EDITORS

    Jean-Pierre Franc ?oise

    Universite ′ P.-M. Curie, Paris VI

    Paris, France

    Gregory L. Naber

    Drexel University

    Philadelphia, PA, USA

    Tsou Sheung Tsun

    University of Oxford

    Oxford, UKEDITORIAL ADVISORY BOARD

    Sergio Albeverio

    Rheinische Friedrich-Wilhelms-Universita ¨ t Bonn

    Bonn, Germany

    Huzihiro Araki

    Kyoto University

    Kyoto, Japan

    Abhay Ashtekar

    Pennsylvania State University

    University Park, PA, USA

    Andrea Braides

    Universita ` di Roma ‘‘Tor Vergata’’

    Roma, Italy

    Francesco Calogero

    Universita ` di Roma ‘‘La Sapienza’’

    Roma, Italy

    Cecile DeWitt-Morette

    The University of Texas at Austin

    Austin, TX, USA

    Artur Ekert

    University of Cambridge

    Cambridge, UK

    Giovanni Gallavotti

    Universita ` di Roma ‘‘La Sapienza’’

    Roma, Italy

    Simon Gindikin

    Rutgers University

    Piscataway, NJ, USA

    Gennadi Henkin

    Universite ′ P.-M. Curie, Paris VI

    Paris, France

    Allen C. Hirshfeld

    Universita ¨ t Dortmund

    Dortmund, Germany

    Lisa Jeffrey

    University of Toronto

    Toronto, Canada

    T.W.B. Kibble

    Imperial College of Science, Technology and Medicine

    London, UK

    Antti Kupiainen

    University of Helsinki

    Helsinki, Finland

    Shahn Majid

    Queen Mary, University of London

    London, UK

    Barry M. McCoy

    State University of New York Stony Brook

    Stony Brook, NY, USA

    Hirosi Ooguri

    California Institute of Technology

    Pasadena, CA, USA

    Roger Penrose

    University of Oxford

    Oxford, UK

    Pierre Ramond

    University of Florida

    Gainesville, FL, USA

    Tudor Ratiu

    Ecole Polytechnique Federale de Lausanne

    Lausanne, Switzerland

    Rudolf Schmid

    Emory University

    Atlanta, GA, USA

    Albert Schwarz

    University of California

    Davis, CA, USAYakov Sinai

    Princeton University

    Princeton, NJ, USA

    Herbert Spohn

    Technische Universita ¨ tMu ¨nchen

    Mu ¨nchen, Germany

    Stephen J. Summers

    University of Florida

    Gainesville, FL, USA

    Roger Temam

    Indiana University

    Bloomington, IN, USA

    Craig A. Tracy

    University of California

    Davis, CA, USA

    Andrzej Trautman

    Warsaw University

    Warsaw, Poland

    Vladimir Turaev

    Institut de Recherche Mathe ′matique Avance ′e,Strasbourg, France

    Gabriele Veneziano

    CERN, Gene `ve, Switzerland

    Reinhard F. Werner

    Technische Universita ¨ t Braunschweig

    Braunschweig, Germany

    C.N. Yang

    Tsinghua University

    Beijing, China

    Eberhard Zeidler

    Max-Planck Institut fu ¨ r Mathematik in

    den Naturwissenschaften

    Leipzig, Germany

    Steve Zelditch

    Johns Hopkins University

    Baltimore, MD, USAFOREWORD

    I

    n bygone centuries, our physical world appeared to be filled to the brim with mysteries. Divine powers

    could provide for genuine miracles; water and sunlight could turn arid land into fertile pastures, but the

    same powers could lead to miseries and disasters. The force of life, the vis vitalis, was assumed to be the

    special agent responsible for all living things. The heavens, whatever they were for, contained stars and other

    heavenly bodies that were the exclusive domain of the Gods.

    Mathematics did exist, of course. Indeed, there was one aspect of our physical world that was recognised to

    be controlled by precise, mathematical logic: the geometric structure of space, elaborated to become a genuine

    form of art by the ancient Greeks. From my perspective, the Greeks were the first practitioners of ‘mathematical

    physics’, when they discovered that all geometric features of space could be reduced to a small number of

    axioms. Today, these would be called ‘fundamental laws of physics’. The fact that the flow of time could be

    addressed with similar exactitude, and that it could be handled geometrically together with space, was only

    recognised much later. And, yes, there were a few crazy people who were interested in the magic of numbers,but the real world around us seemed to contain so much more that was way beyond our capacities of analysis.

    Gradually, all this changed. The Moon and the planets appeared to follow geometrical laws. Galilei and

    Newton managed to identify their logical rules of motion, and by noting that the concept of mass could be

    applied to things in the sky just like apples and cannon balls on Earth, they made the sky a little bit more

    accessible to us. Electricity, magnetism, light and sound were also found to behave in complete accordance

    with mathematical equations.

    Yet all of this was just a beginning. The real changes came with the twentieth century. A completely new

    way of thinking, by emphasizing mathematical, logical analysis rather than empirical evidence, was pioneered

    by Albert Einstein. Applying advanced mathematical concepts, only known to a few pure mathematicians, to

    notions as mundane as space and time, was new to the physicists of his time. Einstein himself had a hard

    time struggling through the logic of connections and curvatures, notions that were totally new to him, but are

    only too familiar to students of mathematical physics today. Indeed, there is no better testimony of Einstein’s

    deep insights at that time, than the fact that we now teach these things regularly in our university classrooms.

    Special and general relativity are only small corners of the realm of modern physics that is presently being

    studied using advancedmathematicalmethods.We have notoriously complex subjects such as phase transitions in

    condensed matter physics, superconductivity, Bose–Einstein condensation, the quantum Hall effect, particularly

    the fractional quantum Hall effect, and numerous topics from elementary particle physics, ranging from fibre

    bundles and renormalization groups to supergravity, algebraic topology, superstring theory, Calabi–Yau spaces

    and what not, all of which require the utmost of our mental skills to comprehend them.

    The most bewildering observation that we make today is that it seems that our entire physical world

    appears to be controlled by mathematical equations, and these are not just sloppy and debatable models, but

    precisely documented properties of materials, of systems, and of phenomena in all echelons of our universe.

    Does this really apply to our entire world, or only to parts of it? Do features, notions, entities exist that are

    emphatically not mathematical? What about intuition, or dreams, and what about consciousness? What

    about religion? Here, most of us would say, one should not even try to apply mathematical analysis, although

    even here, some brave social scientists are making attempts at coordinating rational approaches.No, there are clear and important differences between the physical world and the mathematical world.

    Where the physical world stands out is the fact that it refers to ‘reality’, whatever ‘reality’ is. Mathematics is

    the world of pure logic and pure reasoning. In physics, it is the experimental evidence that ultimately decides

    whether a theory is acceptable or not. Also, the methodology in physics is different.

    A beautiful example is the serendipitous discovery of superconductivity. In 1911, the Dutch physicist Heike

    Kamerlingh Onnes was the first to achieve the liquefaction of helium, for which a temperature below 4.25 K

    had to be realized. Heike decided to measure the specific conductivity of mercury, a metal that is frozen solid

    at such low temperatures. But something appeared to go wrong during the measurements, since the volt

    meter did not show any voltage at all. All experienced physicists in the team assumed that they were dealing

    with a malfunction. It would not have been the first time for a short circuit to occur in the electrical

    equipment, but, this time, in spite of several efforts, they failed to locate it. One of the assistants was

    responsible for keeping the temperature of the sample well within that of liquid helium, a dull job, requiring

    nothing else than continuously watching some dials. During one of the many tests, however, he dozed off.

    The temperature rose, and suddenly the measurements showed the normal values again. It then occurred to

    the investigators that the effect and its temperature dependence were completely reproducible. Below 4.19

    degrees Kelvin the conductivity of mercury appeared to be strictly infinite. Above that temperature, it is

    finite, and the transition is a very sudden one. Superconductivity was discovered (D. van Delft, ‘‘Heike

    Kamerling Onnes’’, Uitgeverij Bert Bakker, Amsterdam, 2005 (in Dutch)).

    This is not the way mathematical discoveries are made. Theorems are not produced by assistants falling

    asleep, even if examples do exist of incidents involving some miraculous fortune.

    The hybrid science of mathematical physics is a very curious one. Some of the topics in this Encyclopedia

    are undoubtedly physical. High Tc superconductivity, breaking water waves, and magneto-hydrodynamics,are definitely topics of physics where experimental data are considered more decisive than any high-brow

    theory. Cohomology theory, Donaldson–Witten theory, and AdSCFT correspondence, however, are examples

    of purely mathematical exercises, even if these subjects, like all of the others in this compilation, are strongly

    inspired by, and related to, questions posed in physics.

    It is inevitable, in a compilation of a large number of short articles with many different authors, to see quite a

    bit of variation in style and level. In this Encyclopedia, theoretical physicists as well as mathematicians together

    made a huge effort to present in a concise and understandable manner their vision on numerous important

    issues in advanced mathematical physics. All include references for further reading. We hope and expect that

    these efforts will serve a good purpose.

    Gerardt Hooft,Spinoza Institute,Utrecht University,The Netherlands.PREFACE

    Mathematical Physics as a distinct discipline is relatively new. The International Association of

    Mathematical Physics was founded only in 1976. The interaction between physics and mathematics

    has, of course, existed since ancient times, but the recent decades, perhaps partly because we are living

    through them, appear to have witnessed tremendous progress, yielding new results and insights at a dizzying

    pace, so much so that an encyclopedia seems now needed to collate the gathered knowledge.

    Mathematical Physics brings together the two great disciplines of Mathematics and Physics to the benefit of

    both, the relationship between them being symbiotic. On the one hand, it uses mathematics as a tool to

    organize physical ideas of increasing precision and complexity, and on the other it draws on the questions

    that physicists pose as a source of inspiration to mathematicians. A classical example of this relationship

    exists in Einstein’s theory of relativity, where differential geometry played an essential role in the formulation

    of the physical theory while the problems raised by the ensuing physics have in turn boosted the development

    of differential geometry. It is indeed a happy coincidence that we are writing now a preface to an

    encyclopedia of mathematical physics in the centenary of Einstein’s annus mirabilis.

    The project of putting together an encyclopedia of mathematical physics looked, and still looks, to us a

    formidable enterprise. We would never have had the courage to undertake such a task if we did not believe,first, that it is worthwhile and of benefit to the community, and second, that we would get the much-needed

    support from our colleagues. And this support we did get, in the form of advice, encouragement, and

    practical help too, from members of our Editorial Advisory Board, from our authors, and from others as well,who have given unstintingly so much of their time to help us shape this Encyclopedia.

    Mathematical Physics being a relatively new subject, it is not yet clearly delineated and could mean

    different things to different people. In our choice of topics, we were guided in part by the programs of recent

    International Congresses on Mathematical Physics, but mainly by the advice from our Editorial Advisory

    Board and from our authors. The limitations of space and time, as well as our own limitations, necessitated

    the omission of certain topics, but we have tried to include all that we believe to be core subjects and to cover

    as much as possible the most active areas.

    Our subject being interdisciplinary, we think it appropriate that the Encyclopedia should have certain

    special features. Applications of the same mathematical theory, for instance, to different problems in physics

    will have different emphasis and treatment. By the same token, the same problem in physics can draw upon

    resources from different mathematical fields. This is why we divide the Encyclopedia into two broad sections:

    physics subjects and related mathematical subjects. Articles in either section are deliberately allowed a fair

    amount of overlap with one another and many articles will appear under more than one heading, but all are

    linked together by elaborate cross referencing. We think this gives a better picture of the subject as a whole

    and will serve better a community of researchers from widely scattered yet related fields.

    The Encyclopedia is intended primarily for experienced researchers but should be of use also to beginning

    graduate students. For the latter category of readers,we have included eight elementary introductory articles for easy

    reference, with those on mathematics aimed at physics graduates and those on physics aimed at mathematics

    graduates, so that these articles can serve as their first port of call to enable them to embark on any of the main

    articles without the need to consult other material beforehand. In fact, we think these articles may even form thefoundation of advanced undergraduate courses, aswe knowthat some authors have alreadymade such use of them.

    In addition to the printed version, an on-line version of the Encyclopedia is planned, which will allow both

    the contents and the articles themselves to be updated if and when the occasion arises. This is probably a

    necessary provision in such a rapidly advancing field.

    This project was some four years in the making. Our foremost thanks at its completion go to the members

    of our Editorial Advisory Board, who have advised, helped and encouraged us all along, and to all our

    authors who have so generously devoted so much of their time to writing these articles and given us much

    useful advice as well. We ourselves have learnt a lot from these colleagues, and made some wonderful

    contacts with some among them. Special thanks are due also to Arthur Greenspoon whose technical expertise

    was indispensable.

    The project was started with Academic Press, which was later taken over by Elsevier. We thank warmly

    members of their staff who have made this transition admirably seamless and gone on to assist us greatly in

    our task: both Carey Chapman and Anne Guillaume, who were in charge of the whole project and have been

    with us since the beginning, and Edward Taylor responsible for the copy-editing. And Martin Ruck, who

    manages to keep an overwhelming amount of details constantly at his fingertips, and who is never known to

    have lost a single email, deserves a very special mention.

    As a postscript, we would like to express our gratitude to the very large number of authors who generously

    agreed to donate their honorariums to support the Committee for Developing Countries of the European

    Mathematical Society in their work to help our less fortunate colleagues in the developing world.

    Jean-Pierre Franc ?oise

    Gregory L. Naber

    Tsou Sheung TsunPERMISSION ACKNOWLEDGMENTS

    The following material is reproduced with kind permission of Nature Publishing Group

    Figures 11 and 12 of ‘‘Point-vortex Dynamics’’

    http:www.nature.comnature

    The following material is reproduced with kind permission of Oxford University Press

    Figure 1 of ‘‘Random Walks in Random Environments’’

    http:www.oup.co.ukGUIDE TO USE OF THE ENCYCLOPEDIA

    Structure of the Encyclopedia

    The material in this Encyclopedia is organised into two sections. At the start of Volume 1 are eight Introductory Articles.

    The introductory articles on mathematics are aimed at physics graduates; those on physics are aimed at mathematics

    graduates. It is intended that these articles should serve as the first port of call for graduate students, to enable them to

    embark on any of the main entries without the need to consult other material beforehand.

    Following the Introductory Articles, the main body of the Encyclopedia is arranged as a series of entries in alphabetical

    order. These entries fill the remainder of Volume 1 and all of the subsequent volumes (2–5).

    To help you realize the full potential of the material in the Encyclopedia we have provided four features to help you find

    the topic of your choice: a contents list by subject, an alphabetical contents list, cross-references, and a full subject index.

    1. Contents List by Subject

    Your first point of reference will probably be the contents list by subject. This list appears at the front of each volume,and groups the entries under subject headings describing the broad themes of mathematical physics. This will enable the

    reader to make quick connections between entries and to locate the entry of interest. The contents list by subject is divided

    into two main sections: Physics Subjects and Related Mathematics Subjects. Under each main section heading, you will

    find several subject areas (such as GENERAL RELATIVITY in Physics Subjects or NONCOMMUTATIVE GEOMETRY

    in Related Mathematics Subjects). Under each subject area is a list of those entries that cover aspects of that subject,together with the volume and page numbers on which these entries may be found.

    Because mathematical physics is so highly interconnected, individual entries may appear under more than one subject

    area. For example, the entry GAUGE THEORY: MATHEMATICAL APPLICATIONS is listed under the Physics Subject

    GAUGE THEORY as well as in a broad range of Related Mathematics Subjects.

    2. Alphabetical Contents List

    The alphabetical contents list, which also appears at the front of each volume, lists the entries in the order in which they

    appear in the Encyclopedia. This list provides both the volume number and the page number of the entry.

    You will find ‘‘dummy entries’’ where obvious synonyms exist for entries or where we have grouped together related

    topics. Dummy entries appear in both the contents list and the body of the text.

    Example

    If you were attempting to locate material on path integral methods via the alphabetical contents list:

    PATH INTEGRAL METHODS see Functional Integration in Quantum Physics; Feynman Path Integrals

    The dummy entry directs you to two other entries in which path integral methods are covered. At the appropriate

    locations in the contents list, the volume and page numbers for these entries are given.

    If you were trying to locate the material by browsing through the text and you had looked up Path Integral Methods,then the following information would be provided in the dummy entry:

    Path Integral Methods see Functional Integration in Quantum Physics; Feynman Path Integrals3. Cross-References

    All of the articles in the Encyclopedia have been extensively cross-referenced. The cross-references, which appear at the

    end of an entry, serve three different functions:

    i. To indicate if a topic is discussed in greater detail elsewhere.

    ii. To draw the reader’s attention to parallel discussions in other entries.

    iii. To indicate material that broadens the discussion.

    Example

    The following list of cross-references appears at the end of the entry STOCHASTIC HYDRODYNAMICS

    See also: Cauchy Problem for Burgers-Type Equations; Hamiltonian

    Fluid Dynamics; Incompressible Euler Equations: Mathematical Theory;

    Malliavin Calculus; Non-Newtonian Fluids; Partial Differential Equations:

    Some Examples; Stochastic Differential Equations; Turbulence Theories;

    Viscous Incompressible Fluids: Mathematical Theory; Vortex Dynamics

    Here you will find examples of all three functions of the cross-reference list: a topic discussed in greater detail elsewhere

    (e.g. Incompressible Euler Equations: Mathematical Theory), parallel discussion in other entries (e.g. Stochastic Differ-

    ential Equations) and reference to entries that broaden the discussion (e.g. Turbulence Theories).

    The eight Introductory Articles are not cross-referenced from any of the main entries, as it is expected that introductory

    articles will be of general interest. As mentioned above, the Introductory Articles may be found at the start of Volume 1.

    4. Index

    The index will provide you with the volume and page number where the material is located. The index entries

    differentiate between material that is a whole entry, is part of an entry, or is data presented in a figure or table. Detailed

    notes are provided on the opening page of the index.

    5. Contributors

    A full list of contributors appears at the beginning of each volume.

    xii GUIDE TO USE OF THE ENCYCLOPEDIACONTRIBUTORS

    A Abbondandolo

    Universita ` di Pisa

    Pisa, Italy

    M J Ablowitz

    University of Colorado

    Boulder, CO, USA

    S L Adler

    Institute for Advanced Study

    Princeton, NJ, USA

    H Airault

    Universite ′ de Picardie

    Amiens, France

    G Alberti

    Universita ` di Pisa

    Pisa, Italy

    S Albeverio

    Rheinische Friedrich–Wilhelms-Universita ¨ t Bonn

    Bonn, Germany

    S T Ali

    Concordia University

    Montreal, QC, Canada

    R Alicki

    University of Gdan ′sk

    Gdan ′sk, Poland

    G Altarelli

    CERN

    Geneva, Switzerland

    C Amrouche

    Universite ′ de Pau et des Pays de l’Adour

    Pau, France

    M Anderson

    State University of New York at Stony Brook

    Stony Brook, NY, USA

    L Andersson

    University of Miami

    Coral Gables, FL, USA and Albert Einstein Institute

    Potsdam, Germany

    B Andreas

    Humboldt-Universita ¨ t zu Berlin

    Berlin, Germany

    V Arau ′ jo

    Universidade do Porto

    Porto, Portugal

    A Ashtekar

    Pennsylvania State University

    University Park, PA, USA

    W Van Assche

    Katholieke Universiteit Leuven

    Leuven, Belgium

    G Aubert

    Universite ′ de Nice Sophia Antipolis

    Nice, France

    H Au-Yang

    Oklahoma State University

    Stillwater, OK, USA

    M A Aziz-Alaoui

    Universite ′ du Havre

    Le Havre, France

    V Bach

    Johannes Gutenberg-Universita ¨ t

    Mainz, Germany

    C Bachas

    Ecole Normale Supe ′ rieure

    Paris, France

    V Baladi

    Institut Mathe ′matique de Jussieu

    Paris, FranceD Bambusi

    Universita ` di Milano

    Milan, Italy

    C Bardos

    Universite ′ de Paris 7

    Paris, France

    D Bar-Natan

    University of Toronto

    Toronto, ON, Canada

    E L Basor

    California Polytechnic State University

    San Luis Obispo, CA, USA

    M T Batchelor

    Australian National University

    Canberra, ACT, Australia

    S Bauer

    Universita ¨ t Bielefeld

    Bielefeld, Germany

    V Beffara

    Ecole Nomale Supe ′ rieure de Lyon

    Lyon, France

    R Beig

    Universita ¨ t Wien

    Vienna, Austria

    M I Belishev

    Petersburg Department of Steklov Institute

    of Mathematics

    St. Petersburg, Russia

    P Bernard

    Universite ′ de Paris Dauphine

    Paris, France

    D Birmingham

    University of the Pacific

    Stockton, CA, USA

    Jir ˇ? ′ Bic ˇa ′k

    Charles University, Prague, Czech Republic

    and Albert Einstein Institute

    Potsdam, Germany

    C Blanchet

    Universite ′ de Bretagne-Sud

    Vannes, France

    M Blasone

    Universita ` degli Studi di Salerno

    Baronissi (SA), Italy

    M Blau

    Universite ′ de Neucha ? tel

    Neucha ? tel, Switzerland

    S Boatto

    IMPA

    Rio de Janeiro, Brazil

    L V Bogachev

    University of Leeds

    Leeds, UK

    L Boi

    EHESS and LUTH

    Paris, France

    M Bojowald

    The Pennsylvania State University

    University Park, PA, USA

    C Bonatti

    Universite ′ de Bourgogne

    Dijon, France

    P Bonckaert

    Universiteit Hasselt

    Diepenbeek, Belgium

    F Bonetto

    Georgia Institute of Technology

    Atlanta, GA, USA

    G Bouchitte ′

    Universite ′ de Toulon et du Var

    La Garde, France

    A Bovier

    Weierstrass Institute for Applied Analysis and Stochastics

    Berlin, Germany

    H W Braden

    University of Edinburgh

    Edinburgh, UK

    H Bray

    Duke University

    Durham, NC, USA

    Y Brenier

    Universite ′ de Nice Sophia Antipolis

    Nice, France

    xiv CONTRIBUTORSJ Bros

    CEADSMSPhT, CEASaclay

    Gif-sur-Yvette, France

    R Brunetti

    Universita ¨ t Hamburg

    Hamburg, Germany

    M Bruschi

    Universita ` di Roma ‘‘La Sapienza’’

    Rome, Italy

    T Brzezin ′ski

    University of Wales Swansea

    Swansea, UK

    D Buchholz

    Universita ¨ tGo ¨ ttingen

    Go ¨ ttingen, Germany

    N Burq

    Universite ′ Paris-Sud

    Orsay, France

    F H Busse

    Universita ¨ t Bayreuth

    Bayreuth, Germany

    G Buttazzo

    Universita ` di Pisa

    Pisa, Italy

    P Butta `

    Universita ` di Roma ‘‘La Sapienza’’

    Rome, Italy

    S L Cacciatori

    Universita ` di Milano

    Milan, Italy

    P T Callaghan

    Victoria University of Wellington

    Wellington, New Zealand

    Francesco Calogero

    University of Rome, Rome, Italy and Institute

    Nazionale di Fisica Nucleare

    Rome, Italy

    A Carati

    Universita ` di Milano

    Milan, Italy

    J Cardy

    Rudolf Peierls Centre for Theoretical Physics

    Oxford, UK

    R Caseiro

    Universidade de Coimbra

    Coimbra, Portugal

    A S Cattaneo

    Universita ¨ tZu ¨ rich

    Zu ¨ rich, Switzerland

    A Celletti

    Universita ` di Roma ‘‘Tor Vergata’’

    Rome, Italy

    D Chae

    Sungkyunkwan University

    Suwon, South Korea

    G-Q Chen

    Northwestern University

    Evanston, IL, USA

    L Chierchia

    Universita ` degli Studi ‘‘Roma Tre’’

    Rome, Italy

    S Chmutov

    Petersburg Department of Steklov

    Institute of Mathematics

    St. Petersburg, Russia

    M W Choptuik

    University of British Columbia

    Vancouver, Canada

    Y Choquet-Bruhat

    Universite ′ P.-M. Curie, Paris VI

    Paris, France

    P T Chrus ′ciel

    Universite ′ de Tours

    Tours, France

    Chong-Sun Chu

    University of Durham

    Durham, UK

    F Cipriani

    Politecnico di Milano

    Milan, Italy

    CONTRIBUTORS xvR L Cohen

    Stanford University

    Stanford, CA, USA

    T H Colding

    University of New York

    New York, NY, USA

    J C Collins

    Penn State University

    University Park, PA, USA

    G Comte

    Universite ′ de Nice Sophia Antipolis

    Nice, France

    A Constantin

    Trinity College

    Dublin, Republic of Ireland

    D Crowdy

    Imperial College

    London, UK

    A B Cruzeiro

    University of Lisbon

    Lisbon, Portugal

    G Dal Maso

    SISSA

    Trieste, Italy

    F Dalfovo

    Universita ` di Trento

    Povo, Italy

    A S Dancer

    University of Oxford

    Oxford, UK

    P D’Ancona

    Universita ` di Roma ‘‘La Sapienza’’

    Rome, Italy

    S R Das

    University of Kentucky

    Lexington, KY, USA

    E Date

    Osaka University

    Osaka, Japan

    N Datta

    University of Cambridge

    Cambridge, UK

    G W Delius

    University of York

    York, UK

    G F dell’Antonio

    Universita ` di Roma ‘‘La Sapienza’’

    Rome, Italy

    C DeWitt-Morette

    The University of Texas at Austin

    Austin, TX, USA

    L Dio ′si

    Research Institute for Particle and Nuclear Physics

    Budapest, Hungary

    A Doliwa

    University of Warmia and Mazury in Olsztyn

    Olsztyn, Poland

    G Dolzmann

    University of Maryland

    College Park, MD, USA

    S K Donaldson

    Imperial College

    London, UK

    T C Dorlas

    Dublin Institute for Advanced Studies

    Dublin, Republic of Ireland

    M R Douglas

    Rutgers, The State University of New Jersey

    Piscataway, NJ, USA

    MDu ¨ tsch

    Universita ¨ tZu ¨ rich

    Zu ¨ rich, Switzerland

    B Dubrovin

    SISSA-ISAS

    Trieste, Italy

    J J Duistermaat

    Universiteit Utrecht

    Utrecht, The Netherlands

    S Duzhin

    Petersburg Department of Steklov Institute of

    Mathematics

    St. Petersburg, Russia

    xvi CONTRIBUTORSG Ecker

    Universita ¨ t Wien

    Vienna, Austria

    M Efendiev

    Universita ¨ t Stuttgart

    Stuttgart, Germany

    T Eguchi

    University of Tokyo

    Tokyo, Japan

    J Ehlers

    Max Planck Institut fu ¨ r Gravitationsphysik

    (Albert-Einstein Institut)

    Golm, Germany

    P E Ehrlich

    University of Florida

    Gainesville, FL, USA

    D Einzel

    Bayerische Akademie der Wissenschaften

    Garching, Germany

    G A Elliott

    University of Toronto

    Toronto, Canada

    G F R Ellis

    University of Cape Town

    Cape Town, South Africa

    C L Epstein

    University of Pennsylvania

    Philadelphia, PA, USA

    J Escher

    Universita ¨ t Hannover

    Hannover, Germany

    J B Etnyre

    University of Pennsylvania

    Philadelphia, PA, USA

    G Falkovich

    Weizmann Institute of Science

    Rehovot, Israel

    M Farge

    Ecole Normale Supe ′ rieure

    Paris, France

    B Ferrario

    Universita ` di Pavia

    Pavia, Italy

    R Finn

    Stanford University

    Stanford, CA, USA

    D Fiorenza

    Universita ` di Roma ‘‘La Sapienza’’

    Rome, Italy

    A E Fischer

    University of California

    Santa Cruz, CA, USA

    A S Fokas

    University of Cambridge

    Cambridge, UK

    J-P Franc ? oise

    Universite ′ P.-M. Curie, Paris VI

    Paris, France

    S Franz

    The Abdus Salam ICTP

    Trieste, Italy

    L Frappat

    Universite ′ de Savoie

    Chambery-Annecy, France

    J Frauendiener

    Universita ¨ tTu ¨bingen

    Tu ¨bingen, Germany

    K Fredenhagen

    Universita ¨ t Hamburg

    Hamburg, Germany

    S Friedlander

    University of Illinois-Chicago

    Chicago, IL, USA

    M R Gaberdiel

    ETH Zu ¨ rich

    Zu ¨ rich, Switzerland

    G Gaeta

    Universita ` di Milano

    Milan, Italy

    CONTRIBUTORS xviiL Galgani

    Universita ` di Milano

    Milan, Italy

    G Gallavotti

    Universita ` di Roma ‘‘La Sapienza’’

    Rome, Italy

    R Gambini

    Universidad de la Repu ′blica

    Montevideo, Uruguay

    G Gentile

    Universita ` degli Studi ‘‘Roma Tre’’

    Rome, Italy

    A Di Giacomo

    Universita ` di Pisa

    Pisa, Italy

    P B Gilkey

    University of Oregon

    Eugene, OR, USA

    R Gilmore

    Drexel University

    Philadelphia, PA, USA

    S Gindikin

    Rutgers University

    Piscataway, NJ, USA

    A Giorgilli

    Universita ` di Milano

    Milan, Italy

    G A Goldin

    Rutgers University

    Piscataway, NJ, USA

    G Gonza ′ lez

    Louisiana State University

    Baton Rouge, LA, USA

    R Gopakumar

    Harish-Chandra Research Institute

    Allahabad, India

    D Gottesman

    Perimeter Institute

    Waterloo, ON, Canada

    H Gottschalk

    Rheinische Friedrich-Wilhelms-Universita ¨ t Bonn

    Bonn, Germany

    O Goubet

    Universite ′ de Picardie Jules Verne

    Amiens, France

    T R Govindarajan

    The Institute of Mathematical Sciences

    Chennai, India

    A Grassi

    University of Pennsylvania

    Philadelphia, PA, USA

    P G Grinevich

    L D Landau Institute for

    Theoretical Physics

    Moscow, Russia

    Ch Gruber

    Ecole Polytechnique Fe ′de ′ rale de Lausanne

    Lausanne, Switzerland

    J-L Guermond

    Universite ′ de Paris Sud

    Orsay, France

    F Guerra

    Universita ` di Roma ‘‘La Sapienza’’

    Rome, Italy

    T Guhr

    Lunds Universitet

    Lund, Sweden

    C Guillope ′

    Universite ′ Paris XII – Val de Marne

    Cre ′ teil, France

    C Gundlach

    University of Southampton

    Southampton, UK

    S Gutt

    Universite ′ Libre de Bruxelles

    Brussels, Belgium

    K Hannabuss

    University of Oxford

    Oxford, UK

    xviii CONTRIBUTORSM Haragus

    Universite ′ de Franche-Comte ′

    Besanc ? on, France

    S G Harris

    St. Louis University

    St. Louis, MO, USA

    B Hasselblatt

    Tufts University

    Medford, MA, USA

    P Hayden

    McGill University

    Montreal, QC, Canada

    D C Heggie

    The University of Edinburgh

    Edinburgh, UK

    B Helffer

    Universite ′ Paris-Sud

    Orsay, France

    G M Henkin

    Universite ′ P.-M. Curie, Paris VI

    Paris, France

    M Henneaux

    Universite ′ Libre de Bruxelles

    Bruxelles, Belgium

    S Herrmann

    Universite ′ Henri Poincare ′ , Nancy 1

    Vandoeuvre-le `s-Nancy, France

    C P Herzog

    University of California at Santa Barbara

    Santa Barbara, CA, USA

    J G Heywood

    University of British Columbia

    Vancouver, BC, Canada

    A C Hirshfeld

    Universita ¨ t Dortmund

    Dortmund, Germany

    A S Holevo

    Steklov Mathematical Institute

    Moscow, Russia

    T J Hollowood

    University of Wales Swansea

    Swansea, UK

    D D Holm

    Imperial College

    London, UK

    J-W van Holten

    NIKHEF

    Amsterdam, The Netherlands

    A Huckleberry

    Ruhr-Universita ¨ t Bochum

    Bochum, Germany

    K Hulek

    Universita ¨ t Hannover

    Hannover, Germany

    D Iagolnitzer

    CEADSMSPhT, CEASaclay

    Gif-sur-Yvette, France

    R Illge

    Friedrich-Schiller-Universita ¨ t Jena

    Jena, Germany

    P Imkeller

    Humboldt Universita ¨ t zu Berlin

    Berlin, Germany

    G Iooss

    Institut Non Line ′aire de Nice

    Valbonne, France

    M Irigoyen

    Universite ′ P.-M. Curie, Paris VI

    Paris, France

    J Isenberg

    University of Oregon

    Eugene, OR, USA

    R Ivanova

    University of Hawaii Hilo

    Hilo, HI, USA

    E M Izhikevich

    The Neurosciences Institute

    San Diego, CA, USA

    R W Jackiw

    Massachusetts Institute of Technology

    Cambridge, MA, USA

    J K Jain

    The Pennsylvania State University

    University Park, PA, USA

    CONTRIBUTORS xixM Jardim

    IMECC–UNICAMP

    Campinas, Brazil

    L C Jeffrey

    University of Toronto

    Toronto, ON, Canada

    J Jime ′nez

    Universidad Politecnica de Madrid

    Madrid, Spain

    S Jitomirskaya

    University of California at Irvine

    Irvine, CA, USA

    P Jizba

    Czech Technical University

    Prague, Czech Republic

    A Joets

    Universite ′ Paris-Sud

    Orsay, France

    K Johansson

    Kungl Tekniska Ho ¨gskolan

    Stockholm, Sweden

    G Jona-Lasinio

    Universita ` di Roma ‘‘La Sapienza’’

    Rome, Italy

    V F R Jones

    University of California at Berkeley

    Berkeley, CA, USA

    N Joshi

    University of Sydney

    Sydney, NSW, Australia

    D D Joyce

    University of Oxford

    Oxford, UK

    CDJa ¨kel

    Ludwig-Maximilians-Universita ¨ tMu ¨nchen

    Mu ¨nchen, Germany

    G Kasperski

    Universite ′ Paris-Sud XI

    Orsay, France

    L H Kauffman

    University of Illinois at Chicago

    Chicago, IL, USA

    R K Kaul

    The Institute of Mathematical Sciences

    Chennai, India

    Y Kawahigashi

    University of Tokyo

    Tokyo, Japan

    B S Kay

    University of York

    York, UK

    R Kenyon

    University of British Columbia

    Vancouver, BC, Canada

    M Keyl

    Universita ` di Pavia

    Pavia, Italy

    T W B Kibble

    Imperial College

    London, UK

    S Kichenassamy

    Universite ′ de Reims Champagne-Ardenne

    Reims, France

    J Kim

    University of California at Irvine

    Irvine, USA

    S B Kim

    Chonnam National University

    Gwangju, South Korea

    A Kirillov

    University of Pennsylvania

    Philadelphia, PA, USA

    A Kirillov, Jr.

    Stony Brook University

    Stony Brook, NY, USA

    K Kirsten

    Baylor University

    Waco, TX, USA

    xx CONTRIBUTORSF Kirwan

    University of Oxford

    Oxford, UK

    S Klainerman

    Princeton University

    Princeton, NJ, USA

    I R Klebanov

    Princeton University

    Princeton, NJ, USA

    Y Kondratiev

    Universita ¨ t Bielefeld

    Bielefeld, Germany

    A Konechny

    Rutgers, The State University of New Jersey

    Piscataway, NJ, USA

    K Konishi

    Universita ` di Pisa

    Pisa, Italy

    T H Koornwinder

    University of Amsterdam

    Amsterdam, The Netherlands

    P Kornprobst

    INRIA

    Sophia Antipolis, France

    V P Kostov

    Universite ′ de Nice Sophia Antipolis

    Nice, France

    R Kotecky ′

    Charles University

    Prague, Czech Republic and the

    University of Warwick, UK

    Y Kozitsky

    Uniwersytet Marii Curie-Sklodowskiej

    Lublin, Poland

    P Kramer

    Universita ¨ tTu ¨bingen

    Tu ¨bingen, Germany

    C Krattenthaler

    Universita ¨ t Wien

    Vienna, Austria

    M Krbec

    Academy of Sciences

    Prague, Czech Republic

    D Kreimer

    IHES

    Bures-sur-Yvette, France

    A Kresch

    University of Warwick

    Coventry, UK

    D Kretschmann

    Technische Universita ¨ t Braunschweig

    Braunschweig, Germany

    P B Kronheimer

    Harvard University

    Cambridge, MA, USA

    B Kuckert

    Universita ¨ t Hamburg

    Hamburg, Germany

    Y Kuramoto

    Hokkaido University

    Sapporo, Japan

    J M F Labastida

    CSIC

    Madrid, Spain

    G Labrosse

    Universite ′ Paris-Sud XI

    Orsay, France

    C Landim

    IMPA, Rio de Janeiro, Brazil and UMR 6085

    and Universite ′ de Rouen

    France

    E Langmann

    KTH Physics

    Stockholm, Sweden

    S Laporta

    Universita ` di Parma

    Parma, Italy

    O D Lavrentovich

    Kent State University

    Kent, OH, USA

    CONTRIBUTORS xxiG F Lawler

    Cornell University

    Ithaca, NY, USA

    C Le Bris

    CERMICS – ENPC

    Champs Sur Marne, France

    A Lesne

    Universite ′ P.-M. Curie, Paris VI

    Paris, France

    D Levi

    Universita ` ‘‘Roma Tre’’

    Rome, Italy

    J Lewandowski

    Uniwersyte Warszawski

    Warsaw, Poland

    R G Littlejohn

    University of California at Berkeley

    Berkeley, CA, USA

    R Livi

    Universita ` di Firenze

    Sesto Fiorentino, Italy

    R Longoni

    Universita ` di Roma ‘‘La Sapienza’’

    Rome, Italy

    J Lowengrub

    University of California at Irvine

    Irvine, USA

    C Lozano

    INTA

    Torrejo ′n de Ardoz, Spain

    TTQLe ?

    Georgia Institute of Technology

    Atlanta, GA, USA

    B Lucquin-Desreux

    Universite ′ P.-M. Curie, Paris VI

    Paris, France

    V Lyubashenko

    Institute of Mathematics

    Kyiv, Ukraine

    M Lyubich

    University of Toronto

    Toronto, ON, Canada and Stony Brook University

    NY, USA

    RLe ′andre

    Universite ′ de Bourgogne

    Dijon, France

    PLe ′vay

    Budapest University of Technology and Economics

    Budapest, Hungary

    R Maartens

    Portsmouth University

    Portsmouth, UK

    N MacKay

    University of York

    York, UK

    J Magnen

    Ecole Polytechnique

    France

    F Magri

    Universita ` di Milano Bicocca

    Milan, Italy

    J Maharana

    Institute of Physics

    Bhubaneswar, India

    S Majid

    Queen Mary, University of London

    London, UK

    C Marchioro

    Universita ` di Roma ‘‘La Sapienza’’

    Rome, Italy

    K Marciniak

    Linko ¨ping University

    Norrko ¨ping, Sweden

    M Marcolli

    Max-Planck-Institut fu ¨ r Mathematik

    Bonn, Germany

    M Marin ?o

    CERN

    Geneva, Switzerland

    xxii CONTRIBUTORSJ Marklof

    University of Bristol

    Bristol, UK

    C-M Marle

    Universite ′ P.-M. Curie, Paris VI

    Paris, France

    L Mason

    University of Oxford

    Oxford, UK

    V Mastropietro

    Universita ` di Roma ‘‘Tor Vergata’’

    Rome, Italy

    V Mathai

    University of Adelaide

    Adelaide, SA, Australia

    J Mawhin

    Universite ′ Catholique de Louvain

    Louvain-la-Neuve, Belgium

    S Mazzucchi

    Universita ` di Trento

    Povo, Italy

    B M McCoy

    State University of New York at Stony Brook

    Stony Brook, NY, USA

    E Meinrenken

    University of Toronto

    Toronto, ON, Canada

    I Melbourne

    University of Surrey

    Guildford, UK

    J Mickelsson

    KTH Physics

    Stockholm, Sweden

    W P Minicozzi II

    University of New York

    New York, NY, USA

    S Miracle-Sole ′

    Centre de Physique The ′orique, CNRS

    Marseille, France

    A Miranville

    Universite ′ de Poitiers

    Chasseneuil, France

    P K Mitter

    Universite ′ de Montpellier 2

    Montpellier, France

    V Moncrief

    Yale University

    New Haven, CT, USA

    Y Morita

    Ryukoku University

    Otsu, Japan

    P J Morrison

    University of Texas at Austin

    Austin, TX, USA

    J Mund

    Universidade de Sa ?o Paulo

    Sa ?o Paulo, Brazil

    F Musso

    Universita ` ‘‘Roma Tre’’

    Rome, Italy

    G L Naber

    Drexel University

    Philadelphia, PA, USA

    B Nachtergaele

    University of California at Davis

    Davis, CA, USA

    C Nash

    National University of Ireland

    Maynooth, Ireland

    S ˇ Nec ˇasova ′

    Academy of Sciences

    Prague, Czech Republic

    A I Neishtadt

    Russian Academy of Sciences

    Moscow, Russia

    N Neumaier

    Albert-Ludwigs-University in Freiburg

    Freiburg, Germany

    S E Newhouse

    Michigan State University

    E. Lansing, MI, USA

    CONTRIBUTORS xxiiiC M Newman

    New York University

    New York, NY, USA

    S Nikc ˇevic ′

    SANU

    Belgrade, Serbia and Montenegro

    M Nitsche

    University of New Mexico

    Albuquerque, NM, USA

    R G Novikov

    Universite ′ de Nantes

    Nantes, France

    J M Nunes da Costa

    Universidade de Coimbra

    Coimbra, Portugal

    S O’Brien

    Tyndall National Institute

    Cork, Republic of Ireland

    A Okounkov

    Princeton University

    Princeton, NJ, USA

    A Onuki

    Kyoto University

    Kyoto, Japan

    J-P Ortega

    Universite ′ de Franche-Comte ′

    Besanc ? on, France

    H Osborn

    University of Cambridge

    Cambridge, UK

    Maciej P Wojtkowski

    University of Arizona

    Tucson, AZ, USA and Institute of Mathematics PAN

    Warsaw, Poland

    J Palmer

    University of Arizona

    Tucson, AZ, USA

    J H Park

    Sungkyunkwan University

    Suwon, South Korea

    P E Parker

    Wichita State University

    Wichita KS, USA

    S Paycha

    Universite ′ Blaise Pascal

    Aubie ` re, France

    P A Pearce

    University of Melbourne

    Parkville VIC, Australia

    P Pearle

    Hamilton College

    Clinton, NY, USA

    M Pedroni

    Universita ` di Bergamo

    Dalmine (BG), Italy

    B Pelloni

    University of Reading

    UK

    R Penrose

    University of Oxford

    Oxford, UK

    A Perez

    Penn State University,University Park, PA, USA

    J H H Perk

    Oklahoma State University

    Stillwater, OK, USA

    T Peternell

    Universita ¨ t Bayreuth

    Bayreuth, Germany

    D Petz

    Budapest University of Technology and Economics

    Budapest, Hungary

    M J Pflaum

    Johann Wolfgang Goethe-Universita ¨ t

    Frankfurt, Germany

    B Piccoli

    Istituto per le Applicazioni del Calcolo

    Rome, Italy

    C Piquet

    Universite ′ P.-M. Curie, Paris VI

    Paris, France

    xxiv CONTRIBUTORSL P Pitaevskii

    Universita ` di Trento

    Povo, Italy

    S Pokorski

    Warsaw University

    Warsaw, Poland

    E Presutti

    Universita ` di Roma ‘‘Tor Vergata’’

    Rome, Italy

    E Previato

    Boston University

    Boston, MA, USA

    B Prinari

    Universita ` degli Studi di Lecce

    Lecce, Italy

    J Pullin

    Louisiana State University

    Baton Rouge, LA, USA

    M Pulvirenti

    Universita ` di Roma ‘‘La Sapienza’’

    Rome, Italy

    O Ragnisco

    Universita ` ‘‘Roma Tre’’

    Rome, Italy

    P Ramadevi

    Indian Institute of Technology Bombay

    Mumbai, India

    S A Ramakrishna

    Indian Institute of Technology

    Kanpur, India

    J Rasmussen

    Princeton University

    Princeton, NJ, USA

    L Rastelli

    Princeton University

    Princeton, NJ, USA

    T S Ratiu

    Ecole Polytechnique Fe ′de ′ rale de Lausanne

    Lausanne, Switzerland

    S Rauch-Wojciechowski

    Linko ¨ping University

    Linko ¨ping, Sweden

    K-H Rehren

    Universita ¨ tGo ¨ ttingen

    Go ¨ ttingen, Germany

    E Remiddi

    Universita ` di Bologna

    Bologna, Italy

    J E Roberts

    Universita ` di Roma ‘‘Tor Vergata’’

    Rome, Italy

    L Rey-Bellet

    University of Massachusetts

    Amherst, MA, USA

    R Robert

    Universite ′ Joseph Fourier

    Saint Martin D’He ` res, France

    F A Rogers

    King’s College London

    London, UK

    R M S Rosa

    Universidade Federal do Rio de Janeiro

    Rio de Janeiro, Brazil

    C Rovelli

    Universite ′ de la Me ′diterrane ′e et Centre

    de Physique The ′orique

    Marseilles, France

    S N M Ruijsenaars

    Centre for Mathematics and Computer Science

    Amsterdam, The Netherlands

    F Russo

    Universite ′ Paris 13

    Villetaneuse, France

    L H Ryder

    University of Kent

    Canterbury, UK

    S Sachdev

    Yale University

    New Haven, CT, USA

    H Sahlmann

    Universiteit Utrecht

    Utrecht, The Netherlands

    CONTRIBUTORS xxvM Salmhofer

    Universita ¨ t Leipzig

    Leipzig, Germany

    P M Santini

    Universita ` di Roma ‘‘La Sapienza’’

    Rome, Italy

    A Sarmiento

    Universidade Federal de Minas Gerais

    Belo Horizonte, Brazil

    R Sasaki

    Kyoto University

    Kyoto, Japan

    A Savage

    University of Toronto

    Toronto, ON, Canada

    M Schechter

    University of California at Irvine

    Irvine, CA, USA

    D-M Schlingemann

    Technical University of Braunschweig

    Braunschweig, Germany

    R Schmid

    Emory University

    Atlanta, GA, USA

    G Schneider

    Universita ¨ t Karlsruhe

    Karlsruhe, Germany

    K Schneider

    Universite ′ de Provence

    Marseille, France

    B Schroer

    Freie Universita ¨ t Berlin

    Berlin, Germany

    T Schu ¨cker

    Universite ′ de Marseille

    Marseille, France

    S Scott

    King’s College London

    London, UK

    P Selick

    University of Toronto

    Toronto, ON, Canada

    M A Semenov-Tian-Shansky

    Steklov Institute of Mathematics

    St. Petersburg, Russia and and Universite ′ de Bourgogne

    Dijon, France

    A N Sengupta

    Louisiana State University

    Baton Rouge LA, USA

    S Serfaty

    New York University

    New York, NY, USA

    E R Sharpe

    University of Utah

    Salt Lake City, UT, USA

    D Shepelsky

    Institute for Low Temperature Physics and Engineering

    Kharkov, Ukraine

    S Shlosman

    Universite ′ de Marseille

    Marseille, France

    A Siconolfi

    Universita ` di Roma ‘‘La Sapienza’’

    Rome, Italy

    V Sidoravicius

    IMPA

    Rio de Janeiro, Brazil

    J A Smoller

    University of Michigan

    Ann Arbor MI, USA

    M Socolovsky

    Universidad Nacional Auto ′noma de Me ′xico

    Me ′xico DF, Me ′xico

    J P Solovej

    University of Copenhagen

    Copenhagen, Denmark

    A Soshnikov

    University of California at Davis

    Davis, CA, USA

    J M Speight

    University of Leeds

    Leeds, UK

    xxvi CONTRIBUTORSH Spohn

    Technische Universita ¨ tMu ¨nchen

    Garching, Germany

    J Stasheff

    Lansdale, PA, USA

    D L Stein

    University of Arizona

    Tucson, AZ, USA

    K S Stelle

    Imperial College

    London, UK

    G Sterman

    Stony Brook University

    Stony Brook, NY, USA

    S Stringari

    Universita ` di Trento

    Povo, Italy

    S J Summers

    University of Florida

    Gainesville, FL, USA

    V S Sunder

    The Institute of Mathematical Sciences

    Chennai, India

    Y B Suris

    Technische Universita ¨ tMu ¨nchen

    Mu ¨nchen, Germany

    R J Szabo

    Heriot-Watt University

    Edinburgh, UK

    S Tabachnikov

    Pennsylvania State University

    University Park, PA, USA

    H Tasaki

    Gakushuin University

    Tokyo, Japan

    M E Taylor

    University of North Carolina

    Chapel Hill, NC, USA

    R Temam

    Indiana University

    Bloomington, IN, USA

    B Temple

    University of California at Davis

    Davis, CA, USA

    R P Thomas

    Imperial College

    London, UK

    U Tillmann

    University of Oxford

    Oxford, UK

    K P Tod

    University of Oxford

    Oxford, UK

    J A Toth

    McGill University

    Montreal, QC, Canada

    C A Tracy

    University of California at Davis

    Davis, CA, USA

    A Trautman

    Warsaw University

    Warsaw, Poland

    D Treschev

    Moscow State University

    Moscow, Russia

    L Triolo

    Universita ` di Roma ‘‘Tor Vergata’’

    Rome, Italy

    J Troost

    Ecole Normale Supe ′ rieure

    Paris, France

    Tsou Sheung Tsun

    University of Oxford

    Oxford, UK

    V Turaev

    IRMA

    Strasbourg, France

    D Ueltschi

    University of Arizona

    Tucson, AZ, USA

    A M Uranga

    Consejo Superior de Investigaciones Cientificas

    Madrid, Spain

    CONTRIBUTORS xxviiA Valentini

    Perimeter Institute for Theoretical Physics

    Waterloo, ON, Canada

    M Vaugon

    Universite ′ P.-M. Curie, Paris VI

    Paris, France

    P Di Vecchia

    Nordita

    Copenhagen, Denmark

    A F Verbeure

    Institute for Theoretical Physics

    KU Leuven, Belgium

    Y Colin de Verdie ` re

    Universite ′ de Grenoble 1

    Saint-Martin d’He ` res, France

    M Viana

    IMPA

    Rio de Janeiro, Brazil

    G Vitiello

    Universita ` degli Studi di Salerno

    Baronissi (SA), Italy

    D-V Voiculescu

    University of California at Berkeley

    Berkeley, CA, USA

    S Waldmann

    Albert-Ludwigs-Universita ¨ t Freiburg

    Freiburg, Germany

    J Wambsganss

    Universita ¨ t Heidelberg

    Heidelberg, Germany

    R S Ward

    University of Durham

    Durham, UK

    E Wayne

    Boston University

    Boston, MA, USA

    F W Wehrli

    University of Pennsylvania

    Philadelphia, PA, USA

    R F Werner

    Technische Universita ¨ t Braunschweig

    Braunschweig, Germany

    H Widom

    University of California at Santa Cruz

    Santa Cruz, CA, USA

    C M Will

    Washington University

    St. Louis, MO, USA

    N M J Woodhouse

    University of Oxford

    Oxford, UK

    Siye Wu

    University of Colorado

    Boulder, CO, USA

    VWu ¨nsch

    Friedrich-Schiller-Universita ¨ t Jena

    Jena, Germany

    D R Yafaev

    Universite ′ de Rennes

    Rennes, France

    M Yamada

    Kyoto University

    Kyoto, Japan

    M Yuri

    Hokkaido University

    Sapporo, Japan

    DZ ˇubrinic ′

    University of Zagreb

    Zagreb, Croatia

    VZ ˇupanovic ′

    University of Zagreb

    Zagreb, Croatia

    R Zecchina

    International Centre for Theoretical Physics (ICTP)

    Trieste, Italy

    S Zelditch

    Johns Hopkins University

    Baltimore, MD, USA

    xxviii CONTRIBUTORSS Zelik

    Universita ¨ t Stuttgart

    Stuttgart, Germany

    S-C Zhang

    Stanford University

    Stanford, CA, USA

    M B Ziane

    University of Southern California

    Los Angeles, CA, USA

    M R Zirnbauer

    Universita ¨ tKo ¨ ln

    Ko ¨ ln, Germany

    A Zumpano

    Universidade Federal de Minas Gerais

    Belo Horizonte, Brazil

    CONTRIBUTORS xxixCONTENTS LIST BY SUBJECT

    Location references refer to the volume number and page number (separated by a colon).

    INTRODUCTORY ARTICLES

    Classical Mechanics 1:1

    Differential Geometry 1:33

    Electromagnetism 1:40

    Equilibrium Statistical Mechanics 1:51

    Functional Analysis 1:88

    Minkowski Spacetime and Special Relativity 1:96

    Quantum Mechanics 1:109

    Topology 1:131

    PHYSICS SUBJECTS

    Classical Mechanics

    Boundary Control Method and Inverse Problems of

    Wave Propagation 1:340

    Constrained Systems 1:611

    Cotangent Bundle Reduction 1:658

    Gravitational N-body Problem (Classical) 2:575

    Hamiltonian Fluid Dynamics 2:593

    Hamiltonian Systems: Obstructions to

    Integrability 2:624

    Infinite-Dimensional Hamiltonian Systems 3:37

    Inverse Problem in Classical Mechanics 3:156

    KAM Theory and Celestial Mechanics 3:189

    Peakons 4:12

    Poisson Reduction 4:79

    Stability Problems in Celestial Mechanics 5:20

    Symmetry and Symplectic Reduction 5:190

    Classical, Conformal and Topological

    Field Theory

    Topological Quantum Field Theory:

    Overview 5:278

    AdSCFT Correspondence 1:174

    Axiomatic Approach to Topological Quantum Field

    Theory 1:232

    BF Theories 1:257

    Boundary Conformal Field Theory 1:333

    Chern–Simons Models: Rigorous Results 1:496

    Donaldson–Witten Theory 2:110

    Duality in Topological Quantum Field

    Theory 2:118

    Finite-Type Invariants 2:340

    Four-Manifold Invariants and Physics 2:386

    Gauge Theoretic Invariants of 4-Manifolds 2:457

    h-Pseudodifferential Operators and

    Applications 2:701

    The Jones Polynomial 3:179

    Knot Theory and Physics 3:220

    Kontsevich Integral 3:231

    Large-N and Topological Strings 3:263

    Mathai–Quillen Formalism 3:390

    Mathematical Knot Theory 3:399

    Operator Product Expansion in Quantum Field

    Theory 3:616

    Schwarz-Type Topological Quantum Field

    Theory 4:494

    Solitons and Other Extended Field

    Configurations 4:602

    Topological Defects and Their Homotopy

    Classification 5:257

    Topological Gravity, Two-Dimensional 5:264

    Topological Knot Theory and Macroscopic

    Physics 5:271

    Topological Sigma Models 5:290

    Two-Dimensional Conformal Field Theory and

    Vertex Operator Algebras 5:317

    WDVV Equations and Frobenius

    Manifolds 5:438

    Condensed Matter and Optics

    Bose–Einstein Condensates 1:312

    Falicov–Kimball Model 2:283

    Fractional Quantum Hall Effect 2:402

    High Tc Superconductor Theory 2:645

    Hubbard Model 2:712

    Liquid Crystals 3:320

    Negative Refraction and Subdiffraction

    Imaging 3:483

    Nuclear Magnetic Resonance 3:592Optical Caustics 3:620

    Quantum Phase Transitions 4:289

    Quasiperiodic Systems 4:308

    Renormalization: Statistical Mechanics and

    Condensed Matter 4:407

    Short-Range Spin Glasses: The Metastate

    Approach 4:570

    Topological Defects and Their Homotopy

    Classification 5:257

    Disordered Systems

    Cellular Automata 1:455

    Lagrangian Dispersion (Passive Scalar) 3:255

    Mean Field Spin Glasses and Neural

    Networks 3:407

    Percolation Theory 4:21

    Random Matrix Theory in Physics 4:338

    Random Walks in Random Environments 4:353

    Short-Range Spin Glasses: The Metastate

    Approach 4:570

    Spin Glasses 4:655

    Stochastic Loewner Evolutions 5:80

    Two-Dimensional Ising Model 5:322

    Wulff Droplets 5:462

    Dynamical Systems

    Averaging Methods 1:226

    Bifurcations of Periodic Orbits 1:285

    Billiards in Bounded Convex Domains 1:296

    Central Manifolds, Normal Forms 1:467

    Cellular Automata 1:455

    Chaos and Attractors 1:477

    Cotangent Bundle Reduction 1:658

    Diagrammatic Techniques in Perturbation

    Theory 2:54

    Dissipative Dynamical Systems of Infinite

    Dimension 2:101

    Dynamical Systems and Thermodynamics 2:125

    Dynamical Systems in Mathematical Physics:

    An Illustration from Water Waves 2:133

    Entropy and Quantitative Transversality 2:237

    Ergodic Theory 2:250

    Fractal Dimensions in Dynamics 2:394

    Generic Properties of Dynamical Systems 2:494

    Gravitational N-Body Problem (Classical) 2:575

    Hamiltonian Fluid Dynamics 2:593

    Hamiltonian Systems: Stability and Instability

    Theory 2:631

    Holomorphic Dynamics 2:652

    Homeomorphisms and Diffeomorphisms of the

    Circle 2:665

    Homoclinic Phenomena 2:672

    h-Pseudodifferential Operators and

    Applications 2:701

    Hyperbolic Billiards 2:716

    Hyperbolic Dynamical Systems 2:721

    Isomonodromic Deformations 3:173

    KAM Theory and Celestial Mechanics 3:189

    Lyapunov Exponents and Strange Attractors 3:349

    Multiscale Approaches 3:465

    Normal Forms and Semiclassical

    Approximation 3:578

    Point-Vortex Dynamics 4:66

    Poisson Reduction 4:79

    Polygonal Billiards 4:84

    Quasiperiodic Systems 4:308

    Random Dynamical Systems 4:330

    Regularization For Dynamical -Functions 4:386

    Resonances 4:415

    Riemann–Hilbert Problem 4:436

    Semiclassical Spectra and Closed Orbits 4:512

    Separatrix Splitting 4:535

    Stability Problems in Celestial Mechanics 5:20

    Stability Theory and KAM 5:26

    Symmetry and Symmetry Breaking in Dynamical

    Systems 5:184

    Symmetry and Symplectic Reduction 5:190

    Synchronization of Chaos 5:213

    Universality and Renormalization 5:343

    Weakly Coupled Oscillators 5:448

    Equilibrium Statistical Mechanics

    Bethe Ansatz 1:253

    Cluster Expansion 1:531

    Dimer Problems 2:61

    Eight Vertex and Hard Hexagon Models 2:155

    Falicov–Kimball Model 2:283

    Fermionic Systems 2:300

    Finitely Correlated States 2:334

    Holonomic Quantum Fields 2:660

    Hubbard Model 2:712

    Large Deviations in Equilibrium Statistical

    Mechanics 3:261

    Metastable States 3:417

    Phase Transitions in Continuous Systems 4:53

    Pirogov–Sinai Theory 4:60

    Quantum Central-Limit Theorems 4:130

    Quantum Phase Transitions 4:289

    Quantum Spin Systems 4:295

    Quantum Statistical Mechanics: Overview 4:302

    Reflection Positivity and Phase Transitions 4:376

    Short-Range Spin Glasses: The Metastate

    Approach 4:570

    Statistical Mechanics and Combinatorial

    Problems 5:50

    Statistical Mechanics of Interfaces 5:55

    Superfluids 5:115

    Toeplitz Determinants and Statistical

    Mechanics 5:244

    Two-Dimensional Ising Model 5:322

    Wulff Droplets 5:462

    Fluid Dynamics

    Bifurcations in Fluid Dynamics 1:281

    Breaking Water Waves 1:383

    xxxii CONTENTS LIST BY SUBJECTCapillary Surfaces 1:431

    Cauchy Problem for Burgers-Type Equations 1:446

    Compressible Flows: Mathematical Theory 1:595

    Fluid Mechanics: Numerical Methods 2:365

    Geophysical Dynamics 2:534

    Hamiltonian Fluid Dynamics 2:593

    Incompressible Euler Equations: Mathematical

    Theory 3:10

    Interfaces and Multicomponent Fluids 3:135

    Intermittency in Turbulence 3:144

    Inviscid Flows 3:160

    Korteweg–de Vries Equation and Other Modulation

    Equations 3:239

    Lagrangian Dispersion (Passive Scalar) 3:255

    Magnetohydrodynamics 3:375

    Newtonian Fluids and Thermohydraulics 3:492

    Non-Newtonian Fluids 3:560

    Partial Differential Equations: Some Examples 4:6

    Peakons 4:12

    Stability of Flows 5:1

    Superfluids 5:115

    Turbulence Theories 5:295

    Variational Methods in Turbulence 5:351

    Viscous Incompressible Fluids: Mathematical

    Theory 5:369

    Vortex Dynamics 5:390

    Wavelets: Application to Turbulence 5:408

    Gauge Theory

    Abelian and Nonabelian Gauge Theories Using

    Differential Forms 1:141

    Abelian Higgs Vortices 1:151

    AdSCFT Correspondence 1:174

    Aharonov–Bohm Effect 1:191

    Anomalies 1:205

    BRST Quantization 1:386

    Chern–Simons Models: Rigorous Results 1:496

    Dirac Fields in Gravitation and Nonabelian Gauge

    Theory 2:67

    Donaldson–Witten Theory 2:110

    Effective Field Theories 2:139

    Electric–Magnetic Duality 2:201

    Electroweak Theory 2:209

    Exact Renormalization Group 2:272

    Gauge Theories from Strings 2:463

    Gauge Theory: Mathematical Applications 2:468

    Instantons: Topological Aspects 3:44

    Large-N and Topological Strings 3:263

    Lattice Gauge Theory 3:275

    Measure on Loop Spaces 3:413

    Noncommutative Geometry and the Standard

    Model 3:509

    Nonperturbative and Topological Aspects of Gauge

    Theory 3:568

    Perturbative Renormalization Theory and

    BRST 4:41

    Quantum Chromodynamics 4:144

    Quantum Electrodynamics and Its Precision

    Tests 4:168

    Renormalization: General Theory 4:399

    Seiberg–Witten Theory 4:503

    Standard Model of Particle Physics 5:32

    Supergravity 5:122

    Supersymmetric Particle Models 5:140

    Symmetry Breaking in Field Theory 5:198

    Twistor Theory: Some Applications 5:303

    Two-Dimensional Models 5:328

    General Relativity

    General Relativity: Overview 2:487

    Asymptotic Structure and Conformal

    Infinity 1:221

    Black Hole Mechanics 1:300

    Boundaries for Spacetimes 1:326

    Brane Worlds 1:367

    Canonical General Relativity 1:412

    Critical Phenomena in Gravitational

    Collapse 1:668

    Computational Methods in General Relativity:

    The Theory 1:604

    Cosmology: Mathematical Aspects 1:653

    Dirac Fields in Gravitation and Nonabelian Gauge

    Theory 2:67

    Einstein–Cartan Theory 2:189

    Einstein’s Equations with Matter 2:195

    Einstein Equations: Exact Solutions 2:165

    Einstein Equations: Initial Value

    Formulation 2:173

    General Relativity: Experimental Tests 2:481

    Geometric Analysis and General Relativity 2:502

    Geometric Flows and the Penrose

    Inequality 2:510

    Gravitational Lensing 2:567

    Gravitational Waves 2:582

    Hamiltonian Reduction of Einstein’s

    Equations 2:607

    Minimal Submanifolds 3:420

    Newtonian Limit of General Relativity 3:503

    Quantum Field Theory in Curved

    Spacetime 4:202

    Relativistic Wave Equations Including Higher Spin

    Fields 4:391

    Shock Wave Refinement of the Friedman–

    Robertson–Walker Metric 4:559

    Spacetime Topology, Causal Structure and

    Singularities 4:617

    Spinors and Spin Coefficients 4:667

    Stability of Minkowski Space 5:14

    Stationary Black Holes 5:38

    Twistors 5:311

    Integrable Systems

    Integrable Systems: Overview 3:106

    Abelian Higgs Vortices 1:151

    Affine Quantum Groups 1:183

    Ba ¨ cklund Transformations 1:241

    CONTENTS LIST BY SUBJECT xxxiiiBethe Ansatz 1:253

    Bi-Hamiltonian Methods in Soliton Theory 1:290

    Boundary-Value Problems For Integrable

    Equations 1:346

    Calogero–Moser–Sutherland Systems of

    Nonrelativistic and Relativistic Type 1:403

     -Approach to Integrable Systems 2:34

    Eigenfunctions of Quantum Completely Integrable

    Systems 2:148

    Functional Equations and Integrable Systems 2:425

    Holonomic Quantum Fields 2:660

    Instantons: Topological Aspects 3:44

    Integrability and Quantum Field Theory 3:50

    Integrable Discrete Systems 3:59

    Integrable Systems and Algebraic Geometry 3:65

    Integrable Systems and Discrete Geometry 3:78

    Integrable Systems and Recursion Operators on

    Symplectic and Jacobi Manifolds 3:87

    Integrable Systems and the Inverse Scattering

    Method 3:93

    Integrable Systems in Random Matrix

    Theory 3:102

    Isochronous Systems 3:166

    Nonlinear Schro ¨dinger Equations 3:552

    Painleve ′ Equations 4:1

    Peakons 4:12

    Quantum Calogero–Moser Systems 4:123

    Riemann–Hilbert Methods in Integrable

    Systems 4:429

    Sine-Gordon Equation 4:576

    Solitons and Kac–Moody Lie Algebras 4:594

    Toda Lattices 5:235

    Twistor Theory: Some Applications 5:303

    Yang–Baxter Equations 5:465

    M-Theory see String Theory and

    M-Theory

    Nonequilibrium Statistical Mechanics

    Nonequilibrium Statistical Mechanics (Stationary):

    Overview 3:530

    Adiabatic Piston 1:160

    Boltzmann Equation (Classical and

    Quantum) 1:306

    Glassy Disordered Systems: Dynamical

    Evolution 2:553

    Fourier Law 2:374

    Interacting Particle Systems ......