure andreactivity, modeling of solvation, are swarming withproblems whose solutions require sophisticated math-ematical techniques. Mathematics also plays a centralrole in many areas of “applied” chemistry and chemicalengineering. Important examples include atmosphericchemistry, biochemistry, and the broad field of com-puter simulations. The development of faster and moreaccurate spectroscopic techniques, the design of molec-ular devices, biomolecular computers, and of new em-pirical methods to predict reliable chemical data, andthe conception of more efficient chemical reactors arejust a few of a vast number of other topics that havestrong links to applied mathematics. A closer interac-tion between chemists and mathematicians may there-fore lead to significant progress in many key problemsin chemistry. The proposed workshop will foster thatinteraction since it will identify a number of importantresearch issues which will benefit from a joint effort.

Intended Audience are Researchers and post-graduatestudents on mathematics or chemical sciences.

The event will be held at the Complexo Interdisciplinarof the University of Lisbon.

Invited Speakers

Sylvio Canuto (Sao Paulo, Brazil)

David C. Clary (Oxford, UK)

Irene Fonseca (Pittsburgh, USA)

James T. Hynes (Paris, France and Boulder, USA)

Claude Leforestier (Montpellier, France)

John A. Perdew (New Orleans, USA)

Piotr Piecuch (Michigan, USA)

Jean-Louis Rivail (Nancy, France)

Mario Nuno Berberan e Santos (Lisboa, Portugal)

Joao Aires de Sousa (Lisboa, Portugal)

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Antonio Varandas (Coimbra, Portugal)

Marcelo Viana (Rio de Janeiro, Brazil)

Hans-Joachim Werner (Stuttgart, Germany)

J. H. Zhang (New York, USA)

For more information about the event, see

www.math-chem.org

September 4-8: 3rd International Workshopon Mathematical Techniques and Problems inTelecommunications

Organizers

Antonio Navarro (Univ. of Aveiro), Carlos Rocha andCarlos Salema (Technical Univ. of Lisbon).

Aims

The goals are three fold. Firstly, to identify and possi-bly find solutions for a number of mathematical prob-lems in the field of Telecommunications. Secondly todisseminate among telecommunications engineers somemathematical techniques which are not widely known inthis community even if they are being applied in mod-ern communication techniques. Thirdly, to improvemutual understanding and recognition between math-ematicians and telecommunication engineers, heavyusers of mathematical techniques in the field of engi-neering.

The intended audience includes telecommunications en-gineers, mostly those providing the problems and beingintroduced to new mathematical tools, and mathemati-cians, mainly providing solutions and being introducedto real life problems that may influence the direction oftheir research. A strong participation of young scien-tists, mainly those attending undergraduate degrees isalso expected.

The event is a Satellite Conference of the ICM and willtake place at the Polytechnic Institute of Leiria.

Invited speakers

V. Poor (Princeton University, USA)Cross-Layer Issues in Wireless Networks

J. Rosenthal (Notre Dame University, USA)Encryption

J. C. Pedro (University of Aveiro, Portugal)Mathematical Needs for Behavioural Modelling ofTelecommunication Circuits and Systems

J. Craveirinha (University of Coimbra, Portugal)A multiobjective routing optimisation framework formultiservice networks - a heuristic approach

C. Guillemot (INRIA, France)Signal Processing And Compression

For more information about the event, see

www.mtpt.it.pt

September 11-15: 2nd Summer School on Math-ematics in Biology and Medicine

Organizers

J. Carneiro, F. Dionısio, G. Gomes and I. Gordo (Gul-benkian Institute of Science, Oeiras).

Aims

The aim of this event is to promote the use of mathe-matical modelling in biology and medicine. This will beaccomplished by bringing some of international expertsto give a short course on their area of expertise. Thelecturing team combines researchers with a diversityof backgrounds in mathematics, biology and medicine,who will share their experience with the participants.

The school is aimed at postgraduate students frommathematics,physics, biology or medicine, who aremotivated to develop biomathematical research ap-proaches.

The role of mathematical formalisms in providing in-sight into biological and medical processes became ap-parent at the beginning of the 20th century. The ap-proach has since increased in popularity, especially dur-ing the past 10-20 years. This new phase of expansionis, to a large degree, stimulated by new developmentsin molecular biology and computation. Appropriatemathematical models are in great demand in many ar-eas of biology.

Given the high stands of mathematical and biomedicalresearch in Portugal, it is disappointing that only a fewresearch groups integrate the two disciplines. This canbe promoted by organizing interdisciplinary activitiesas proposed here.

The school will include a broad range of areas in biologyand medicine where mathematical modelling is estab-lished. We plan to include six courses covering severalresearch areas such as evolution, populations genetics,

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epidemiology, population biology, developmental biol-ogy and immunology. Each course will consist of threelectures.

The event will take place at the Gulbenkian Instituteof Science, Oeiras.

Short Courses

Developmental BiologyR. Azevedo (University of Houston, USA)

NeurobiologyC. Brody (Cold Spring Harbor, USA)

ImmunologyD. Coombs (Univ. of British Columbia, Canada)

Evolutionary BiologyT. Day (Queens University, Ontario, Canada)

Epidemiology and Population BiologyS. Levin (Princeton University, USA)

Population Genetics and Disease MappingG. McVean (University of Oxford, UK)

For more information about the event, see

eao.igc.gulbenkian.pt/mbm2006

October 20-21: CompIMAGE: InternationalSymposium on Computational Modelling of Ob-jects Represented in Images

Organizers

Joao Manuel Tavares and Renato Natal Jorge (Facultyof Engineering, Univ. of Porto, Portugal).

Aims

In our days the research related with objects modellinghas been a source of hard work in several distinct ar-eas of science as, for example, mathematics, mechani-cal, physics, informatics, etc. One major application ofobjects modelling is in medical area. For instance, itis possible to consider the use of statistical or physicalprocedures on medical images in order to model the rep-resented objects. Its modelling can have different goalslike 3D shapes reconstruction, organs segmentation in

3D or 2D images, etc. Examples of others applicationsare: temporal tracking of objects, tracking and analy-ses of objects deformation, objects recognition, objectssimulation, etc.

The main goal of the CompIMAGE consists in the pro-vision of a comprehensive forum for discussion on thecurrent state-of-the-art in these fields. The Symposiumwill cover (but is not limited to): Image Processingand Analysis, Image Segmentation, Data Interpolation,Registration, Acquisition and Compression 3D Recon-struction, Objects Tracking, Motion and DeformationAnalysis, Objects Simulation, Medical Imaging, Com-putational Bioimaging and Visualization.

The Symposium will bring together several researchersrepresenting several fields related to Computational Vi-sion, Computer Graphics, Computational Mechanical,Mathematics, Statistics, Medical Imaging, etc. Theexpertise will span a broad range of techniques, suchas finite element method, modal analyses, stochasticmethods, principal components analyses, independentcomponents analyses, distribution models, etc.

The Symposium will be held at Hotel D. Luıs, Coimbra.

Invited Speakers

Francisco J. Perales (Univ. Illes Balears, Spain)Perceptual Users Interfaces. Virtual & Augmented Re-ality Applications

Helder Araujo (University of Coimbra, Portugal)Active and Catadioptric Vision Systems for RoboticsApplications

Mario Forjaz Secca (Univ. Nova de Lisboa, Portugal)MR Diffusion Tensor Imaging

Sonia Isabel Goncalves (University of Lisbon, Portugaland Vrije Universiteit Medical Centre, Netherlands)Multimodality in Brain Imaging: Methodologic aspectsand applications

Hemerson Pistori (Dom Bosco Catholic Univ., Brasil)Computer Vision and Digital Inclusion of Persons withSpecial Needs: Overview and State of Art

Tony Chan (UCLA, USA)Capturing illusory objects in images via level set meth-ods

For more information about the event, see

paginas.fe.up.pt/CompIMAGE

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Other CIM events in 2006:

Working Afternoons SPM/CIM

CIM, Coimbra

A joint initiative of the Portuguese Mathematical Soci-ety (SPM) and the International Center for Mathemat-ics (CIM). Next meetings:

October 7, 2006 - Numerical AnalysisOrganizer: Maria Joana Soares (Univ. of Minho)

December 16, 2006 - AlgebraOrganizer: Gracinda Gomes (Univ. of Lisbon)

For more information, see

www.spm.pt/investigacao/spmcim/spmcim.phtml

Iberian Conference in Optimization

November 16-18, CIM, Coimbra

CIM Events for 2007

The CIM Scientific Council, in a meeting held in Coim-bra on February 11, approved the CIM scientific pro-gramme for 2007.

The list of events is the following:

Workshop onMathematical Control Theory and Finance

April 10-14, Instituto Superior de Economia e Gestao,Lisboa

Workshop on Eigenvalue Problems:Software and Applications

June 27-29, University of Porto

Lisbon Quantum Computation, Informationand Logic 2007

July 18-20, Instituto Superior Tecnico, Lisboa

CIM / UC Summer School:Topics in Nonlinear PDEs

July 22-28, University of Coimbra

Workshop on Robotics and Mathematics

September 17-19, University of Coimbra

For updated information on these events, see

www.cim.pt/?q=events

5

CIM News

Meeting of theGeneral Assembly of CIM

The General Assembly of CIM met on May 27, 2006,

during the morning, in the CIM premises at the Astro-nomical Observatory of the University of Coimbra.

In the afternoon of the same day, the members of theGeneral Assembly had the opportunity to attend a talkby Doutor Jorge Barros Luıs (Montepio Geral).

Research in Pairs at CIM

CIM has facilities for research work in pairs and welcomes applications for their use for limited periods.

These facilities are located at Complexo do Observatorio Astronomico in Coimbra and include:

• office space, computing facilities, and some secretarial support;

• access to the library of the Department of Mathematics of the Univ. of Coimbra (30 minutes away by bus);

• lodging: a two room flat.

At least one of the researchers should be affiliated with an associate of CIM, or a participant in a CIM event.

Applicants should fill in the electronic application form in

www.cim.pt/?q=research

CIM on the Web

For updated information about CIM and its activities, see

www.cim.pt

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News from our Associates

(from UA)

Views on ODEs

June 21-24, 2006

Aveiro, Portugal

Organizers

Vasile Staicu (Chairman), Alexandre Almeida, AntonioCaetano, Luıs Castro, Jose Mendes, Eugenio Rocha andJoao Santos (all Univ. of Aveiro).

The conference “Views on ODEs” will celebrate the65th birthday of Professors Arrigo Cellina (Milan,Italy) and James A. Yorke (Maryland, USA) and aimsto bring together those enrolled in research activitiesrelated with ordinary differential equations, differen-tial inclusions and their applications. The main topicsare: Dynamical systems, Bifurcations, Invariant mea-sures, Chaotic attractors, Prevalence, Population dy-namics, Markov operators, Semigroups, Viscosity solu-tions, Hamilton-Jacobi equations, Hyperbolic systems,Optimal control and differential inclusions, Variationaland topological methods.

Invited Speakers

Zvi Artstein (Weizmann Inst. of Science, Rehovot, Is-rael), Jean-Pierre Aubin (CREA - Ecole Polytechnique,Paris, France), Stefano Bianchini (SISSA-ISAS, Trieste,Italy), Alberto Bressan (Penn State Univ., USA), Fran-cis Clarke (Univ. Claude Bernard Lyon 1, France),Constantin Corduneanu (Univ. of Texas at Arling-ton, USA), Francesco S. De Blasi (Univ. degli Studidi Roma Tor Vergata, Italy), Rui Dilao (IST, TechnicalUniv. of Lisbon, Portugal), Helene Frankowska (CREA- Ecole Polytechnique, Paris, France), Andrzej Lasota(Univ. of Silesia, Katowice, Poland), Jean Mawhin(Univ. Catholique de Louvain, Louvain-la-Neuve, Bel-gium), Stefan Mirica (Univ. of Bucharest, Bucharest,Romania), Josef Myjak (Univ. degli Studi dell’Aquila,Italy), Boris S. Mordukhovich (Wayne State Univ., De-troit, USA), Farruh Mukhamedov (Depart. of Physics,Univ. of Aveiro, Portugal), Rafael Ortega (Univ. ofGranada, Granada, Spain), Mitsuharu Otani (WasedaUniv., Tokyo, Japan), Carlos Rocha (IST, TechnicalUniv. of Lisbon, Portugal), Biagio Ricceri (Univ. of

Catania, Catania, Italy), Nikolaos Papageorgiou (Na-tional Technical Univ. of Athens, Greece), Giulio Pi-anigiani (Univ. of Florence, Florence, Italy), Miguel A.F. Sanjuan (Univ. Rey Juan Carlos, Madrid, Spain),Susanna Terracini (Univ. of Milano, Bicocca, Italy).

For more information, see

www.divp-proj.org

(from UC)

12th European Intensive Course onComplex Analysis and Applications toPartial Differential Equations

June 26-July 7, 2006

Coimbra and Aveiro, Portugal

Organizers

Amılcar Branquinho and J. Carvalho e Silva (Univ. ofCoimbra), Ana Foulquie and Maria Isabel Cacao (Univ.of Aveiro).

This intensive course follows the eleven held at the Uni-versities of Coimbra and Aveiro from 1995 to 2005. Thisintensive course will have a total of 40 hours of lecturesand is at postgraduate level. Lecturers will have timeavailable to discuss with the students. This course isorganized by the Universities of Coimbra and Aveirowith the same goals as the ones organized under theSocrates/Erasmus Intensive Program of Higher Educa-tion, and is opened to all young mathematicians inter-ested in Complex Analysis and its applications.

Invited Lecturers

Andrei Martınez Finkelshtein (Almerıa Univ., Spain)The Riemann-Hilbert technique for Polynomials Or-thogonal on the Unit Circle

Guillermo Lopez Lagomasino (Carlos III Univ., Spain)Multi-Orthogonal Polynomials

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Irene Falcao (Minho Univ., Braga, Portugal)Numerical conformal mappings and generalizations:part I

Sebastian Bock (Bauhaus-Univ. Weimar, Germany)Numerical conformal mappings and generalizations:part II

There will be a Workshop on “Applications and Gen-eralizations of Complex Analysis” on the 1st of July,sponsored by CIM (www.mat.uc.pt/~ajplb/9th.htm).

For more information, see

www.mat.uc.pt/~ajplb/12.htm

(from IST)

ICORS 2006International Conference on Robust

Statistics

July 16-21, 2006

Lisbon, Portugal

Organizers

Conceicao Amado, Rosario Oliveira, Ana Pires andIsabel Rodrigues (Technical Univ. of Lisbon), CarlaPereira (Univ. of Porto), Peter Rousseeuw (Univ.of Antwerp), Manuela Souto de Miranda (Univ. ofAveiro).

The ICORS 2006 conference aims to join people work-ing on robust statistics and related fields. It intendsto bring together both leading experts and young re-searchers, thus creating a forum to cover recent progressand to stimulate exchanges among active researchers.The conference will also encourage informal contactsand discussions between participants.

Contributions to applied statistics are welcome as wellas theoretical ones, introducing new problems relatedto or interacting with robust statistics.

Previous ICORS meetings were held in: Vorau (2001),Vancouver (2002), Antwerp (2003), Beijing (2004) andin 2005 in Jyvaskyla.

Invited Speakers

Jorge Adrover (National Univ. of Cordoba, Ar-gentina), Claudia Becker (Martin-Luther-Univ. Halle-Wittenberg, Germany), Jose Berrendero (Univ.

Autonoma de Madrid, Spain), Ana Bianco (Univ. ofBuenos Aires, Argentina), Javier Cabrera (RutgersUniv., USA), Andreas Christman (Vrije Univ. Brussel,Belgium), Christophe Croux (Catholic Univ. Leuven,Belgium), Laurie Davies (Univ. of Essen, Germany),Peter Filzmoser (Vienna Univ. of Technology, Aus-tria), Ursula Gather (Univ. of Dortmund, Germany),Marc Genton (Texas A & M Univ., USA), Ivette Gomes(Univ. of Lisbon, Portugal), Alfonso Gordaliza (Univ.of Valladolid, Spain), Xuming He (Univ. of Illinoisat Urbana-Champaign, USA), Christian Hennig (Univ.College London, UK), Alfio Marazzi (Univ. of Lau-sanne, Switzerland), Ricardo Maronna (National Univ.of La Plata, Argentina), Ivan Mizera (Univ. of Alberta,Canada), Stephan Morgenthaler (Ecole PolytechniqueFederale de Lausanne, Switzerland), Hannu Oja (Univ.of Tampere, Finland), Marco Riani (Univ. of Parma,Italy), Elvezio Ronchetti (Univ. of Geneve, Switzer-land), Anne Ruiz-Gazen (Univ. of Toulouse I, France),Stefan Van Aelst (Ghent Univ., Belgium), Maria-PiaVictoria Feser (Univ. of Geneve, Switzerland), VictorYohai (Univ. of Buenos Aires, Argentina), Ruben Za-mar (Univ. of British Columbia, Vancouver, Canada).

For more information, see

www.math.ist.utl.pt/icors2006

(from UC)

Workshop “Geometric Aspects ofIntegrable Systems”

July 17-19, 2006

Coimbra, Portugal

Organizers

Joana Nunes da Costa, Joana Teles and Raquel Ca-seiro (Univ. of Coimbra), Nenad Manojlovic (Univ. ofAlgarve), Patrıcia Santos (Inst. Sup. Eng. Coimbra).

This is a satellite workshop of the ICM, and is intendedto focus on the following topics: Hamiltonian IntegrableSystems, Bi-Hamiltonian Systems and Poisson Geom-etry, Quantum Integrable Systems, Superintegrability,Separability, Reduction, Lie Algebroids in Mechanics.

In addition to the invited lectures there will be contri-butions from the participants. The scientific committeewill review the applications for contributions from theparticipants. PhD students are invited to participate.

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Invited Speakers

Francesco Calogero (Univ. Roma “La Sapienza”, Italy)

Jose Carinena (Univ. Zaragoza, Spain)

Pantelis Damianou (Univ. of Cyprus, Nicosia, Cyprus)

Jean-Pierre Francoise (Univ. Pierre et Marie Curie,Paris 6, France)

Franco Magri (Univ. Milano, Bicocca, Italy)

Juan Carlos Marrero (Univ. of La Laguna, Spain)

Yavuz Nutku (Feza Gurst Institute, Turkey)

Orlando Ragnisco (Univ. Degli Studi Roma, Italy)

For more information, see

www.mat.uc.pt/~geomis

(from UP)

XVth OPORTO MEETING onGEOMETRY, TOPOLOGY AND

PHYSICS

July 20-23, 2006

Porto, Portugal

Organizers

Miguel Costa, Carlos Herdeiro and Joao Nuno Tavares(Univ. of Porto), Marco Mackaay (Univ. of Algarve),Jose Mourao and Roger Picken (IST, Lisbon).

The aim of the Oporto meetings is to bring togethermathematicians and physicists interested in the inter-relation between geometry, topology and physics and toprovide them with a pleasant and informal environmentfor scientific interchange. The XVth Oporto Meeting isa Satellite Meeting of the ICM.

As in previous years, the meeting is focussed on theshort courses given by the main speakers, which aresupplemented by seminars by other participants. Thetalks are at the advanced graduate or postdoctoral level,and should be of interest to all researchers wishing tolearn about recent developments in the overlap betweengeometry, topology and physics.

Main Speakers

Frank Ferrari (Univ. Libre de Bruxelles, Belgium)Supersymmetric Gauge Theories, Matrix Models andGeometric Transitions

Dan Freed (University of Texas at Austin, USA)The Geometry of Abelian Gauge Fields

Nicholas Manton (Cambridge University, UK)Supersymmetric Systems - Investigation and Interpre-tation

Veeravalli Varadarajan (UCLA, USA)Unitary representations of Super Lie Groups

Albert Schwarz (Univ. of California at Davis, USA)Supergeometry and Number Theory

For more information, see

www.fc.up.pt/cfp/omgtp2006

(from UA)

Communicating Mathematics in theDigital Era

August 15-18, 2006

Aveiro, Portugal

Organizers

Eugenio Rocha (Chairman, Univ. of Aveiro), AntonioBatel, Carlos Ferreira and Joaquim Pinto (Univ. ofAveiro), Jose Borbinha (IST-Lisbon).

The Digital Era has brought dramatical changes tothe way researchers search, produce, publish and dis-seminate scientific work, in particular, mathematicians.This process is still evolving due to improvements inInformation Science (Information Arquitecture, Archiv-ing, Long-term Preservation), new achivements in Com-puter Science Technologies (ex. XML, Data Min-ing, Clustering, Recovery, Visualization, Web Tools)and initiatives as DML, Open Access Journals (ex.DOAJ), Digitisation Projects (ex. EMANI, GDZ, JS-TOR, NUMDAM), Scientific Catalogues (ex. GoogleScholar), or OS Digital Repositories (ex. EPRINTS,DSpace).

CMDE2006 is a Satellite Conference of the ICMand aims to bring together researchers from areasof Mathematics, Information Sciences and ComputerSciences, providing a forum for presenting and dis-cussing new ideas that may contribute to improveparadigms/mechanisms of producing, searching and us-ing scientific and technical scholarship in Mathematics.

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The main topics of CMDE2006 include: Data Mining,Clustering and Recovery, Digital Libraries and Archiv-ing Networks, E-Mathematics Resources, ElectronicPublishing, Free and Open Source Initiatives, Infor-mation Representation and Visualization, InternationalCopyrights and Author’s Rights, Math-networking andElectronic Communication, Mathematics E-Learning,Metadata Models and Standards, Multimedia Tools,Retrodigitisation, and Web Searching.

Presentations will be divided in four categories: Philo-sophical Implications on Scholarly Publishing, Theoret-ical and Technical, Projects and Initiatives, Softwareand Demonstrations.

Invited Speakers

J. Borwein (CEIC/IMU chair)

T. Bouche (NUMDAM, mini-DML project, Gallica-Math, CEDRAM)

B. Cipra (JPBM 2005 Communication Awarded)

J. Ewing (American Mathematical Society ExecutiveDirector)

P. Ion (W3C HTML-Math Working Group co-chair)

J. Kiernan (IBM Almaden Research Center)

B. Wegner (ELibM/EMIS chair)

For more information, see

www.cmde2006.org

(from IST)

International Summer School andWorkshop of Operator Algebras,

Operator Theory and its Applications

September 1-5, 2006

Lisbon, Portugal

Organizers

Amelia Bastos (Coordinator), Antonio Bravo, AmarinoLebre, Lina Oliveira, Paulo Pinto, Ana Moura Santos,Frank-Olme Speck (all Technical University of Lisbon).

Aims

Operator algebras are presently one of the dynamic ar-eas of mathematics. Nowadays Operator Algebras and

Operator Theory play an important role in different ar-eas of Mathematics and its Applications, particularlyin Mathematical Physics and Numerical Analysis.

This event is being organized at Instituto SuperiorTecnico, Universidade Tecnica de Lisboa, Portugal, andit is a Satellite Conference of the ICM. It consists of asummer school with short courses on Operator Alge-bras and its Applications to Mathematical Physics andNumerical Analysis, and a workshop that will bring to-gether researchers in the Operator Algebras and Oper-ator Theory areas.

The main topics of the workshop include: Crossed prod-uct C*-algebras; C*-algebras of operators on Hardy andBergman spaces; Invertibility theory for non-local C*-algebras; Von Neumann algebras; Approximate meth-ods in operator algebras; Asymptotic properties of ap-proximation operators; Toeplitz, Hankel, and convo-lution type operators and algebras; Symbol calculi;Invertibility and index theory; Operator TheoreticalMethods in Diffraction Theory; Factorization Theoryand Integrable Systems; Applications to MathematicalPhysics.

Invited Speakers

Summer School Courses

S. Power (Lancaster Univ., UK)Subalgebras of Graph C*-Algebras

Konrad Schmudgen (Leipzig Univ., Germany)C*-Algebras - Selected Topics

B. Silbermann (Chemnitz Univ., Germany)C*-Algebras and Asymptotic Spectral Theory

H. Upmeier (Marburg Univ., Germany)Toeplitz Operator Algebras and Multivariable ComplexAnalysis

Workshop plenary lecturers

Mikhail Agranovich (Moscow Institute of Electronicsand Mathematics, Russia)

Florin Boca (Univ. of Illinois, USA)

Lewis Coburn (State Univ. of New York at Bufallo,USA)

David Evans (Cardiff Univ., UK)

Israel Gohberg (Tel Aviv Univ., Israel)

Yuri Karlovich (Univ. Autonoma del Estado de More-los, Mexico)

Naum Krupnik (Univ. of Toronto, Canada)

Vladimir Manuilov (Moscow State Univ., Russia)

Nikolai Nikolski (Univ. of Bordeaux, France)

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Vladimir Rabinovich (Instituto Politecnico Nacional,Mexico)

Steffen Roch (Technical Univ. Darmstadt, Germany)

Stefan Samko (Univ. of Algarve, Portugal)

Ilya Spitkovski (William and Mary College, USA)

For more information, see

woat2006.ist.utl.pt

(from UNL)

SCRA2006/FIM XIIIInternational Conference on

Interdisciplinary Mathematical andStatistical Techniques

September 1-4, 2006

Lisbon/Tomar, Portugal

Local Organizers

Carlos A. Coelho (Chair) and Joao T. Mexia (Co-Chair)(New University of Lisbon), Luıs M. Grilo and LuısMerca (Polytechnic Institute of Tomar), Teresa Oliveira(Open Univ.-Lisbon).

The Forum for Interdisciplinary Mathematics, the De-partment of Mathematics, New University of Lisbonand the Polytechnic Institute of Tomar, are proud toco-organize a four-day International Conference in Lis-bon/Tomar, Portugal. This year’s conference themeis Interdisciplinary Mathematical and Statistical Tech-niques. The major concentration of the conference’sacademic activities will be in mathematical and sta-tistical sciences including, but not limited to, Actuar-ial and Financial Mathematics, Statistics and Applica-tions, Biostatistics, Combinatorics, Computer and In-formation Sciences, Differential Equations, DistributionTheory and Near Exact Distributions, Environmental

Statistics, Experimental Designs, Extreme Values, For-est Mensuration Modeling, Forest Economics, GraphTheory, Linear Statistical Inference, Mathematical Eco-nomics, Mathematics, Multivariate Statistics, Non-parametric Statistical Inference, Operations Research,Probability/Stochastic Processes, Public Health, Qual-ity Control, Reliability and Life Testing, Sampling,Semigroups, and partner areas. Those wishing to con-tribute outside these areas are welcome to submit theirabstracts. Suggestions for further topics and proposalsto organize a symposium should be sent to the organiz-ers.

The conference will feature separately a section onstudent paper competition. A selection panel willjudge the presentations and make recommendations forawards. In addition, the conference will devote a sessionon “Editors Round Table” to help scholars appreciatethe “current trends and techniques” of scholarly publi-cations.

Invited Speakers

C. R. Rao (Penn State Univ., USA)

Barry C. Arnold (Univ. of California, Riverside, USA)

Francine Blanchet-Sadri (Univ. of North Carolina,Greensborough, USA)

Carlos Braumann (Univ. of Evora, Portugal)

Tadeusz Calinski (Agricultural Univ. of Poznan,Poland)

Richard Davis (Colorado State Univ., USA)

Angela Dean (Ohio State Univ., USA)

Malay Ghosh (Univ. of Florida, Gainesville, USA)

Steven Gilmour (Queen Mary Univ., London, UK)

Ivette Gomes (Univ. of Lisbon, Portugal)

Samad Hedayat (Univ. of Illinois at Chicago, USA)

Benjamin Kedem (Univ. of Maryland, USA)

John Stufken (Univ. of Georgia, Athens, USA)

For more information, see

scra2006.southalabama.edu

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Past Events - Scientific Reports

Follow-Up Workshop on Mathematics and the Environment

Scientific Report

The Follow-Up Workshop on Mathematics and the En-vironment took place in the CIM headquarters in Coim-bra on January 27-28, 2006.

The meeting was intended as an informal gathering ofsome of the participants of the 2004 CIM ThematicTerm on Mathematics and the Environment and otherscientists with interest in this topic.

The two scientific sessions (on Oceanography and At-mospheric Sciences) were each organized under the fol-lowing format: an invited talk, three short communi-cations and an open discussion (see the complete pro-gram on www.aim.estt.ipt.pt/~jmmp/CIM/wme2006/program.htm). The main goal of the event has beento strengthen the connections and to plan future col-laboration between mathematicians, geophysicists andengineers.

After the talks and the discussions it became clear thatOceanography and Atmospheric Sciences represent animmense land of opportunity for researchers and grad-uate students in Mathematics. Among the possible in-teractions we would like to highlight the following:

• modelling and derivation of systems of equations(asymptotic analysis);

• validation of boundary conditions;

• data analysis - model/observations (stochasticanalysis);

• error propagation (numerical analysis);

• small scale phenomena;

• simulation of flows (computational mathematics).

These interactions may develop through the joint super-vision of Master and PhD students and in the frame-work of national and international research projects.

The crucial role played by CIM was acknowledged andit was suggested that a web page gathering relevant andstimulating mathematical problems in the area of theenvironmental sciences would be kept in the CIM website.

Jose Miguel Urbano (Universidade de Coimbra) andJuha Videman (Instituto Superior Tecnico)

Aveiro Workshop on Graph Spectra

Scientific Report

The theory of graph spectra is a strong field of re-search in mathematics and in several applied sciences(e.g. chemistry), combining different areas, like linearalgebra, algebraic combinatorics and algebraic graphtheory. The first explicit mathematical paper in thetheory of graph spectra was published in 1957 by L.Collatz and U. Sinogowitz [1]. However, we may saythat the theory of graph spectra began not after 1931,

when E. Huckel used graph spectra in an implicit wayin quantum theoretical treatment of the chemistry ofbenzenoide hydrocarbons [3]. According to the list ofpublications referred in [4], between 1931 and 1957, afew papers dealing with graph spectra theory implicittechnics applied to chemistry and molecular physics aswell as some papers relating matrix theory and graphs,with applications in mathematical physics, economics

12

and geometry, were published. After 1957 an increas-ing number of papers on graph spectra has been pub-lished (at least 60 papers during the sixties, 158 duringthe seventies, 297 during the eighties, 425 during thenineties and, so far, more than 400 during the currentdecade).

The first comprehensive book on the topic, “Spectra ofGraphs - Theory and Applications” by D. Cvetkovic,M. Doob and H. Sachs, appeared in 1979 [2], becominga land mark and a priority reading for everyone inter-ested in this field. In the last two decades several spec-tral techniques for treating graph theory problems havebeen developed: e.g. the use of graph eigenspaces, thestar complement technique and many others. There areconnections of the theory of graph spectra with otherparts of combinatorics and also with algebra and ge-ometry. It is very much used in theoretical chemistrybut also has some relevance to other applied fields, e.g.physics, electrical engineering and computer science.

The Aveiro Workshop on Graph Spectra was the firstinternational meeting organized as a forum for re-searchers on graph spectra and related topics. Thisthree-day meeting took place in the Mathematics De-partment of University of Aveiro, between 10 and12 of April 2006 (the web page is still active atceoc.mat.ua.pt/conf/graph2006). This workshopdeserved the attention of the Portuguese MathematicalCommunity, promoting the contact of some of his mem-bers (coming from several parts of the country) with thestate of the art and with most of the recent advancesin the topic. Indeed, 26 among the 60 participants,coming from Aveiro (10), Braganca (1), Coimbra (2),Covilha (2), Lisbon (8), Porto (2), Setubal (1), werein contact with international experts on graph spec-tra and their presentations. The foreign participants,including renowned specialist, came from Brazil (8),Canada (2), Germany (1), Italy (4), Malta (1), Poland(2), Serbia and Montenegro (3), Spain (4), The Nether-lands (2), UK (4) and USA (3). The plenary presen-tations ranged through the following: Spectral radiusof tournaments and bipartite graphs (Richard Brualdi,Univ. of Wisconsin - Madison, USA); Graph spectraand graph isomorphism (Chris Godsil, Univ. of Water-loo, Canada); Star complement in finite graphs (PeterRowlinson, Univ. of Stirling, UK); The Laplacian andCheeger inequalities for direct graphs (Fan Chung Gra-ham, Univ. of California, San Diego, USA); Old andnew results on algebraic connectivity of graphs (NairAbreu, UFRJ, Rio de Janeiro, Brazil); Constructinggraphs with integral Laplacian spectra (Steve Kirkland,

Univ. of Regina, Canada); Spectral characterizationof distance-regular graphs (Edwin van Dam, TilburgUniv., The Netherlands); Generalized adjacency matri-ces (Willem Haemers, Tilburg Univ., The Netherlands);Signless Laplacians of finite graphs (Dragos Cvetkovic,Univ. of Belgrade, Serbia and Montenegro). The sci-entific program was complemented with 18 contributedtalks with many recent results (part of them deliveredby leader researchers on the theory of graph spectra)and with a problems session (coordinated by DraganStevanovic from University of Nis, Serbia and Montene-gro). Taking into account the high level of most of thepresentations, their authors were invited to submit thepapers to be referred for publication in a special issueof Linear Algebra and Its Applications, to be edited byDragos Cvetkovic, Willem Haemers and Peter Rowlin-son.

For the organization of this workshop was crucial themain support of the Chairman of Scientific Commit-tee, Dragos Cvetkovic and the very important advicealso received, during the preparation of the workshop,from the other colleagues of Scientific Committee: NairAbreu, Richard Brualdi, Chris Godsil, Willem Hamersand Perter Rowlinson. Finally, I can not forget mycolleagues from the Organizing and Local Commit-tees: Raul Cordovil, Leal Duarte, Carlos Luz, Guedesde Oliveira, Agostinho Agra, Paula Carvalho, RosaAmelia, Paula Rama and Cristina Requejo from whom,since the very beginning, all the commitment and re-quested collaboration was received.

References

[1] Collatz, L.; Sinogowitz, U. Spektren endlicherGrafen.Abhandlungen aus dem MathematischenSeminar der Universitat Hamburg 21 (1957): 63-77.

[2] Cvetkovic, D.; Doob, M.; Sachs, H. Spectra ofgraphs. Academic Press, New York, (1979).

[3] Huckel, E. Quantentheoretische Beitrage zum Ben-zolproblem, Z. Phys. 70 (1931): 2004-286.

[4] Spectral Graph Theory Home Page - SGTHP,www.sgt.pep.ufrj.br

Domingos Moreira Cardoso (Organizing Committee’sChairman)

13

CIM Scientific Council Meeting 2006

Scientific Report

The 2006 Meeting of the CIM Scientific Council tookplace in Hotel Quinta das Lagrimas in Coimbra onFebruary 11, with the following timetable:

10:30-16:30 Meeting of the Scientific Council (in-cluding a work lunch).

17:00-18:00 Lecture by Mario Martinez (StateUniv. of Campinas, Brazil):Lower-sum order-value optimization.

18:30-19:30 Lecture by Enrique Zuazua (Univ.Autonoma de Madrid, Spain):Propagation, dispersion, control and numericalapproximation of waves.

20:00 Dinner.

The Scientific Council Meeting took place in a pleas-ant environment and was attended by thirteen of itsfifteen Members. The CIM Activity Plan for 2007 was

approved. Due to the outstanding scientific quality ofits participants, the Scientific Council Meeting was theideal opportunity for a broad discussion with the Pres-ident about the strategies and future activities of theCentre.

The Seminars were attended by 50 participants, whichis exactly the maximum number expected by CIM intheir website announcement. This public session of theScientific Council Meeting was highly participated andstimulating and many and interesting questions wereput by the audience to the lecturers. A dinner wasserved to all the participants in this last event, whichhad the opportunity to socialize with the members ofthe scientific council. The Rector of the University ofCoimbra also accepted the invitation of the Directionof CIM to participate in this dinner.

Joaquim Joao Judice (President of the Executive Boardof CIM)

14

Feature Article

Poisson vs. Symplectic Geometry

Rui Loja Fernandes1

Departamento de Matematica, Instituto Superior Tecnico

1049-001 Lisboa, PORTUGAL

rfern@math.ist.utl.pt

Abstract

We survey the relationship between symplectic and Poisson geometries, emphasizingthe construction of the symplectic groupoid associated with a Poisson manifold.

1 Introduction

Poisson and symplectic geometries are usually referredto as the geometries underlying classical mechanics. Letus recall briefly why this is so. In every introductorycourse in mechanics one learns Hamilton’s equations de-scribing the motion of a mechanical system with Hamil-tonian H:

qi = ∂H∂pi

pi = −∂H∂qi

(i = 1, . . . , n) (1.1)

where (q1, . . . , qn) are the position and (p1, . . . , pn) arethe momenta, which together form coordinates on thephase space of the system.

Symplectic geometry originates from the following in-terpretation of equations (1.1). Introduce the closed2-form:

ω :=n∑

i=1

dpi ∧ dqi. (1.2)

Since this 2-form is non-degenerate, every smooth func-tionH determines a vector fieldXH by the requirement:

iXHω = dH.

Now (1.1) is just the equation for the integral curvesof this vector field. More generally, one defines a sym-plectic manifold to be a manifoldM equipped with asymplectic form, i.e., a closed, non-degenerate, 2-formω. Then every smooth function H : M → R deter-mines a Hamiltonian vector field XH by exactly thesame procedure. Darboux’s theorem (see [9]) states

that, around any point, there exist local coordinates(qi, pi) such that ω takes the form (1.2), so locally werecover the standard formulation.

A slightly different interpretation of equations (1.1)leads to Poisson geometry. One defines a bilinear, skew-symmetric bracket of functions on the phase space bysetting for any pair of functions F and G:

F,G :=n∑

i=1

(∂F

∂qi

∂G

∂pi− ∂F

∂pi

∂G

∂qi

)(1.3)

and observes that Hamilton’s equations can be writtenin the form: qi = qi,H

pi = pi,H(i = 1, . . . , n).

More geometrically, any smooth function H determinesa Hamiltonian vector field XH by:

XH(·) = ·,H,

and Hamilton’s equations are just the equations for theintegral curves of this vector field. Another justifica-tion for the introduction of the Poisson bracket is thestudy of first integrals of the system: if F and G aretwo first integrals, then their Poisson bracket F,G isalso a first integral. This is because, for any triple offunctions F , G and H, we have the Jacobi identity:

F, G, H+ G, H,F+ H, F,G = 0.1Supported in part by FCT/POCTI/FEDER and by grants POCI/MAT/55958/2004 and POCI/MAT/57888/2004.

15

All this motivates defining a Poisson manifold to bea manifold M equipped with a Poisson bracket , ,i.e., a Lie bracket on the algebra of smooth functionsC∞(M) which satisfies the Leibniz identity:

F,GH = F,GH +GF,G.

Then any smooth function H : M → R determinesa Hamiltonian vector field XH by the procedureabove. But now, contrary to the symplectic situation,locally we may not recover anymore the standard form(1.3) of the Poisson bracket. In fact, we have the fol-lowing theorem, which maybe consider as the first sig-nificant result in Poisson geometry:

Theorem 1.1 (Weinstein [10]). Let (M, , ) be aPoisson manifold. For every x0 ∈ M there exists coor-dinates (q1, . . . , qn, p1, . . . , pn, y1, . . . , yl) centered at x0such that:

F,G =n∑

i=1

(∂F

∂qi

∂G

∂pi− ∂F

∂pi

∂G

∂qi

)+

l∑j,k=1

πjk(y)∂F

∂yj

∂G

∂yk,

where πjk(y) = −πkj(y) are certain functions of theyj’s alone which vanish at 0.

Therefore, contrary to symplectic geometry, wherethere are no local invariants, in Poisson geometry itis important to understand the local structure as well.

More generally, what can be said about the relation-ship between symplectic and Poisson geometry? In onedirection we have that every symplectic manifold is aPoisson manifold: to every symplectic form ω on a man-ifold M , one associates a Poisson bracket by setting:

F,G := ω(XF , XG).

Conversely, the Darboux-Weinstein theorem above im-plies that a Poisson manifold is (singular) foliated bysymplectic (immersed) submanifolds.

However, there is a much more subtle and deeper rela-tion between Poisson and symplectic geometry: to ev-ery Poisson manifold one can associate a canonical sym-plectic object. Moreover, its properties encode both thelocal and global behavior of a Poisson manifold. In theremainder of this paper we will explain how this objectarises naturally, and we will discuss briefly its relevancein the study of both local and global properties of aPoisson manifold.

2 Contravariant geometry

Let (M, , ) be a Poisson manifold. What kind ofpaths should one consider inM which take into accountthe Poisson geometry? Because M is foliated into sym-plectic submanifolds, paths in M should preserve this

foliation. However, this is a singular foliation and totake care of this we must give some “internal geome-try” to the paths. More precisely, let π : T ∗M → M bethe cotangent bundle and consider the the bundle mapdefined by:

# : T ∗M → TM, dH 7→ XH .

It is easy to check that the symplectic leaves ofM are infact the integral leaves of the distribution Im# ⊂ TM .

Definition 2.1. A cotangent path is a path a :[0, 1]→ T ∗M such that:

#a(t) =ddt

π(a(t)).

The space of cotangent paths will be denoted byPΠ(M).

Note that the base path γ(t) = π(a(t)) of a cotangentpath lies in a symplectic leaf. These kind of paths whereintroduce first by A. Weinstein, and they show up invirtual every global construction in Poisson geometry.For example, in [7] one studies connections in Poissongeometry and shows that parallel transport is definedalong cotangent paths.

There is a general principle in Poisson geometry thatevery construction in standard (covariant) geometrycan be dualized to a (contravariant) construction inPoisson geometry. The cotangent paths we have in-troduced is just one instance of this principle. Anotherinstance is Poisson cohomology. Recall that de Rhamcohomology of a manifold is the cohomology of the com-plex of differential forms (Ω•(M),d), where the differ-ential of a r-form is given by the usual formula:

dω(X0, . . . , Xr) =r∑

k=0

(−1)k+1Xk(Q(X0, . . . , Xk, . . . , Xr))+∑k<l

(−1)k+l+1ω([Xk, Xl], X0, . . . , Xk, . . . , Xl, . . . , Xr),

(2.1)

where X0, . . . , Xr ∈ X(M) are vector fields, [ , ] denotesthe usual Lie bracket of vector fields, and the hat overa factor means omitting that factor. Following the gen-eral principle above, in Poisson geometry one considersthe dual objects to differential forms, i.e., the multivec-tor fields Xr(M), and defines a contravariant exteriordifferential dΠ : Xr(M) −→ Xr+1(M) by:

dΠQ(α0, . . . , αr) =r∑

k=0

(−1)k+1#αk(Q(α0, . . . , αk, . . . , αr))+∑k<l

(−1)k+l+1Q([αk, αl]Π, α0, . . . , αk, . . . , αl, . . . , αr),

(2.2)

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where α0, . . . , αr ∈ Ω1(M). Here [ , ] is a Lie bracketon 1-forms induced from the Poisson bracket, which onexact 1-forms is given by

[dF,dG]Π = dF,G,

and extends to any pair of 1-forms by requiring that

[α, Fβ]Π = F [α, β]Π +#α(F )β.

It is easy to see that dΠ is indeed a differential: d2Π = 0.Hence it defines the Poisson cohomology H•

Π(M) ofthe Poisson manifold. It is not hard to see that for asymplectic manifold the Poisson cohomology is isomor-phic to the usual de Rham cohomology. However, ingeneral, the Poisson cohomology is quite hard to com-pute.

Observe that the defining equation of a Hamiltonianvector field can be written in the form XH = dΠH. Itfollows that dΠXH = 0 for any Hamiltonian vector field.More generally, any vector field X such that dΠX = 0is called a Poisson vector field. It is easy to checkthat X is a Poisson vector field iff its flow preservesPoisson brackets. The first Poisson cohomology groupis just the quotient of the Poisson vector fields by theHamiltonian vector fields.

In geometry one integrates 1-forms over curves. Du-ally, in Poisson geometry one integrates vector fieldsover cotangent paths: if X ∈ X(M) is a vector fieldand a ∈ PΠ(M) is a cotangent path with base path γ,then one defines:∫

a

X :=∫ 1

0〈X(γ(t)), a(t)〉dt.

The usual integral of closed 1-forms is invariant underhomotopy and depends only on the end-points of thecurve provided the form is exact. In Poisson geom-etry there is also a notion of cotangent homotopybetween cotangent paths (the precise definition can befound in [6]), and we have:

Proposition 2.2 ([6]). The integral of a Poisson vec-tor field is invariant under cotangent homotopies. Fora Hamiltonian vector field the integral depends only onthe end-points of the cotangent path.

In ordinary topology one defines the fundamental groupπ1(M,x0) of a pointed space (M,x0) to be the loopsbased at x0 modulo homotopies, where the group mul-tiplication arises from concatenation of paths. If M isconnected, changing the base point leads to isomorphicfundamental groups. If one considers (not necessarilyclosed) paths modulo homotopy then we do not get agroup anymore because we cannot always multiply twopaths. We get instead a groupoid Π1(M) ⇒ M : thereare source and target maps

s([γ])) = γ(0), t([γ])) = γ(1),

and the product [γ] · [τ ] is defined provided s([γ]) =t([τ ]).

In Poisson geometry we consider the analogous Pois-son fundamental groupoid Σ(M) ⇒ M formed bycotangent paths modulo cotangent homotopies:

Σ(M) := PΠ(M)/ ∼

Of course we can consider only cotangent paths whosebase paths are loops based at x0, and these form theisotropy group

Σ(M,x0) = s−1(x0) ∩ t−1(x0),

which should be thought of as the Poisson fundamentalgroup based at x0. The Poisson fundamental groupsat different base points are isomorphic provided thebase points lie in the same symplectic leaf ofM (other-wise, they may be non-isomorphic). Also, contrary tothe fundamental group of a space, Poisson fundamen-tal groups are usually non-discrete topological groups.This is because they contain information about the localbehavior at x0 of our Poisson structure. More gener-ally, the groupoid Σ(M) provides both local and globalinformation about the Poisson structure. But before weturn into that we need to study its geometry.

3 Symplectic Groupoids

The Poisson fundamental groupoid Σ(M) is a topo-logical groupoid since it is a quotient of the Banachmanifold PΠ(M). Moreover, Σ(M) has at most onesmooth structure compatible with the quotient topol-ogy for which the projection PΠ(M) → Σ(M) is sub-mersion. Whenever this smooth structure exists Σ(M)becomes a Lie groupoid (i.e., the groupoid structureis compatible with the smooth structure) and we saythat M is an integrable Poisson manifold. The ob-structions to integrability where determined recently in[6, 5], solving a long standing problem in Poisson geom-etry (and Lie groupoid theory). Let us explain brieflyhow they arise.

First of all, for each x ∈ M there exists attached to(M,π) a certain Lie algebra gx. As a vector space, wehave gx := Ker#x ⊂ T ∗x M and the Lie bracket is therestriction of the Lie bracket on 1-forms. We call gx

isotropy Lie algebra at x. Now, if Σ(M) is smooth,each isotropy Lie group Σ(M,x) is a Lie group with Liealgebra gx which, in general, is neither connected norsimply connected. If we denote by Gx the 1-connectedLie group with Lie algebra gx, the connected componentof the identity Σ(M,x)0 is isomorphic to Gx/Nx whereNx ⊂ Gx is a certain normal discrete subgroup calledthe monodromy group at x. One can show that themonodromy group can also be described as the image

17

of a certain homomorphism ∂ : π2(S, x)→ Gx, where Sdenotes the symplectic leaf through x. Moreover, thisdescription is still valid in the non-integrable case, butnow the monodromy groups Nx = im(∂) need not bediscrete subgroups anymore. In fact, the main theoremof [5] implies:

Theorem 3.1. A Poisson manifold is integrable iff themonodromy groups Nx ⊂ Gx are uniformly discrete asx ∈ M varies.

From now on we will assume that (M,π) is an inte-grable Poisson manifold so that Σ(M) is a Lie groupoid.We will show now that Σ(M) is a symplectic manifoldand that the symplectic form ω is compatible with thegroupoid multiplication, i.e., that

m∗ω = π∗1ω + π∗2ω (3.1)

where m : G (2) → G is the multiplication in G , definedon the space G (2) ⊂ G × G of composable arrows, andπ1, π2 : G (2) → G are the (restrictions of the) projec-tions to the first and second factors.

In order to understand how the symplectic form appearwe recall an alternative construction of Σ(M) due toCattaneo and Felder [1], and which is related with thePoisson-sigma model of string theory. Let us denote byP (T ∗M) the set of all paths in the cotangent bundle,so that PΠ ⊂ P (T ∗M). Since P (T ∗M) ' T ∗P (M) isthe cotangent bundle of the manifold of paths in M , itcarries a natural symplectic form ωcan. Now the resultsin [1] (see the explanations in [6]) show that there existsa Lie algebra action

P0Ω1(M)→ X(P (T ∗M))

where P0Ω1(M) denotes the Lie algebra of time-dependent 1-forms αt satisfying α0 = α1 = 0, withLie bracket [ , ]Π. The cotangent paths PΠ ⊂ P (T ∗M)form an invariant submanifold and two cotangent pathslie in the same orbit iff they are cotangent homotopic.

Now observe that the space of cotangent paths is pre-cisely the level set J−1(0) of the map J : P (T ∗M) →P0Ω1(M)∗ given by:

〈J(a), η〉 =∫ 1

0〈 d

dtπ(a(t))− π]a(t), η(t, γ(t))〉dt.

We have the following result due to Cattaneo and Felder[1]:

Theorem 3.2. The Lie algebra action of P0Ω1(M) onP (T ∗M) is Hamiltonian, with equivariant moment mapJ : P (T ∗M)→ P0Ω1(M)∗.

Hence the groupoid Σ(M) can be described alterna-tively as a Marsden-Weinstein reduction:

Σ(M) = P (T ∗M)//P0Ω(M). (3.2)

We deduce:

Corollary 3.3. If Σ(M) is smooth, then it admits asymplectic form which turns Σ(M) into a symplecticgroupoid.

Proof. We only need to check the compatibility of thesymplectic form with the product. First note that wehave the following explicit formula for the symplecticform ωcan in P (T ∗M):

ωcan(U1, U2)a =∫ 1

0ωcan(U1(t), U2(t))dt,

for all U1, U2 ∈ TaP (T ∗M), where ωcan is the canonicalsymplectic form on T ∗M . The additivity of the inte-gral shows that that condition (3.1) holds at the levelof P (T ∗M), hence it must hold also on the reducedsymplectic space Σ(M).

4 Local vs. Global properties

The symplectic groupoid Σ(M) encodes both local andglobal properties of a Poisson manifold, which otherwisewould be difficult or impossible to understand. We willillustrate these with two examples.

Let (M, , ) be a Poisson manifold and assume thatthe Poisson bracket vanishes at a point x0. This meansthat in the local form given by Theorem 1.1 there isonly the second term (no (p, q) coordinates), so for xclose to x0 we have:

yi, yj(x) = ckijyk + · · ·

where the dots represent higher order terms. Here theckij are just the structure constants of the isotropy Liealgebra gx0 = T ∗x0M in the basis dx0y1, . . . ,dx0ym.The linearization problem asks for new local coordinateswhere the higher order terms vanish (see [8] for a surveyof this problem). We have the following deep theorem:

Theorem 4.1 (Conn [2, 3]). If the isotropy Lie al-gebra is semisimple of compact type then there existslinearizing coordinates.

The original proof due to Conn is a hard analysis proofbased on the Nash-Moser method. He constructs suc-cessive coordinate systems which give better approx-imations to the linearizing coordinates and which doconverge to the linearizing coordinates. Using the sym-plectic groupoid Σ(M), Crainic and the author gave asoft geometric proof of this result, along the followinglines:

• The hypothesis of the theorem is equivalent to theisotropy Lie group Σ(M,x0) being a compact, 1-connected Lie group.

18

• By Reeb stability, the s-fibers are compact andQ-homological 2-connected. Hence the groupoidΣ(M) is a proper 2-groupoid with 2-connectedfibers.

• By the vanishing cohomology theorem of Crainic[4], the differentiable groupoid cohomology ofΣ(M) vanishes. By the Van Est Theorem (see[4]), it follows that the 2nd Poisson cohomologygroup H2Π(M) also vanishes.

• The vanishing of H2Π(M) allows one to applya contravariant version of Moser’s Path Method(see [9]) to obtain the linearizing coordinates.

Notice how this proof makes clear the relevance of theassumption, while in the original proof the assumptionis hidden in the analysis (it is used to build certainnorms necessary for the Nash-Moser method to work).

The previous example was about local properties of aPoisson manifold. Let us give a different example whereboth global and local properties are present. We claimthat the following result holds:

Theorem 4.2. If a Poisson manifold (M, , ) inte-grates to a compact symplectic groupoid Σ(M) then thePoisson bracket cannot vanish at any point.

Proof. Let us assume that the Poisson bracket vanishesat some point x0. Just like we observed in the proofabove, it follows that Σ(M) is a compact groupoidwith 2-connected fibers, and from that we conclude thatH2Π(M) vanishes. Now the Poisson bracket always de-fines a class [Π] ∈ H2Π(M), and hence this class mustbe trivial. At the level of the groupoid, this means thatthe cohomology class [ω] ∈ H2(Σ(M)) of the symplecticform in Σ(M) is trivial. But this is not possible since,by assumption, Σ(M) is a compact manifold.

As examples of Poisson manifolds (M,π) with Σ(M)compact, we can take compact symplectic manifoldswith finite fundamental group. I don’t know of anyother examples, and either they do not exist or theywill provide an extremely interesting class of Poissonmanifolds. So I believe it is important to solve the fol-lowing:

Open Problem. Are there (non-symplectic) Pois-son manifolds which integrate to a compact symplecticgroupoid Σ(M)?

References

[1] A.S. Cattaneo and G. Felder, Poisson sigma mod-els and symplectic groupoids, Quantization of sin-gular symplectic quotients, Progr. Math., vol. 198,Birkhauser, Basel, 2001, pp. 61–93.

[2] J. Conn, Normal forms for analytic Poisson struc-tures, Annals of Math. 119 (1984), 576–601.

[3] , Normal forms for smooth Poisson struc-tures, Annals of Math. 121 (1985), 565–593.

[4] M. Crainic, Differentiable and algebroid coho-mology, van Est isomorphisms, and characteristicclasses, Comment. Math. Helv. 78 (2003), no. 4,681–721.

[5] M. Crainic and R.L. Fernandes, Integrability of Liebrackets, Ann. of Math. (2) 157 (2003), no. 2, 575–620.

[6] , Integrability of Poisson brackets, J. Dif-ferential Geom. 66 (2004), no. 1, 71–137.

[7] R.L. Fernandes, Connections in Poisson geome-try. I. Holonomy and invariants, J. DifferentialGeom. 54 (2000), no. 2, 303–365.

[8] R.L. Fernandes and P. Monnier, Linearization ofPoisson brackets, Lett. Math. Phys. 69 (2004), 89–114.

[9] D. McDuff and D. Salamon, Introduction to sym-plectic topology, 2nd edition, Oxford MathematicalMonographs. Oxford University Press, New York,1998.

[10] A. Weinstein, The local structure of Poisson man-ifolds, J. Differential Geometry 18 (1983), 523–557.

19

Math in the Media

Originally published by the American Mathematical Society in MATH in the MEDIA, a section of the AMS Website,www.ams.org/mathmedia. Reprinted with permission.

“A Helix with a Handle”. That’s the title ofFenella Saunders’ piece in the May-June 2006 Amer-ican Scientist. The subject is what its authors –Mathias Weber (Indiana), David Hoffman (Stanford)and Michael Wolf (Rice) – describe as “the firstproperly embedded minimal surface with infinite to-tal curvature and finite topology to be found since1776, when Meusnier showed that the helicoid wasa minimal surface.” (Their paper, which appeared inthe November 15, 2005 PNAS, is available online –www.indiana.edu/~minimal/research/helicoid.pdf).In fact the new surface is closely related to Meusnier’s;it is properly described as a helicoid with a handle, or a“genus-one helicoid”, and is asymptotic to the helicoidat infinity.

The helicoid and the genus-one helicoid. This picture shows a

segment of a cylindrical core through each of the surfaces, which

actually extend to infinity in every direction. Image courtesy of

Indiana University.

Saunders tries to start her readers off gently: “Dip aloop of wire into a soapy solution, and the film that cov-ers the loop will be what mathematicians call a minimalsurface.” But soon we hear: “At any point, a minimalsurface is maximally curved in one direction and mini-mally curved in the opposite direction, but the amountof curvature in each direction is exactly the same.” Thereaders may have better luck with the project’s inter-esting history. “Over a decade ago, Hoffman, withFusheng Wei ... and Hermann Karcher ... , had cre-ated computer simulations of such handled helicoids,but an airtight demonstration of minimal surfacehoodeluded them.” They knew what it looked like, but they

could not prove that it really was an embedded min-imal surface. Saunders quotes Hoffman: “I think theinformation about how to solve this problem was lurk-ing in the pictures all the time, but we just had to thinkabout it for a long time and have the theory catch upwith the evidence we had.”

Dynamics of Roach Congregation. “Group-livinganimals are often faced with choosing between one ormore alternative resource sites.” Thus begins the ab-stract of a paper published April 11, 2006 in the Pro-ceedings of the National Academy of Sciences (1055835-5840). The authors, a French-Belgian team led byJean-Marc Ame and Jose Halloy, report on “an exper-imental and theoretical study of groups of cockroaches(Blattella germanica) tested in a circular arena ... withidentical shelters.” When the number of shelters is two,the phenomenon can be described by the graph below,giving the occupancy of shelter 1 as a function of sheltersize. Until the size of a shelter is enough for the wholepopulation, the roaches split between the two shelters.But as soon as there is room for everyone in each of theshelters, then the roaches all occupy one and not theother.

Occupancy of two shelters as a function of shelter size S for a

fixed number N of individuals. When S/N is less than .5, both

shelters are filled; for S/N between .5 and 1, the animals split

evenly between the shelters; if S/N is 1 or more, all the animals

congregate in one of the shelters. Adapted from PNAS 105

5835-5840, from which the equations below are taken. Image

courtesy of Jose Halloy ULB.

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This behavior is predicted by a mathematical model.First, the researchers determined from experiment thatthe probability Qi of an individual leaving shelter ivaries inversely with the crowdedness (the ratio of thenumber xi of animals in the shelter to the shelter sizeS):

Qi =θ

1 + ρ(

xi

S

)n

where θ, ρ and n are experimentally derived param-eters. On the other hand, the probability Ri for anexploring cockroach to join shelter i decreases linearlywith the crowdedness:

Ri = µ(1− xi

S

)where µ is experimentally derived. These two laws canbe combined into a system of differential equations:

dxi

dt= µxe

(1− xi

S

)− θxi

1 + ρ(

xi

S

)n

(here xe is the number of unattached individuals) sub-ject to the constraint

xe + x1 + x2 + ...+ xp = N

if there are p shelters and a total of N individuals. Theauthors solved numerically for the steady states, whichfor p = 2 appear schematically in the graph above.Larger numbers of shelters give more complex bifurca-tion. The authors explain why this collective behaviorgives the optimal outcome for each individual roach,and speculate that “the collective decision-making pro-cess studied here should have its equivalent in manygregarious animals ... . ” This work was picked up inthe April 6 2006 Nature Research Highlights.

Mathematical Incompleteness in the ScientificAmerican. The March 2006 Scientific American fea-tures a report by Gregory Chaitin, entitled “The Lim-its of Reason,” describing his own work on the incom-pleteness of mathematics. “Unlike Godel’s approach,mine is based on measuring information and showingthat some mathematical facts cannot be compressedinto a theory because they are too complicated” andthat therefore “... a theory of everything for all ofmathematics cannot exist.” Chaitin outlines his theory,including the irreducible number Omega: the first Ndigits of Omega cannot be computed using a programsignificantly shorter than N bits long. He sketches theargument that computing the first N binary digits ofOmega would solve the halting problem for all programsof length up to N ; so the uncomputability of Omega fol-lows from Turing’s proof of the unsolvability of the halt-ing problem. It follows from its definition that “an infi-nite number of bits of Omega constitute mathematical

facts ... that cannot be derived from any principles sim-pler than the string of bits itself. Mathematics there-fore has infinite complexity, whereas any individual the-ory of everything would only have finite complexity andcould not capture all the richness of the full world ofmathematical truth.” Chaitin then spends some timepondering the scientific and philosophical consequencesof his work. “Irreducible principles – axioms – have al-ways been part of mathematics. Omega just shows thata lot more of them are out there than we suspected.”“If Hilbert had been right, ... there would be a static,closed theory of everything for all of mathematics, andthis would be like a dictatorship. ... I much prefer anopen system. I do not like rigid, authoritarian waysof thinking.” “Extensive computer calculations can beextremely persuasive, but do they render proof unnec-essary? Yes and no. In fact, they provide a differentkind of evidence. In important situations, I would ar-gue that both kinds of evidence are required, as proofsmay be flawed, and conversely computer searches mayhave the bad luck to stop just before encountering acounterexample that disproves the conjectured result.”

Fractals finger suspect Pollocks. Alison Abbottreports in the February 9 2006 Nature on a new mathe-matical development in the saga of the 32 small “possi-ble Pollocks” recently discovered on Long Island. Largepoured works by Jackson Pollock bring prices in thetens of millions of dollars; if these paintings are au-thentic they are very valuable. But their authenticity,accepted by some experts, has been challenged by oth-ers. Enter the physicist Richard Taylor. Taylor hadpublished in 1999 (Nature 399 422) his group’s discov-ery that Pollock’s poured works showed (as Abbott ex-plains it) “two distinct sets of fractal patterns. Onewas on a scale larger than 5 cm; the other showed upon scales between 1 mm and 5 cm.” and furthermore“that the fractal dimension of Pollock’s works ... in-creased through the years as the artist refined his tech-nique.” In a later experiment, he analyzed “14 Pollockpaintings, 37 imitations created by students at the Uni-versity of Oregon, and 46 paintings of unknown origin.”Abbott quotes Taylor: “The only shared thing in Pol-lock’s very different poured paintings is a fractal com-position that was systematic through the years.” Thenon-Pollocks, when they had fractal structure, had dif-ferent fractal characteristics. So it was natural for theKrasner-Pollock foundation to send six of the putative32 for Taylor to examine. His diagnosis: “I found sig-nificant deviation from Pollock’s characteristics.” Thefoundation’s final judgment has not yet been promul-gated. The Nature piece has several echoes in the NewYork Times. It gets picked up as a news item by RandyKennedy (“Computer Analysis Suggests Paintings AreNot Pollocks”) on February 9. Their art critic MichaelKimmelman weighs in with “A Drip by Any OtherName” on February 12: “... the curious truth is that

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while a few drips and splashes can imitate Pollock’stouch ... it is nearly impossible to replicate ... the full-scale complex rhythms and overlapping patterns, theall-over, depthless, balletic and irregular space he cre-ated.” And then on the February 19 Op-Ed page, Pro-fessor Don Foster (English, Vassar, “Mind Over Splat-ter”) brings us an academic perspective. He starts outserious: “At the heart of the controversy lie criticalquestions about artistic meaning and value that havevexed literary scholars no less than art historians.” Buthe leaves us with: “Meanwhile, Jackson Pollock may bechuckling in his grave: if the object of Abstract Expres-sionist work is to embody the rebellious, the anarchic,the highly idiosyncratic – if we embrace Pollock’s workfor its anti-figurative aesthetic – may faux-Pollock notbe quintessential Pollock? May not a Pollock forgerythat passes for authentic be the best Pollock of all?”

The differential geometry of quantum compu-tation. “Quantum computers have the potential tosolve efficiently some problems that are considered in-tractable on conventional classical computers.” This isthe start of “Quantum Computation as Geometry,” areport in the February 4 2006 issue of Science. Theauthors are four members of the School of Physical Sci-ences, University of Queensland; a team led by MichaelNielsen. They continue: “Despite this great promise,as yet there is no general method for constructing goodquantum algorithms, and very little is known about thepotential power (or limitations) of quantum comput-ers.” What they propose in this report is an alternativeapproach to understanding the difficulty of an n-qubitcomputation, i.e. the complexity of the quantum al-gorithm that would be needed to carry it out. Sucha computation corresponds to a unitary operator U (a2n × 2n matrix with complex entries). The authors’definition of difficulty is the length d(I, U) of the short-est path from the identity matrix to U , where length ismeasured in a metric which penalizes all computationalmoves which require gates with more than two inputs.They show that this distance is “essentially equivalentto the number of gates required to synthesize U .” “Ourresult allows the tools of Riemannian geometry to beapplied to understand quantum computation. In par-ticular we can use a powerful tool – the calculus ofvariations – to find the geodesics of the space.” Theyremark that thinking of an algorithm as a geodesic “isin contrast with the usual case in circuit design, eitherclassical or quantum, where being given part of an opti-mal circuit does not obviously assist in the design of therest of the circuit.” Finally they show how “to constructexplicitly a quantum circuit containing a number of [oneand two-cubit] gates that is polynomial in d(I, U) andwhich approximates U closely.”

Chaos in the deep. “Reduced mixing generates oscil-

lations and chaos in the oceanic deep chlorophyll max-imum” appeared in the January 19 2006 Nature. Theauthors, an Amsterdam-Honolulu collaboration led byJef Huisman and Nga N. Pham Thi, investigated thestability of deep chlorophyll maxima (DCMs) – layers ofhigh concentration of phytoplankton who flourish wherethere are sufficient nutrients welling up from the bot-tom and sufficient light filtering down from the top. Thepoint of the article: “we extend recent phytoplanktonmodels to show that the phytoplankton populations ofDCMs can show sustained fluctuations.” The authorsset up a mathematical model, a reaction-advection-diffusion equation for the phytoplankton populationdensity P coupled to a partial differential equation forthe nutrient availability N . A common parameter inboth equations is the “turbulent diffusivity” κ , thecoefficient of the second-derivative terms. If κ is suffi-ciently large, “nutrients in the top layer are graduallydepleted by the phytoplankton. The nutricline movesdownwards, tracked by the phytoplankton population,until the population settles at a stable equilibrium atwhich the downward flux of consumed nutrients equalsthe upward flux of new nutrients.” To investigate thebehavior for lower κ, the authors ran “numerous sim-ulations using a wide range of turbulent diffusivities.”“The model simulations predict that the DCM becomesunstable when turbulent diffusivity is in the lower endof the realistic range. By a cascade of period doublings,reduced turbulent mixing can even generate chaos in theDCM.”

The numerical solution of the coupled P −N differential

equations shows bifurcation and eventually chaos as the mixing

parameter is decreased. This is a close-up picture of the

evolution of the local maxima and minima of the phytoplankton

population as a function of turbulent diffusivity, near the low

end of the realistic range 0.1 < κ < 1. Image from Nature 439

324, used with permission.

Their explanation for the periodic behavior: if κ is low,the phytoplankton sink faster than the nutrients arewelling up; without sufficient light their numbers de-cline. This lets more nutrients through up to more lu-minous layers, and “fuels the next peak in the DCM.”

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An ominous note: “Climate models predict that globalwarming will reduce vertical mixing in the oceans.”

Plant growth and the Golden Ratio, re-evaluated. The Botanical Journal of the LinnaeanSociety ran in its January 2006 issue an article byTodd Cooke (Maryland) with the title: “Do Fibonaccinumbers reveal the involvement of geometrical impera-tives or biological interactions in phyllotaxis?” From theabstract: “This paper reviews the fundamental prop-erties of number sequences, and discusses the under-appreciated limitations of the Fibonacci sequence fordescribing phyllotactic patterns.” Apparently golden-ratio giddiness has spread to botany, and this paperaims to be a corrective. Prof. Cooke’s main point isthat although “it is inescapable that the spiral phyl-lotaxes of vegetative shoots are overwhelmingly char-acterized by low Fibonacci numbers,” the common be-lief that “such spiral arrangements are attributable tothe leaf primordia being positioned in optimal pack-ing” must be questioned and ultimately rejected. Theargument, as I understand it, runs as follows. Sup-pose consecutive primordia are arranged at an exactlyconstant angular difference. If that difference is ex-actly the golden angle, here given as 137.5, then onedoes indeed achieve optimal packing. But “even slightvariation from the Fibonacci angle disrupt[s] optimalpacking.” E.g. constant angle 137.45 or constant angle137.92 don’t work. “It is difficult, if not impossible, toimagine any biological system being capable of organiz-ing itself with such discriminating accuracy as a directresponse to a hypothetical geometrical imperative foroptimal packing. It seems more likely that the spiralphyllotaxes observed ... are the outcome of some bio-logical process, the consequence of which is that suchstructures tend to approach optimal packing.” Thereare two points here. The mathematical one is shaky.The golden ratio is (supremely) irrational, and the evi-dence for its occurrence in the likeliest interval betweenconsecutive primordia (viz., the appearance of numbersof spirals corresponding to its rational approximators2/3, 3/5, 5/8, etc.) is excellent. On the other hand thequestion whether optimal packing is an “imperative” ora “consequence” does not seem to me to be one that sci-ence can answer. The end of this article addresses theidentification of the biological process governing phyl-lotaxis. Cooke refers to the 1992 Physics Review Letterspaper (68, 2089-2010) by Stephane Douady and YvesCouder, where they “managed to create spiral phyl-lotaxis on a lab bench” working with mutually repellingferrofluid drops floating on silicon oil in a varying mag-netic field. Presumably something analogous is happen-ing at the growing tip of a plant. “The ... mechanism... appears to involve the interaction of mathematicalrules, generating process, and overall geometry. In par-ticular, it seems quite plausible that the mathematicalrules for phyllotaxis arise from local inhibitory interac-

tions among existing primordia. These interactions areapparently mediated by the expression of specific geneswhose products regulate growth hormones ... .” Thiswork was picked up in the “Research Highlights” of theFebruary 9 Nature.

Cuboctahedral vesicles in eukaryotic cells. A eu-karyotic cell is a complex, three-dimensional organism.Just as our food is ingested in one place and movedto another for processing, with the nutrients then fer-ried about the body by the bloodstream, so in a cell’sinternal economy a critical role is played by transporta-tion. The agents of intracellular transport are vesicles:molecular cages that enclose their cargo and move itfrom A to B. A paper in the January 12 2006 Na-ture explores the structure of one type of vesicle: thosewhose skin is made from the coat protein complex II,or COPII. The authors (a Scripps Research Instituteteam of 8, led by Scott Stagg) explain that the struc-tural part of COPII consists of a lattice formed by theprotein complex Sec13/31.

Part of a micrograph of Sec13/31 cages preserved in vitreous ice.

The cages are approximately 600A(0.06 microns) in diameter;

their images show the planar projection of their cuboctahedral

structure. Image from Nature 439 235, used with permission.

Using electron cryo-microscopy, they determined thatthe most elementary cages formed by Sec13/31 havethe structure of a cuboctahedron, but they suggestthat in order to enclose larger cargoes, the same unitscould organize into the small rhombicuboctahedron,the icosidodecahedron or the small rhombicosidodec-ahedron. These semi-regular solids all share with thecuboctahedron (and the octahedron) the property thatfour edges meet at each vertex. The condition corre-sponds to the assymmetry in the molecular realizationof the Sec13/31 complex: the two ends are different,so it cannot assemble into a network with odd-orderedvertices.

23

The three axes of symmetry of the Sec13/31 cage. Each edge is

a Sec13/31 protein complex. The color (blue-green-yellow)

encodes distance from the cage center. Note the assymmetry in

the edges. Cage diameter approximately 600A. Image from

Nature 439 236, used with permission.

Deal or No Deal? The New York Times ran a“Critic’s Notebook” column by Virginia Heffernan onDecember 24, 2005. The subject was the popular gameshow “Deal or No Deal,” and the title was “A GameShow for the Probability Theorist in Us All.” Here’show the game works (you can test it out on the NBCwebsite (www.nbc.com/Deal or No Deal/game) – clickon “Start game!” once it has uploaded).

• Twenty-six known amounts of money, rangingfrom one cent to one million dollars, are (symbol-ically) randomly placed in 26 numbered, sealedbriefcases. The contestant chooses a briefcase.The unknown sum in the briefcase is the contes-tant’s.

• In the first round of play, the contestant chooses 6of the remaining 25 briefcases to open. Then the“banker” offers to buy the contestant’s briefcasefor a sum based on its expected value, given theinformation now at hand, but tweaked sometimesto make the game more interesting. The contes-tant can accept (“Deal”) or opt to continue play(“No Deal”).

• If the game continues, 5 more briefcases areopened in the second round, another offer is made,and accepted or refused. If the contestant con-tinues to refuse the banker’s offers, subsequentrounds open 4, 3, 2, 1, 1, 1, 1 briefcases until onlytwo are left.

• The banker makes one last offer; the contestantaccepts that offer or takes whatever money is inthe initially chosen briefcase.

The psychology is what makes the game fun. As Hef-fernan explains: “So far, no game theorist from theInstitute for Advanced Study has appeared to try his

hand at ‘Deal or No Deal’ and play as a cool-headed ra-tionalist. Instead the players on the American show are,like most game-show contestants, hysterics.” In fact thethree scientists at Erasmus University who conductedan exhaustive analysis of the decisions made by con-testants in the Dutch version of the game (jackpot 5million Euros) remark that “For analyzing risky choice,‘Deal or No Deal’ has a number of favorable designfeatures. The stakes are very high: ... the game showcan send contestants home multimillionaires – or prac-tically empty-handed. Unlike other game shows, ‘Dealor No Deal’ involves only simple stop-go decisions thatrequire minimal skill or strategy. Also, the probabilitydistribution is simple and known with near-certainty.Because of these features, ‘Deal or No Deal’ seems well-suited for analyzing real-life decisions involving realand large risky stakes.” Their report is available online(papers.ssrn.com/sol3/papers.cfm?abstract id=636508).

Mathematical patterns in asthma attacks. Amathematically powered breakthrough in the study ofthe incidence of asthma attacks, with potentially im-portant therapeutic implications, was reported in theDecember 1 2005 Nature. Urs Frey (University Hospitalof Berne) works in pediatric respiratory medicine; BelaSuki (Boston University) is a physicist who “analysescomplex nonlinear systems, such as the factors that con-tribute to avalanches” (quote from an “Authors” sketchat the beginning of the issue). With their collaborators,they analyzed the records of a “previously published,randomised, placebo-controlled, double-blind crossoverstudy” following 80 asthmatic subjects for 3 six-monthtreatment periods. In that study, the PEF (peak expi-ratory flow) of each subject was measured twice daily;the subject was also assigned a daily asthma symptomscore. The team’s strategy was to “examine whetherthe statistical and correlation properties of the timeseries of PEF recordings can be used to predict therisk of subsequent exaggeration of airway instability.”They can. To disentangle the effects the authors cre-ated a “nonlinear stochastic model of the PEF fluctua-tions” (“a cascade connection of a linear dynamic sys-tem followed by a second order nonlinear system withno memory. ...”) They were able to tune this modelto match the statistical characteristics of the experi-mental data, and then use it to measure the impact ofthe characteristics separately. One startling conclusionfrom their analysis is that short-acting bronchodilators,such as the popular drug albuterol, can actually aggra-vate medium-term risk of an asthmatic attack.

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An Interview with Marcelo Viana

First, we congratulate you for being awarded, recently,the first ever Ramanujan Prize, which “distinguishesyoung mathematicians for conducting outstanding re-search in developing countries”. What does this prizemean to you ?

It was a great pleasure and honor to be awarded thefirst Ramanujan Prize, of course. The prize is a greatrecognition of the quality of what I, and people close tome, have been doing. I do perceive it as a distinctiongranted to the individual, but I am aware that it alsopays tribute to the Mathematics that is being done inBrazil and, more broadly, in Latin America.

Marcelo Viana after receiving the Ramanujan Prize for 2005(ICTP, Trieste, Italy, December 15, 2005).

Photo: ICTP Photo Archives, Massimo Silvano.

How is it to pursue a top mathematics career in a de-veloping country ?

It is really very exciting. Economical, social, and edu-cational conditions are much less favorable than in de-veloped countries. That is frustrating sometimes, butit also means that our work must have a broader scopethan in other parts of the world. It is not only aboutproving theorems and teaching graduate courses, wemust be involved in improving education at all levels,

spreading good mathematical activity across the coun-try, helping bring mathematical “literacy” to everyone.That is what it makes it so exciting: there are manythings to be done, many opportunities to make a realimpact.

Your main workplace has been the IMPA in Rio deJaneiro. You are now its Deputy Director. Can youdescribe what makes this institute a very special place ?

IMPA is a small institute, with a very well defined mis-sion, which is to create mathematics, to train high levelresearchers, and to disseminate mathematical knowl-edge at all levels. Since it was founded, in 1952, IMPAhas developed a tradition of scientific excellency that, Ibelieve, is now part of its essence. Being small meansthere is a very personal contact between faculty stu-dents and staff, which adds to the great scientific am-biance, as do the many visitors coming around the year.And, being located in one of the nicest places I have everseen, between the Botanical Garden and what is said tobe the largest urban forest in the world, can only help,right ?

What has been the impact of the activities of IMPA forMathematics in Brazil and, more generally, in LatinAmerica ?

For one thing, IMPA is the main graduate school inMathematics in Latin America. It has granted morethan 250 doctoral degrees and nearly 500 master de-grees so far. Most of these students came from LatinAmerican countries, including all the regions of Brazil,and what they have been doing for the developmentof Mathematics in the continent can not be overesti-mated. Most university departments in Brazil includeformer students from IMPA, as do many institutionsin Argentina, Chile, Colombia, Cuba, Ecuador, Mex-ico, Peru, Uruguay, Venezuela. More recently, we havebeen attracting students from Bolivia and Paraguay aswell.

Incidentally, Portugal is among the countries with thelargest number of researchers with a doctoral degreefrom IMPA: I remember 10, at least.

Another important aspect are our collaboration agree-ments with several universities in Brazil. Through theseagreements, IMPA and its faculty support the graduateteaching, help organize meetings and Summer Schools,and interact scientifically with researchers in those insti-tutions. In the Latin American scenario we have strong

25

ties with the most important centers, both directly andthrough UMALCA, the Latin American MathematicalUnion. Researchers from IMPA have been very activein organizing Schools, as a means to attract the beststudents of the region to a career in Mathematics, andother scientific events in various countries in the re-gion. For instance, I have just come back from Santi-ago de Chile, where I helped organize the first Inter-national Congress on the Applications of Mathematics,held jointly by UMALCA, the European Mathemati-cal Society - EMS, and the Society for Industrial andApplied Mathematics - SIAM.

Do institutes like IMPA influence the way mathemati-cians do research ?

To some extent, yes. To begin with, we keep a vigor-ous visiting program. Around the year and, specially,during our Summer Program in January-February, wehost a large number of researchers, both from Braziland from abroad. This gives them a chance to get awayfrom their daily obligations and benefit from the work-ing conditions at IMPA, which are great.

There is also a certain style of doing Mathematics.A Dutch colleague once told me that interacting withcolleagues from IMPA he discovered that “research isfun!”. I have heard similar comments from other col-leagues, and this is something I try to convey to myown students. Successfully, I have reasons to believe.

There is yet another point. Recently, a famous Ameri-can mathematician wrote that a place like IMPA is “abeacon to the mathematics community” in the develop-ing world. I do believe that letting our talented youthknow that personal and professional realization is pos-sible in our countries through Mathematics may be oneof the most beneficial influences.

Let us now talk a little about you. When did you firstbecome interested in Mathematics ? What was it aboutMathematics that attracted you ?

It happened gradually in school. We moved quite a lotwhen I was a kid, because my mother was a teacherand she was appointed in various places by the govern-ment. Eventually we settled down in Povoa de Varzim,which is where I went to high school, in the Liceu Ecade Queiros. I was quite successful in most topics, but Ihad a clear preference for exact sciences. Among them,Mathematics was the “neatest”, the most “proper”. SoI was more and more inclined to it and when the timecame to make a definite decision, that was easy.

Were any people or events particularly influential inyour choice of Mathematics as a career ? Were thereteachers who made a particular difference ?

I always had very good Mathematics teachers, whoadded to the general feeling that everything about thesubject was quality stuff. In that sense they all were in-fluential in my picking Mathematics as my first choicewhen I applied for college.

At later stages, Prof. Arala Chaves, from the Univer-sity of Porto, and Prof. Jacob Palis, from IMPA, playedkey parts.

You were born in Brazil but moved with your parentsto Portugal almost immediately after. Can you tell ussomething about your youth in Portugal ? How wereyou as a student ?

We moved around the North of Portugal for a while,then we settled down in Povoa de Varzim, which iswhere my parents are from. As I said, I was gener-ally quite successful at school. This is not always agood thing, because somehow it puts you a little bitapart from the others. At those ages, school is the fore-most ground for social interaction and I was eager toestablish friendships.

Did you find that your undergraduate education in Por-tugal prepared you well academically ?

Yes, definitely. The education I got was quite solid andI did not experience any particular difficulties when Imoved to graduate school. I believe it would have beenthe same if I had joined any other graduate school, inany part of the world. But I did find out later that partof my education was somehow out of touch with “real-ity”. Many advanced topics I was taught were not ascentral as they had been, and could have been replaced.Also, I came to miss a more “experimental” approach tomathematical knowledge. These things have improvedsubstantially over the last two decades, because there isnowadays much more research activity in Portugal. ButI still notice in some of my Portuguese students a ten-dency to view mathematical issues as finished objects,a difficulty at realizing that (re)formulating questionsis an important part of the game.

After graduating from the University of Porto, you gotyour PhD under Jacob Palis at IMPA, with a thesis en-titled “Strange attractors in higher dimensions”. Howdid you first become interested in Dynamical Systems ?What fascinates you in the area ?

That I chose Dynamical Systems as my research topicwas primarily due to Prof. Manuel Arala Chaves, fromthe University of Porto. He introduced me and my col-leagues to the subject in the last year of the Licen-ciatura, at the Faculdade de Ciencias do Porto. Hedirected me to visit the Ecole Normale Superieure inParis, in 1984, and it was also at his advice that I at-tended a scientific meeting at the University of Coimbrain 1985, that was a crucial event in my career.

26

At that meeting in Coimbra I met Prof. Jacob Palis,who would become my doctoral advisor. He convincedme to come to IMPA, and is largely responsible for myremaining here after two decades (rather than the twoor three years I hoped to get away with, when I came).

After your PhD you decided to leave definitely Portugal.Why ? Did you notice big differences when you arrivedat IMPA?

I did come back to Portugal for a while, but somehowI did not feel happy. A good part of it was due topersonal reasons, to some important changes that weretaking place in my personal life. Somehow, it was diffi-cult to separate things. It is true that I already felt verywell at IMPA, professionally. But, all in all, personalissues ended up being crucial.

Mathematical research in Portugal is mostly based at theuniversities. Do you have any suggestions to improvethe working environment at universities ?

Science is done by individuals or groups of individuals,and even more so in the case of Mathematics. Institu-tions are often more of a nuisance, if I may say so. Inthe last couple of decades, Portugal has moved a gooddeal in the direction of acknowledging the role of indi-viduals and groups, for instance in the way funding isbeing distributed, and I have no doubt that was fun-damental for the excellent evolution experienced in thisperiod. I believe there is still room to proceed in thisdirection. A fine scientific ambiance is often the workof one or more people with strong leadership. As manyother institutions with a past and tradition, the Por-tuguese universities give limited room for the action ofsuch leaders.

I also gather from conversations that the teaching andadministrative load has been rising, at least in someinstitutions. Such duties are certainly part of a sci-entist’s job description, as are other academic activi-ties beyond the teaching plus writing papers binomial.However, creative thinking does demand a lot of timeand tranquility. A few years ago, during a meeting heldin Lisbon by the Minister for Science and Technology,I suggested the creation of a mechanism for “teachingbuy-outs” aimed, specially, at the most talented youngresearchers.

A common trace of your work has been the search forvery general theorems and global theories suitable forthe majority of dynamical systems. Is this the kind ofapproach to mathematics that attracts you (generaliza-tion instead of specialization ...) ?

In my experience, Mathematics is done from examples.I really do not understand those areas (and there area few honorable ones!) where experts are not able to

indicate simple interesting examples to motivate theirstatements. One of my favorite advices to the studentsis “do not try to solve the whole thesis problem at thesame time: look for an interesting particular case, andfocus on it for a while.” My own work has been muchdriven by examples, the Henon map, the Lorenz attrac-tor, DA diffeomorphisms, and so on. But I also believethe ultimate goal is to reach statements with the “right”level of generality. For instance, the Role Theorem wasfirst discovered for polynomials; the statement we nowfind in Calculus books is better, not because it is moregeneral, but because the result has nothing to do withthe algebraic structure, it is really only about differen-tiability.

Marcelo Viana delivering his lecture “Lorenz strange attractors”at the Math Colloquium of the University of Coimbra.

(February 14, 2006)

In an article in the book “Mathematics Unlimited —2001 and Beyond” (MU2001) you give an overview ofthe challenges that Dynamical Systems face in this cen-tury. What are some of the most significant unsolvedproblems in the area ?

In the 1960’s people believed that it would be possi-ble to give a global description of how “most” dynam-ical systems behave. Steven Smale, made a concreteproposal, inspired by his previous work in Morse the-ory: he believed that globally most dynamical systemslook very much like gradient flows. It was rapidly clearthat this was not really so: Smale himself gave the firstcounter-example! I think the most exciting questionsare: Can such a global picture be obtained, and how ?In the mid 1990’s, Palis proposed a new program tothis effect, and there has been a lot of progress. In thatarticle, I mentioned some important specific challengesin this direction.

One intriguing issue, which has been blocking progressfor quite some time, is the so-called closing lemma. In

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some cases, one can prove that if a dynamical systemhas a trajectory that is almost closed then, by modify-ing the system a little, one can cause the orbit to actu-ally close. There has been a great deal of effort to try toextend the validity of this statement, which would havetremendous implications. However, after four decades,the results are really meager. At this point, I think weshould change the strategy, and move on developing thefield without the closing lemma. There has been someprogress along these lines, specially in low dimensions.

In MU2001 you say that even Hilbert, in his 1900 fa-mous speech, could not foresee the birth and extraor-dinary development of Dynamical Systems (in spite oftwo of his problems — 16th and 21st — being related toit). Do you have any explanation for this ?

Hilbert’s list is mostly about specific mathematicalproblems. One exception is the sixth problem, wherehe asks for the axiomatization of Physics, and we haveto acknowledge that was premature, at the very best.Hilbert had a remarkable insight and, together withPoincare, he may have been the last “universal” math-ematician. But the kind of forces that led to the devel-opment of Dynamical Systems, very often arising fromthe experimental sciences, were not particularly closeto his heart (they were much closer to Poincare’s).

How would you describe Dynamical Systems to a lay-man ?

I usually describe it as the mathematical discipline thatstudies systems that evolve in time, in order to under-stand and predict this evolution. This definition is abit “imperialistic”, in the sense that it includes almostanything, but a few examples help make it more clear:the motion of planets, the evolution of ecological envi-ronments, the spreading of epidemics, fluids in motion.In all these, and many other systems, one would like tounderstand how and why they evolve, and how externalfactors may affect that evolution. Dynamical Systemsprovides the conceptual tools to model and predict theirbehavior.

You are in Coimbra to participate at the Annual Scien-tific Council meeting of CIM. What do you think aboutCIM and its role in the Portuguese mathematical com-munity ? and about Portuguese mathematics in gen-eral ?

I think it is very important for the country to have aninstitution such as CIM. Limited as its budget is, theCentro has been having a very positive effect at large,that goes much beyond the concrete activities it sup-ports. CIM also serves as a forum and meeting groundat the national level, which I find very important as thePortuguese mathematical community has grown bigger

and more sophisticated. I believe its influence and im-portance will steadily grow in the future.

You are also going to present a talk in Coimbra for thestudents of the Delfos programme. What advice do yougive to young students who are coming to like mathe-matics and to show real talent ?

Just do it. I often try to convey to young people thepleasures of personal and professional realization onecan attain through Mathematics, or Science in general,and how so very few professional activities can boastthe same.

Marcelo Viana delivering his Delfos Lecture at the

University of Coimbra (February 12, 2006).

How can we encourage young people to take up mathe-matics, especially in the schools ?

This is something we are very concerned in Brazil aswell. There are many mechanisms for attracting youngtalents to Mathematics. But they all involve havingresearchers and educators exposing students to Math-ematics in a way that displays its beauty and, at thesame time, gives the students the chance to exhibit theirown capacity to face challenges. We all know this is of-ten not the way Mathematics is presented in the lectureroom, unfortunately.

You are a superb lecturer and seem to enjoy very muchexplaining ideas to others which suggest that you liketeaching. What makes teaching fun for you ? What isyour way of teaching ?

Teaching is a great way to interact with people. I enjoyconducting the audience to the understanding of thesubject, withholding information till the right moment,disclosing the secrets when time is ripe. We mathemati-cians are trained to be rigorous, to do our best not to

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say anything that is wrong. In my opinion, that is amistake. Mathematical knowledge is ultimately storedin an orderly, rigorous fashion. But that is not the wayit is discovered and so it is not the way it ought to betaught.

Tell us about this mysterious link “Bola da Vez” on yourwebsite ?

The expression can not really be translated and I amnot even sure how much sense it makes in Portugal.But, this being Brazil, you may guess it has to do withfootball. It all started with an exchange of teasing re-marks between two of my students. As a result, thereis now an actual football that is passed along from onestudent to another. Whenever a student finishes, (s)hesigns and dates one of the hexagons, and gives the ballto the colleague who’s expected to finish next. This“honor” tends to be more appreciated when the thesisdefense is already in sight, of course. At some point Ithought it would be good to have a replica of that ballon my website, with the students photos.

What are your mathematical plans for the near future,what areas and problems have your attention at the mo-ment ?

I have several unfinished projects, about equilibriumstates, partially hyperbolic systems, linear cocycles, dy-namics of flows. My various coauthors kindly, and reg-ularly, remind me how upset they are we are takingso long. . . So, the first priority will be to finish theseprojects. Then I would like to go deeper into the studyof Teichmuller flows, on which I have been working overthe last couple of years. I would like to really get intoconservative Dynamical Systems, where one studies sys-tems that preserve a volume form. And I keep goingback to problems that I was unable to solve the firsttime.

What things interest you other than Mathematics ?

My family, of course. Including my five deliciousnephews and nieces. I also like reading and listeningto music. I tend to be conservative, and listen to thethings I already know and like, but I have been makingsome discoveries, specially about ancient and medievalmusic. I really like History. Not so much recent His-tory, but I have read about almost anything prior tothe XXth century. Oh yes! I try to do some physicalexercise, but my heart is not really into it.

Interview conducted by Maria Manuel Clementino and Jorge Picado (University of Coimbra)

Marcelo Viana is a Professor of Mathematics and Deputy Director of the Institute of Pure and Applied Mathematics(IMPA), Rio de Janeiro, Brazil.

Born in Rio de Janeiro on March 4, 1962, of Portuguese parents, he was educated in Portugal, for where he came atonly the age of three months. He received his B.S. from the University of Porto in 1984, where he held a position,and got his Ph.D. from the IMPA in 1990, with a thesis entitled “Strange attractors in higher dimensions”, writtenunder the direction of Jacob Palis. He was awarded a Guggenheim fellowship to work at UCLA and Princetonin 1993, and since then he has been a visitor to main centers around the world. Marcelo Viana is a well-knownmathematician worldwide in the fields of dynamical systems where he has made important contributions. His researchfocuses on dynamical systems, ergodic theory, and bifurcation theory. He has published more than 50 researcharticles in international journals (including Acta Mathematica, Annals of Mathematics, Inventiones Mathematicaeand Publications Mathematiques de l’IHES) and has already supervised seventeen Ph.D. thesis.

Marcelo Viana has both Portuguese and Brazilian nationalities. Although he works in Brazil, he keeps strong linkswith mathematics in Portugal, by participating in research evaluation panels, conferences and seminars, and bysupervising several Portuguese Ph.D. students. He is also a member of the Scientific Council of CIM.

Invited to give talks at two consecutive International Congresses of Mathematicians, Zurich-94 and Berlin-98 (theformer as Section Speaker and the latter as Plenary Lecturer) he is considered a superb lecturer. Besides, hewas also invited to give a Plenary Lecture at the International Congress on Mathematical Physics, (Paris, 1994).He is on the editorial board of seven mathematical journals: Ergodic Theory & Dynamical Systems, DynamicalSystems: An International Journal, Portugaliae Mathematica, Discrete and Continuous Dynamical Systems, Journalof Stochastics and Dynamics, Nonlinear Differential Equations and Applications, Dynamics of Partial DifferentialEquations.

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Among his several prizes and distinctions are The Mathematical Union for Latin America and the Caribbean Awardin Mathematics, 2000, the Great Cross of Scientific Merit, granted by the President of Brazil in 2000, Member ofthe Third World Academy of Sciences (elected in 2000), Third World Academy of Sciences Award in Mathematics,1998, Member of the Brazilian Academy of Sciences (elected in 1997).

Last December Marcelo Viana was the first recipient of the Ramanujan Prize, which honors a researcher, youngerthan 45 years old, who has conducted outstanding research in a developing country. The prize is funded by the NielsHenrik Abel Memorial Fund and it is awarded by the Abdus Salam International Centre for Theoretical Physics inconjunction with the International Mathematical Union.

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Editors: Antonio Caetano (acaetano@mat.ua.pt)Jorge Picado (picado@mat.uc.pt).

Address: Departamento de Matematica, Universidade de Coimbra, 3001-454 Coimbra, Portugal.

The CIM Bulletin is published twice a year. Material intended for publication should be sent to one of the editors.The bulletin is available at www.cim.pt.

The CIM acknowledges the financial support of FCT - Fundacao para a Ciencia e a Tecnologia.

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