Festschrift in Honor of Alan Weinstein

Progress in Mathematics Volume 232

Series Editors Hyman Bass Joseph Oesterle´ Alan Weinstein The Breadth of Symplectic and Poisson Geometry Festschrift in Honor of Alan Weinstein

Jerrold E. Marsden Tudor S. Ratiu Editors

Birkhauser¨ Boston • Basel • Berlin Jerrold E. Marsden Tudor S. Ratiu California Institute of Technology Ecole Polytechnique Fed´ erale´ de Lausanne Department of Engineering Departement´ de Mathematiques´ and Applied Science CH-1015 Lausanne Control and Dynamical Systems Switzerland Pasadena, CA 91125 U.S.A.

AMS Subject Classiﬁcations: 53Dxx, 17Bxx, 22Exx, 53Dxx, 81Sxx

Library of Congress Cataloging-in-Publication Data The breadth of symplectic and Poisson geometry : festschrift in honor of Alan Weinstein / Jerrold E. Marsden, Tudor S. Ratiu, editors. p. cm. – (Progress in mathematics ; v. 232) Includes bibliographical references and index. ISBN 0-8176-3565-3 (acid-free paper) 1. Symplectic geometry. 2. Geometric quantization. 3. Poisson manifolds. I. Weinstein, Alan, 1943- II. Marsden, Jerrold E. III. Ratiu, Tudor S. IV. Progress in mathematics (Boston, Mass.); v. 232.

QA665.B74 2004 516.3’.6-dc22 2004046202

ISBN 0-8176-3565-3 Printed on acid-free paper.

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987654321 SPIN10958261 www.birkhauser.com Margo, Alan, and Asha in Paris at the lovely Fontaine des Quatre Parties du Monde. Contents

Preface ...... ix Academic genealogy of Alan Weinstein ...... xiii About Alan Weinstein ...... xv Students of Alan Weinstein...... xv Alan Weinstein’s publications ...... xvi Dirac structures, momentum maps, and quasi-Poisson manifolds Henrique Bursztyn, Marius Crainic ...... 1 Construction of Ricci-type connections by reduction and induction Michel Cahen, Simone Gutt, Lorenz Schwachhöfer ...... 41 A mathematical model for geomagnetic reversals J. J. Duistermaat ...... 59 Nonholonomic systems via moving frames: Cartan equivalence and Chaplygin Hamiltonization Kurt Ehlers, Jair Koiller, Richard Montgomery, Pedro M. Rios ...... 75 Thompson’s conjecture for real semisimple Lie groups Sam Evens, Jiang-Hua Lu ...... 121 The Weinstein conjecture and theorems of nearby and almost existence Viktor L. Ginzburg ...... 139 Simple singularities and integrable hierarchies Alexander B. Givental, Todor E. Milanov ...... 173 Momentum maps and measure-valued solutions (peakons, ﬁlaments, and sheets) for the EPDiff equation Darryl D. Holm, Jerrold E. Marsden ...... 203 viii Contents

Higher homotopies and Maurer–Cartan algebras: Quasi-Lie–Rinehart, Gerstenhaber, and Batalin–Vilkovisky algebras Johannes Huebschmann ...... 237

Localization theorems by symplectic cuts Lisa Jeffrey, Mikhail Kogan ...... 303

Reﬁnements of the Morse stratiﬁcation of the normsquare of the moment map Frances Kirwan ...... 327

Quasi, twisted, and all that… in Poisson geometry and Lie algebroid theory Yvette Kosmann-Schwarzbach ...... 363

Minimal coadjoint orbits and symplectic induction Bertram Kostant ...... 391

Quantization of pre-quasi-symplectic groupoids and their Hamiltonian spaces Camille Laurent-Gengoux, Ping Xu ...... 423 Duality and triple structures Kirill C. H. Mackenzie ...... 455 Star exponential functions as two-valued elements Y. Maeda, N. Miyazaki, H. Omori, A. Yoshioka ...... 483 From momentum maps and dual pairs to symplectic and Poisson groupoids Charles-Michel Marle ...... 493 Construction of spectral invariants of Hamiltonian paths on closed symplectic manifolds Yong-Geun Oh ...... 525 The universal covering and covered spaces of a symplectic Lie algebra action Juan-Pablo Ortega, Tudor S. Ratiu ...... 571 Poisson homotopy algebra: An idiosyncratic survey of homotopy algebraic topics related to Alan’s interests Jim Stasheff ...... 583 Dirac submanifolds of Jacobi manifolds Izu Vaisman ...... 603 Quantum maps and automorphisms Steve Zelditch ...... 623 Preface

Alan Weinstein is one of the top mathematicians in the world working in the area of symplectic and differential geometry. His research on symplectic reduction, La- grangian submanifolds, groupoids, applications to mechanics, and related areas has had a profound influence on the field. This area of research remains active and vi- brant today and this volume is intended to be a reflection of that vigor. In addition to reflecting the vitality of the field, this is a celebratory volume to honor Alan’s 60th birthday. His birthday was also celebrated in August, 2003 with a wonderful week-long conference held at the ESI: the Erwin Schrödinger International Institute for Mathematical Physics in Vienna. Alan was born in New York in 1943. He was an undergraduate at MIT and a graduate student at UC Berkeley, where he was awarded his Ph.D. in 1967 under the direction of S. S. Chern. After spending postdoctoral years at IHES near Paris, MIT, and the University of Bonn, he joined the faculty at UC Berkeley in 1969, becoming a full Professor in 1976. Alan has received many honors, including an Alfred P. Sloan Foundation Fel- lowship, a Miller Professorship (twice), a Guggenheim Fellowship, election to the American Academy of Arts and Sciences in 1992, and an honorary degree at the University of Utrecht in 2003. At the ESI conference, S. S. Chern, Alan’s advisor, sent the following words to celebrate the occasion: “I am glad about this celebration and I think Alan richly deserves it. Alan came to me in the early sixties as a graduate student at the University of California at Berkeley. At that time, a prevailing problem in our geometry group, and the geometry community at large, was whether on a Riemannian manifold the cut locus and the conjugate locus of a point can be disjoint. Alan immediately showed that this was possible. The result became part of his Ph.D. thesis, which was published in the Annals of Mathematics.He received his Ph.D. degree in a short period of two years. I introduced him to IHES and the French mathematical community. He stays close with them and with the mathematical ideas of Charles Ehresmann. He is original and x Preface

often came up with ingenious ideas. An example is his contribution to the solution of the Blaschke conjecture. I am very proud to count him as one of my students and I hope he will remain interested in mathematics up to my age, which is now 91.’’ Alan’s technical contributions are wide ranging and deep. As many of his early papers in his publication list illustrate, he started off in his thesis and the years immediately following in pure differential geometry, a topic he has come back to from time to time throughout his career. Already starting with his postdoc years and his early career at Berkeley, he became interested in symplectic geometry and mechanics. In this area he rapidly established himself as one of the world’s authorities, producing important and deep results ranging from reduction theory to Lagrangian and Poisson manifolds to studies of periodic orbits in Hamiltonian systems. He also did important work in fluid mechanics and plasma physics and through this work, he established warm relations with the Berkeley physicists Allan Kaufman and Robert Littlejohn. Alan’s important work on periodic orbits in Hamiltonian systems led him even- tually to formulate the “Weinstein conjecture,’’ namely that for a given Hamiltonian flow on a symplectic manifold, there must be at least one closed orbit on a regular compact contact type level set of the Hamiltonian. Along with Arnold’s conjecture, the Weinstein conjecture has been one of the driving forces in symplectic topology over the last two decades. Alan kept up his interest in symplectic reduction theory throughout his later work. For instance, he laid some important foundation stones in the theory of semidirect product reduction as well as in singular reduction through his work on Satake’s V -manifolds, along with finding important links with singular structures in moduli spaces. Intertwined with his work on symplectic geometry and mechanics, he did exten- sive work on geometric PDE, eigenvalues, the Schrödinger operator and geometric quantization. Alan took the point of view of microlocal analysis and phase space structures in his work in this area, emphasizing the links with quantum mechanics. Preface xi

His work on the limit distribution of eigenvalue clusters in terms of the geodesic Radon transform of the potential inspired a large number of related articles. He showed that the geodesic flow of a Zoll surface was symplectically equivalent to that of a round sphere, and hence that its Laplacian could be conjugated globally to the round Lapla- cian plus a pseudodifferential potential. This work inspired many other results on conjugacies. One of Alan’s fundamental contributions to Poisson geometry was the introduction of symplectic groupoids in 1987, which marks the official beginning of his “oids’’ period. In these works, he makes sweeping generalizations about a wide variety of constructions in symplectic geometry, including (with Courant) the important notion of Dirac structures. During this period of generalizations he constantly returned to specific topics in symplectic and Poisson geometry, such as geometric phases and Poisson Lie groups, in addition to making other key links. For instance, symplectic groupoids are used to link Poisson geometry to noncommutative geometry, and groupoids are also intimately related to many other areas, including symmetries and reduction, dual pairs, quantization and the theory of sigma models. One of the central ideas is that the usual theory of Hamiltonian actions, momentum maps, and symplectic reduction makes sense in the more general context of actions of symplectic groupoids; in this setting, momentum maps are Poisson maps taking values in general Poisson manifolds, rather than just Lie–Poisson manifolds (that is, duals of Lie algebras). Alan has raised the question of whether this framework can be further extended to include new notions of momentum maps such as quasi-Poisson manifolds with group-valued momentum maps as well as optimal momentum maps. Alan is well known not only for his brilliant papers and conjectures, but also for his general philosophy, such as the symplectic creed: Everything is a Lagrangian submanifold. Those of us who know him well also appreciate his very special insight. For example, in the middle of a discussion (for instance, as we both had in our joint works on semidirect product reduction as well as stability theory) he will say something like what you are really doing is... and then give us some usually very special insight that invariably substantially improves the whole project. Alan also has a very interesting and charming sense of humor that even makes its way into his papers from time to time. For instance, Alan had great fun in his papers with the “East Coast–West Coast’’ discussions of whether one should use the term momentum map or moment map. He also gave us a good laugh with the term symplectic bones as it relates to the French translation of Poisson as Fish. Alan is a great educator. His lectures, even on Calculus, are always a treat and are very inspiring for their special insight, their wit and lively presentation. His enthu- siasm for mathematics is infectious. One story that comes to mind on the education front is this: during the days when he was exceptionally keen about groupoids, he was preparing a lecture for undergraduates on the subject. Some of us convinced him to present it as a colloquium lecture for faculty, keeping in mind the old advice “no colloquium talk can be too simple.’’ It was, in fact, not only a beautiful colloquium talk, but was perfectly pitched for the faculty, and it became a popular article in the Notices of the American Mathematical Society. Part of being a good educator is being xii Preface cognizant of history.Alan excels in this area. For instance, his research into the history of Lie is what led directly to the introduction of the term “Lie–Poisson’’ bracket. The papers in this volume were selected by invitation and all of them underwent a rigorous refereeing process. While this process took some time, it resulted in high quality papers. We thank all of the referees for their diligent and helpful work. The authors of this volume represent some of the best workers in the subject and their contributions span a wide range of the topics covered by symplectic and Poisson geometry and mechanics, broadly interpreted. The intended audience for the book includes active researchers in the general area of symplectic geometry and mechanics, as well as aspiring graduate students who wish to learn where the subject is headed and what some of the current research topics are.

Alan and Margo have a special relationship to Paris. They have spent many happy times there, and we wish them all the best and many more happy visits in the years to come. We wish to thank Ann Kostant for her expert editorial guidance throughout the production of this volume. Of course, we also thank all the authors for their contributions as well as their helpful guidance and advice. The referees are also thanked for their valuable comments and suggestions.

Jerry Marsden and Tudor Ratiu September, 2004 Academic genealogy of Alan Weinstein

Otto Mencken Universität Leipzig, 1668 Thomae Hobbesii Epicureismum historice delineatum sistit

Johann C. Wichmannshausen Universität Leipzig, 1685 Disputationem Moralem De Divortiis Secundum Jus Naturae

Christian A. Hausen Martin-Luther-Universität Halle-Wittenberg, 1713 De corpore scissuris ﬁgurisque non cruetando ductu

Abraham G. Kaestner Universität Leipzig, 1739 Theoria radicum in aequationibus

Johann F. Pfaff Georg-August-Universität Gottingen, 1786 Commentatio de ortibus et occasibus siderum apud auctores classicos commemoratis

August F. Möbius Universität Leipzig, 1815 De computandis occultationibus ﬁxarum per planetas

Otto W. Fiedler Johannes Frischauf Karl Friesach Universität Leipzig, 1859 Universität Wien, 1861 Universität Wien, 1846

Gustav Ritter von Escherich Emil Weyr Technische Universität Graz, 1873 University of Prague, 1870 Die Geometrie auf Flachen constanter negativer Krummung

Wilhelm Wirtinger Universität Wien, 1887 Uber eine spezielle Tripelinvolution in der Ebene

Wilhelm Blaschke Universität Wien, 1908

Shiing-Shen Chern Universität Hamburg, 1936 Eine Invariantentheorie der Dreigewebe aus r-dimensionalen Mannigfaltigkeiten im R2r

Alan D. Weinstein University of California at Berkeley, 1967 The Cut Locus and Conjugate Locus of a Riemannian Manifold About Alan Weinstein

Alan David Weinstein

Ph.D.: University of California at Berkeley, 1967 Dissertation: The Cut Locus and Conjugate Locus of a Riemannian Manifold Advisor: Shiing-Shen Chern

Students of Alan Weinstein

1. Jair Koiller, Studies on the Spring-Pendulum, 1975 2. Otto Ruiz, Existence of Brake-Orbits in Finsler Mechanical Systems, 1975 3. Yilmaz Akyildiz, Dynamical Symmetries of the Kepler Problem, 1976 4. Gerald Chachere, Numerical Experiments Concerning the Eigenvalues of the Laplacian on a Zoll Surface, 1977 5. John Jacob, Geodesic Symmetries of Homogeneous Kahler Manifolds, 1977 6. Steven Zelditch, Reconstruction of Singularities for Solutions of Schrödinger’s Equations, 1981 7. Enrique Planchart, Analogies in Symplectic Geometry of Some Results of Cartan in Representation Theory, 1982 8. Barry Fortune, A Symplectic Fixed Point Theorem for Complex Projective Spaces, 1984 9. Stephen Omohundro (Department of Physics), Geometric Perturbation Theory in Physics, 1985 10. Theodore Courant, Dirac Manifolds, 1987 11. Yong-Geun Oh, Nonlinear Schrödinger Equations with Potentials: Evolution, Existence, and Stability of Semi-Classical Bound States, 1988 12. Viktor Ginzburg, On Closed Characteristics of 2-Forms, 1990 13. Milton Lopes Filho, Microlocal Regularity and Symbols for Distributions, 1990 14. Jiang-Hua Lu, Multiplicative and Afﬁne Poisson Structures on Lie Groups, 1990 15. Ping Xu, Morita Equivalence of Poisson Manifolds, 1990 xvi About Alan Weinstein

16. Sean Bates, Symplectic End Invariants and C0 Symplectic Topology, 1994 17. Agust Egilsson, On Embedding a Stratiﬁed Symplectic Space in a Smooth Poisson Manifold, 1995 18. Dong Yan, Yang–Mills Theory on Symplectic Manifolds, 1995 19. Zhao-Hui Qian, Groupoids, Midpoints and Quantizations, 1997 20. Vinay Kathotia, Universal Formulae for Deformation Quantization and The Campbell–Baker–Hausdorff Formula, 1998 21. Dmitry Roytenberg, Courant Algebroids, Derived Brackets And Even Symplectic Supermanifolds, 1999 22. Mélanie Bertelson-Volckaert(Stanford University), FoliationsAssociated to Reg- ular Poisson Structures, 2000 23. Benjamin Davis, On Poisson Spaces Associated to Finitely Generated Poisson R-Algebras, 2001 24. Henrique Bursztyn, Morita Equivalence in Deformation Quantization, 2001 25. Olga Radko, Some Invariants of Poisson Manifolds, 2002 26. Xiang Tang, Quantization of Noncommutative Poisson Manifolds, 2003 27. Marco Zambon, Submanifold Averaging in Riemannian, Symplectic, and Contact Geometry, 2003 28. Chenchang Zhu, Integrating Lie Algebroids via Stacks and Applications to Jacobi Manifolds, 2003

Alan Weinstein’s publications

[1] Weinstein, A., On the homotopy type of positively pinched manifolds, Arch. Math., 18 (1967), 523–524. [2] Weinstein, A., A ﬁxed point theorem for positively curved manifolds, J. Math. Mech., 18 (1968), 149–153. [3] Weinstein, A., The cut locus and conjugate locus of a riemannian manifold, Ann. Math. 87 (1968), 29–41. [4] Weinstein, A., Symplectic structures on Banach manifolds, Bull. Amer. Math. Soc., 75 (1969), 1040–1041. [5] Weinstein,A., and Simon, U., Anwendungen der de rhamschen Zerlegung auf Probleme der localen Flächentheorie, Manuscripta Math., 1 (1969), 139–146. [6] Marsden, J., and Weinstein, A., A comparison theorem for hamiltonian vector ﬁelds, Proc. Amer. Math. Soc., 26 (1970), 629–631. [7] Weinstein,A., The generic conjugate locus, in Global Analysis, Proceedings of Symposia on Pure Mathematics, Vol. 15, American Mathematical Society, Providence, RI, 1970, 299–301. + [8] Weinstein, A., Positively curved n-manifolds in Rn 2, J. Differential Geom., 4 (1970), 1–4. [9] Weinstein, A., Positively curved deformations of invariant Riemannian metrics, Proc. Amer. Math. Soc., 26 (1970), 151–152. [10] Weinstein, A., Sur la non-densité des géodésiques fermées, C. R. Acad. Sci. Paris, 271 (1970), 504. [11] Roels, J., and Weinstein, A., Functions whose Poisson brackets are constants, J. Math. Phys., 12 (1971), 1482–1486. About Alan Weinstein xvii

[12] Weinstein, A., Singularities of families of functions, in Proceedings of the Conference “Differentialgeometrie im Grossen,’’ Mathematische Forschungsinstitut, Oberwolfach, Germany, 1971, 323–330. [13] Weinstein, A., Perturbation of periodic manifolds of Hamiltonian systems, Bull. Amer. Math. Soc., 77 (1971), 814–818. [14] Weinstein, A., Remarks on curvature and the Euler integrand, J. Differential Geom., 6 (1971), 259–262. [15] Weinstein, A., Symplectic manifolds and their lagrangian submanifolds, Adv. Math., 6 (1971), 329–346. [16] Weinstein, A., On the invariance of Poincaré’s generating function for canonical transformations, Invent. Math., 16 (1972), 202–213. [17] Weinstein, A., Distance spheres in complex projective spaces, Proc. Amer. Math. Soc., 39 (1973), 649–650. [18] Weinstein, A., Lagrangian submanifolds and hamiltonian systems, Ann. Math., 98 (1973), 377–410. [19] Weinstein, A., Normal modes for nonlinear hamiltonian systems, Invent. Math., 20 (1973), 47–57. [20] Marsden, J., and Weinstein,A., Reduction of symplectic manifolds with symmetry, Rep. Math. Phys., 5 (1974), 121–130. [21] Weinstein, A., Application des opérateurs intégraux de Fourier aux spectres des variétés riemanniennes, C. R. Acad. Sci. Paris, 279 (1974), 229–230. [22] Weinstein, A., On the volume of manifolds, all of whose geodesics are closed, J. Dif- ferential Geom., 9 (1974), 513–517. [23] Weinstein, A., Quasi-classical mechanics on spheres, Sympos. Math., 14 (1974), 25–32. [24] Weinstein, A., On Maslov’s quantization condition, in Proceedings of the International Symposium on Fourier Integral Operators (Nice, May 1974), Lecture Notes in Mathe- matics, Vol. 469, Springer-Verlag, New York, 1975, 341–372. [25] Guillemin, V., and Weinstein, A., Eigenvalues associated with a closed geodesic, Bull. Amer. Math. Soc., 82 (1976), 92–94. [26] Weinstein, A., Fourier integral operators, quantization, and the spectrum of a Rieman- nian manifold, in Géométrie Symplectique et Physique Mathématique, Colloque Inter- nationale de Centre National de la Recherche Scientiﬁque No. 237, CNRS, Paris, 1976, 289–298. [27] Weinstein, A., The principal symbol of a distribution, Bull. Amer. Math. Soc., 82 (1976), 548–550. [28] Weinstein,A., Symplectic V -manifolds, periodic orbits of Hamiltonian systems, and the volume of certain Riemannian manifolds, Comm. Pure Appl. Math., 30 (1977), 265–271. [29] Weinstein, A., Lectures on Symplectic Manifolds, Regional Conference Series in Math- ematics, Vol. 29, American Mathematical Society, Providence, RI, 1977. [30] Weinstein, A., Asymptotics of eigenvalue clusters for the Laplacian plus a potential, Duke Math. J., 44 (1977), 883–892. [31] Weinstein, A., The order and symbol of a distribution, Trans. Amer. Math. Soc., 241 (1978), 1–54. [32] Weinstein,A., Simple periodic orbits, Amer.Inst. Phys. Conf. Proc., 46 (1978), 260–263. [33] Weinstein, A., Eigenvalues of the laplacian plus a potential, in Proceedings of the Inter- national Congress of Mathematicians, Helsinki, 1978, 803–805. [34] Weinstein, A., A universal phase space for particles in Yang–Mills ﬁelds, Lett. Math. Phys., 2 (1978), 417–420. [35] Weinstein, A., Periodic orbits for convex hamiltonian systems, Ann. Math., 108 (1978), 507–518. xviii About Alan Weinstein

[36] Weinstein, A., Bifurcations and Hamilton’s principle, Math. Z., 159 (1978), 235–248. [37] Weinstein, A., On the hypotheses of Rabinowitz’ periodic orbit theorems, J. Differential Equations 33 (1979), 353–358. [38] Marsden, J. E., and Weinstein, A., Review of Geometric Asymptotics and Symplectic Geometry and Fourier Analysis, Bull. Amer. Math. Soc., 1 (1979), 545–553. [39] Marsden, J., and Weinstein, A., Calculus, Benjamin/Cummings, San Francisco, 1980. [40] Weinstein, A., Fat bundles and symplectic manifolds, Adv. Math., 37 (1980), 239–250. [41] Weinstein, A., Nonlinear stabilization of quasimodes, in Proceedings of the AMS Sym- posium on Geometry of the Laplacian, Hawaii, 1979, Proceedings of Symposia on Pure Mathematics, Vol. 36, American Mathematical Society, Providence, RI, 1980, 301–318. [42] Marsden, J., and Weinstein, A., Calculus Unlimited, Benjamin/Cummings, San Fran- cisco, 1981. [43] Croke, C., and Weinstein, A., Closed curves on convex hypersurfaces and periods of nonlinear oscillations, Invent. Math., 64 (1981), 199–202. [44] Marsden, J., Morrison, P., and Weinstein,A., Comments on: The Maxwell–Vlasov equations as a continuous Hamiltonian system, Phys. Lett., 96A (1981), 235–236. [45] Stanton, R. J., and Weinstein A., On the L4 norm of spherical harmonics, Math. Proc. Cambridge Philos. Soc., 89 (1981), 343–358. [46] Weinstein, A., Symplectic geometry, Bull. Amer. Math. Soc. (N.S.), 5 (1981), 1–13. [47] Weinstein, A., Neighborhood classification of isotropic embeddings, J. Differential Geom., 16 (1981), 125–128. [48] Marsden. J., and Weinstein,A., The hamiltonian structure of the Maxwell–Vlasov equations, Physica, 4D (1982), 394–406. [49] Weinstein, A., Gauge groups and Poisson brackets for interacting particles and fields, Amer. Inst. Phys. Conf. Proc., 88 (1982), 1–11. [50] Weinstein, A., What is microlocal analysis?, Math. Intel., 4 (1982), 90–92. [51] Weinstein, A., The symplectic “category,’’ in Doebner, H.-D., Andersson, S. I., and Petry, H. R., eds., Differential Geometric Methods in Mathematical Physics (Clausthal, Germany 1980), Lecture Notes in Mathematics, Vol.905, Springer-Verlag,Berlin, 1982, 45–50. [52] Weinstein,A., and Zelditch, S., Singularities of solutions of some Schrödinger equations on Rn, Bull. Amer. Math. Soc., (1982). [53] Gotay, M. J., Lashof, R., Sniatycki, J., and Weinstein, A., Closed forms on symplectic fibre bundles, Comm. Math. Helv., 58 (1983), 617–621. [54] Marsden, J., Ratiu, T., Schmid, R., Spencer, R., and Weinstein, A., Hamiltonian systems with symmetry, coadjoint orbits, and plasma physics, Atti Acad. Sci. Torino, 117- Supplemento (1983), 289–340. [55] Marsden, J., and Weinstein, A., Coadjoint orbits, vortices and Clebsch variables for incompressible fluids, Physica, 7D (1983), 305–323. [56] Sniatycki, J., and Weinstein, A., Reduction and quantization for singular momentum mappings, Lett. Math. Phys., 7 (1983), 159–161. [57] Weinstein, A., A symplectic rigidity theorem, Duke Math. J., 50 (1983), 1121–1125. [58] Weinstein, A., Hamiltonian structure for drift waves and geostrophic flow, Phys. Fluids, 26, (1983), 388–390. [59] Weinstein, A., Sophus Lie and symplectic geometry, Expos. Math., 1 (1983), 95–96. [60] Weinstein, A., Removing intersections of lagrangian immersions, Illinois J. Math., 27 (1983), 484–500. [61] Weinstein,A., The local structure of Poisson manifolds, J. Differential Geom., 18 (1983), 523–557. About Alan Weinstein xix

[62] Marsden, J., Morrison, P., and Weinstein, A., The Hamiltonian structure of the BBGKY hierarchy equations, Contemp. Math., 28 (1984), 115–124. [63] Marsden, J., Ratiu, T., and Weinstein,A., Reduction and Hamiltonian structures on duals of semidirect product Lie algebras, Contemp. Math., 28 (1984), 55–100. [64] Marsden. J., Ratiu, T., and Weinstein, A., Semidirect products and reduction in mechanics, Trans. Amer. Math. Soc., 281 (1984), 147–177. [65] Weinstein, A., Equations of plasma physics (notes by Stephen Omohundro), in Chern, S. S., ed., Seminar on Nonlinear Partial Differential Equations, MSRI Pub- lications, Springer-Verlag, New York, 1984, 359–373. [66] Weinstein, A., Stability of Poisson-Hamiltonian equilibria, Contemp. Math., 28 (1984), 3–13. [67] Weinstein, A., C0 perturbation theorems for symplectic fixed points and lagrangian intersections, Travaux en Cours, 3 (1984), 140–144. [68] Fortune, B., and Weinstein,A., A symplectic fixed point theorem for complex projective spaces, Bull. Amer. Math. Soc., 12 (1985), 128–130. [69] Holm. D. D., Marsden, J. E., Ratiu, T., and Weinstein, A., Nonlinear stability of fluid and plasma equilibria, Phys. Rep., 123-1–2 (1985), 1–116. [70] Marsden, J., and Weinstein, A., Calculus I, II, III, 2nd ed., Springer-Verlag, New York, 1985. [71] Weinstein,A.,Asymbol calculus for some Schrödinger equations on Rn, Amer. J. Math., (1985), 1–21. [72] Weinstein, A., A global invertibility theorem for manifolds with boundary, Proc. Roy. Soc. Edinburgh Sect. A, 99 (1985), 283–284. [73] Weinstein, A., Periodic nonlinear waves on a half-line, Comm. Math. Phys., 99 (1985), 385–388. [74] Weinstein, A., Poisson structures and Lie algebras, Astérisque, hors série (1985), 421– 434. [75] Weinstein, A., Symplectic reduction and fixed points, in Séminaire Sud-Rhodanien de Géométrie, Rencontre de Balaruc I, Travaux en Cours, Hermann, Paris, 1985, 140–148. [76] Weinstein, A., Three dimensional contact manifolds with vanishing torsion tensor (appendix to a paper by S. S. Chern and R. Hamilton), in Hirzebruch, F., Schwermer J., and Suter S., eds., Proceedings of the Meeting held by the Max-Planck-Institut für Math- ematik, Bonn. June 15–22, 1984, Lecture Notes in Mathematics, Vol. 1111, Springer- Verlag, Berlin, 1985, 306–308. [77] Floer, A., and Weinstein A., Nonspreading wave packets for the nonlinear Schrödinger equation with a bounded potential, J. Functional Anal., 69 (1986), 397–408. [78] Weinstein, A., On extending the Conley-Zehnder theorem to other manifolds, Proc. Sympos. Pure Math., 45 (1986), 541–544. [79] Weinstein, A., Critical point theory, symplectic geometry, and hamiltonian systems, in Proceedings of the 1983 Beijing Symposium on Differential Geometry and Differential Equations, Science Press, Beijing, 1986, 261–289. [80] Coste, A., Dazord, P., and Weinstein, A., Groupoïdes symplectiques, Publ. Dép. Math. Univ. Claude Bernard-Lyon I, 2A (1987), 1–62. [81] Weinstein, A., ed., Some Problems in Symplectic Geometry, Séminaire Sud-Rhodanien de Géométrie VI, Travaux en Cours, Hermann, Paris, 1987. [82] Weinstein, A., Standing and travelling waves for nonlinear wave equations, Transport Theory Stat. Phys., 16 (1987), 267–277. [83] Weinstein, A., The Geometry of Poisson Brackets (Notes by K. Ono and K. Sugiyama), Surveys in Geometry, Tokyo, 1987. xx About Alan Weinstein

[84] Weinstein, A., Poisson geometry of the principal series and nonlinearizable structures, J. Differential Geom., 25 (1987), 55–73. [85] Weinstein,A., Symplectic groupoids and Poisson manifolds, Bull. Amer. Math. Soc., 16, (1987), 101–104. [86] Courant, T. J., and Weinstein, A., Beyond Poisson structures, in Séminaire Sud- Rhodanien de Géométrie VIII, Travaux en Cours, Vol.27, Hermann, Paris, 1988, 39–49. [87] Mikami, K., and Weinstein,A., Moments and reduction for symplectic groupoid actions, Publ. RIMS Kyoto Univ., 24 (1988), 121–140. [88] Weinstein, A., Coisotropic calculus and Poisson groupoids, J. Math. Soc. Japan, 40 (1988), 705–727. [89] Weinstein, A., Some remarks on dressing transformations, J. Fac. Sci. Univ. Tokyo. Sect. 1A Math., 36 (1988), 163–167. [90] Lu, J.-H., and Weinstein, A., Groupoïdes symplectiques doubles des groupes de Lie- Poisson, C. R. Acad. Sci. Paris, 309 (1989), 951–954. [91] Weinstein, A., Cohomology of symplectomorphism groups and critical values of hamil- tonians, Math. Z., 201 (1989), 75–82. [92] Weinstein, A., Blowing up realizations of Heisenberg–Poisson manifolds, Bull. Sci. Math., 113 (1989), 381–406. [93] Lu, J.-H., and Weinstein, A., Poisson Lie groups, dressing transformations, and the Bruhat decomposition, J. Differential Geom., 31 (1990), 501–526. [94] Weinstein, A., Connections of Berry and Hannay type for moving lagrangian submanifolds, Adv. Math., 82 (1990), 133–159. [95] Weinstein, A., Affine Poisson structures, Internat. J. Math., 1 (1990), 343–360. [96] Dazord, P., Lu, J.-H., Sondaz, D., and Weinstein, A., Affinoïdes de Poisson, C. R. Acad. Sci. Paris, 312 (1991), 523–527. [97] Hofer, H., Weinstein,A., and Zehnder, E.,Andreas Floer, 1956–1991 (obituary), Notices Amer. Math. Soc., 38 (1991), 910–911. [98] Lu, J.-H., and Weinstein, A., Classification of SU(2)-covariant Poisson structures on S2 (appendix to a paper of A. J.-L. Sheu), Comm. Math. Phys., 135 (1991), 229–232. [99] Weinstein, A., Symplectic groupoids, geometric quantization, and irrational rotation algebras, in Symplectic Geometry, Groupoids, and Integrable Systems: Séminaire Sud- Rhodanien de Géométrie à Berkeley (1989), Dazord, P., and Weinstein, A., eds., Springer–MSRI Series, Springer-Verlag, New York, 1991, 281–290. [100] Weinstein, A., Contact surgery and symplectic handlebodies, Hokkaido Math. J., 20 (1991), 241–251. [101] Weinstein, A., Noncommutative geometry and geometric quantization, in Donato, P., Duval, C., Elhadad, J., and Tuynman, G. M., eds., Symplectic Geometry and Mathe- matical Physics: Actes du Colloque en l’Honneur de Jean-Marie Souriau, Progress in Mathematics, Birkhäuser, Basel, 1991, 446–461. [102] Weinstein,A., and Xu, P., Extensions of symplectic groupoids and quantization, J. Reine Angew. Math., 417 (1991), 159–189. [103] Ginzburg, V. L., and Weinstein, A., Lie–Poisson structure on some Poisson Lie groups, J. Amer. Math. Soc., 5 (1992), 445–453. [104] Weinstein, A., and Xu, P., Classical solutions of the quantum Yang–Baxter equation, Comm. Math. Phys., 148 (1992), 309–343. [105] Mardsen, J. E., Tromba,A. J., and Weinstein,A., Basic Multivariable Calculus, Springer- Verlag and W. H. Freeman, New York, 1993. [106] Weinstein, A., Traces and triangles in symmetric symplectic spaces, Contemp. Math., 179 (1994), 261–270. About Alan Weinstein xxi

[107] Maeda, Y., Omori, H., and Weinstein, A., eds., Symplectic Geometry and Quantization: Two Symposia on Symplectic Geometry and Quantization Problems, July 1993, Japan, Contemporary Mathematics, Vol.179, American Mathematical Society, Providence, RI, 1994. [108] Weinstein, A., Classical theta functions and quantum tori, Publ. RIMS Kyoto Univ., 30 (1994), 327–333. [109] Birnir, B., McKean, H., and Weinstein,A., The rigidity of sine-Gordon breathers, Comm. Pure Appl. Math., 47 (1994), 1043–1051. [110] Scovel, C., andWeinsteinA., Finite dimensional Lie–Poisson approximations toVlasov– Poisson equations, Comm. Pure Appl. Math., 47 (1994), 683–709. [111] Emmrich, C., and Weinstein,A., The differential geometry of Fedosov’s quantization, in Brylinski, J. L., Brylinski, R., Guillemin, V.,and Kac, V.,eds., Lie Theory and Geometry: In Honor of B. Kostant, Progress in Mathematics, Birkhäuser, Boston, 1994, 217–239. [112] Weinstein, A., Deformation quantization, Astérisque, 227 (1995) (Séminaire Bourbaki, 46ème année, 1993–94, no. 789), 389–409. [113] Bates, S., and Weinstein,A., Lectures on the Geometry of Quantization, Berkeley Math- ematics Lecture Notes, American Mathematical Society, Providence, RI, 1997. [114] Weinstein, A., The symplectic structure on moduli space, in Hofer, H., Taubes, C., Weinstein, A., and Zehnder, E., eds., The Floer Memorial Volume, Birkhäuser, Basel, 1995, 627–635. [115] Weinstein, A., Lagrangian mechanics and groupoids, in Shadwick, W. F., Krish- naprasad, P. S., and Ratiu, T. S., eds., Mechanics Day, Fields Institute Communications, Vol. 7., American Mathematical Society, Providence, RI, 1995, 207–231. [116] Emmrich, C., and Weinstein, A., Geometry of the transport equation in multicomponent WKB approximations, Comm. Math. Phys., 176 (1996), 701–711. [117] Weinstein,A., Groupoids: Unifying internal and external symmetry, Notices Amer.Math. Soc., 43 (1996), 744–752; reprinted in Contemp. Math., 282 (2001), 1–19. [118] Reshetikhin, N., Voronov, A. A., and Weinstein, A., Semiquantum geometry, Algebraic geometry 5, J. Math. Sci., 82 (1996), 3255–3267. [119] Guruprasad, K., Huebschmann, J., Jeffrey, L., and Weinstein, A., Group systems, groupoids, and moduli spaces of parabolic bundles, Duke Math. J., 89 (1997), 377– 412. [120] Weinstein,A., Tangential deformation quantization and polarized symplectic groupoids, in Gutt, S., Rawnsley, J., and Sternheimer, D., eds., Deformation Theory and Symplectic Geometry, Mathematical Physics Studies, Vol. 20, Kluwer, Dordrecht, the Netherlands, 1997, 301–314. [121] Liu, Z.-J., Weinstein, A., and Xu, P., Manin triples for Lie bialgebroids, J. Differential Geom., 45 (1997), 547–574. [122] Weinstein,A., The modular automorphism group of a Poisson manifold, J. Geom. Phys., 23 (1997), 379–394. [123] Weinstein, A., Some questions about the index of quantized contact transformations, RIMS Kôkyûroku, 1014 (1997), 1–14. [124] Weinstein, A., and Xu, P., Hochschild cohomology and characteristic classes for star- products, in Khovanskii, A., Varchenko, A., and Vassiliev, V., eds., Geometry of Differ- ential Equations, American Mathematical Society, Providence, RI, 1997, 177–194. [125] Liu, Z.-J., Weinstein, A., and Xu, P., Dirac structures and Poisson homogeneous spaces, Comm. Math. Phys., 192 (1998), 121–144. [126] Weinstein, A., Poisson geometry, Differential Geom. Appl., 9 (1998), 213–238. [127] Roytenberg, D., and Weinstein, A., Courant algebroids and strongly homotopy Lie algebras, Lett. Math. Phys., 46 (1998), 81–93. xxii About Alan Weinstein

[128] Weinstein, A., From Riemann Geometry to Poisson Geometry and Back Again, Lecture at Chern Symposium, Mathematical Sciences Research Institute, Berkeley, CA, 1998; available from http://msri.org/publications/video/contents.html and on CD-ROM. [129] Cannas da Silva, A., and Weinstein, A., Geometric Models for Noncommutative Al- gebras, Berkeley Mathematics Lecture Notes, American Mathematical Society, Provi- dence, RI, 1999. [130] Nistor, V., Weinstein, A., and Xu., P., Pseudodifferential operators on differential groupoids, Pacific J. Math. 189 (1999), 117–152. [131] Evens, S., Lu, J.-H., and Weinstein, A., Transverse measures, the modular class, and a cohomology pairing for Lie algebroids, Quart. J. Math., 50 (1999), 417–436. [132] Fuchs, D., Eliashberg, Y., Ratiu, T., and Weinstein, A., eds., Northern California Sym- plectic Geometry Seminar, American Mathematical Society, Providence, RI, 1999. [133] Weinstein, A., Almost invariant submanifolds for compact group actions, J. European Math. Soc., 2 (2000), 53–86. [134] Mikami, K., and Weinstein, A., Self-similarity of Poisson structures on tori, in Pois- son Geometry, Banach Center Publications, Vol. 51, Polish Scientific Publishers PWN, Warsaw, 2000, 211–217. [135] Weinstein, A., Linearization problems for Lie algebroids and Lie groupoids, Lett. Math. Phys., 52 (2000), 93–102. [136] Weinstein, A., Omni-Lie algebras, RIMS Kôkyûroku, 1176 (2000), 95–102. [137] Weinstein,A., Review of Riemannian Geometry During the Second Half of the Twentieth Century by Marcel Berger, Bull. London Math. Soc., 33 (2001), 11. [138] Kinyon, M. K., and Weinstein, A., Leibniz algebras, Courant algebroids, and multipli- cations on reductive homogeneous spaces, Amer. J. Math., 123 (2001), 525–550. [139] Weinstein, A., Poisson geometry of discrete series orbits, and momentum convexity for noncompact group actions, Lett. Math. Phys., 56 (2001), 17–30. [140] Marsden, J., and Weinstein, A., Some comments on the history, theory, and applications of symplectic reduction, in Landsman, N. P., Pflaum, M., and Schlichenmaier, M., eds., Quantization of Singular Symplectic Quotients, Birkhäuser, Basel, 2001, 1–19. [141] Hirsch, M. W., and Weinstein, A., Fixed points of analytic actions of supersoluble Lie groups on compact surfaces, Ergodic Theory Dynam. Systems, 21 (2001), 1783–1787. [142] Ševera, P., and Weinstein, A., Poisson geometry with a 3-form background, Progr. The- oret. Phys. Suppl. Ser., 144 (2002), 145–154. [143] Weinstein,A., Linearization of regular proper groupoids, J. Inst. Math. Jussieu, 1 (2002), 493–511. [144] Newton, P.K., Holmes, P.,andWeinstein,A., eds., Geometry, Mechanics, and Dynamics: Special Volume in Honor of the 60th Birthday of J. E. Marsden, Springer-Verlag, New York, 2002. [145] Bursztyn, H., and Weinstein, A., Picard groups in Poisson geometry, Moscow Math. J., 4 (2004), 39–66. [146] Weinstein, A., The geometry of momentum, in Proceedings of the Conference on “Geometry in the 20th Century: 1930–2000,’’ (Paris, September 2001), to appear; math.SG/0208108. [147] Bursztyn, H., Crainic, M., Weinstein, A., and Zhu, C., Integration of twisted Dirac brackets, Duke Math. J., 123 (2004), 549–607. [148] Tang, X., and Weinstein, A., Quantization and Morita equivalence for constant Dirac structures on tori, Ann. Inst. Fourier, 54 (2004), to appear; math.QA/0305413. [149] Weinstein, A., The Maslov gerbe, Lett. Math. Phys., to appear; math.SG/0312274. About Alan Weinstein xxiii

[150] Bursztyn, H., and Weinstein, A., Poisson geometry and Morita equivalence, in Poisson Geometry, Deformation Quantization, and Group Representations, London Mathemati- cal Society Lecture Note Series, Cambridge University Press, Cambridge, UK, to appear; preprint math.SG/0402347. [151] Weinstein, A., Integrating the nonintegrable, in Proceedings of the Workshop “Feuil- letages: Quantiﬁcation Géométrique,’’ Maison des Science de l’Homme, Paris, 2004.