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# Topic: Symplectic manifolds

###### In the News (Wed 19 Jun 19)

 Symplectic topology - Wikipedia, the free encyclopedia Symplectic topology (also called symplectic geometry) is a branch of differential topology/geometry which studies symplectic manifolds; that is, differentiable manifolds equipped with closed, nondegenerate, 2-forms. Symplectic topology has its origins in the Hamiltonian formulation of classical mechanics where the phase space of certain classical systems takes on the structure of a symplectic manifold. Symplectic topology has a number of similarities and differences with Riemannian geometry, which is the study of differentiable manifolds equipped with nondegenerate, symmetric 2-tensors (called metric tensors). www.wikipedia.org /wiki/Symplectic_topology   (317 words)

 Symplectic manifold - Wikipedia, the free encyclopedia In mathematics, a symplectic manifold is a smooth manifold M equipped with a closed, nondegenerate, 2-form ω called the symplectic form. The study of symplectic manifolds is called symplectic topology. Symplectic manifolds arise naturally in abstract formulations of classical mechanics and analytical mechanics as the cotangent bundles of manifolds, e.g. en.wikipedia.org /wiki/Symplectic_manifold   (317 words)

 Symplectic manifold   (Site not responding. Last check: 2007-11-07) A symplectic manifold is a pair (M,ω) where M is a smooth manifold and ω is a closed, nondegenerate, 2-form on M called the symplectic form. Symplectic topology is that part of mathematics concerned with the study of symplectic manifolds. A Riemann surface mapped into a symplectic manifold compatibly with the almost complex structure is called a pseudoholomorphic curve, and Gromov proved a compactness theorem for such curves; this result has led to the development of a fairly large subdiscipline of symplectic topology. www.sciencedaily.com /encyclopedia/symplectic_manifold   (690 words)

 PlanetMath: examples of symplectic manifolds   (Site not responding. Last check: 2007-11-07) Examples of symplectic manifolds: The most basic example of a symplectic manifold is All orbits in the coadjoint action of a Lie group on the dual of it Lie algebra are symplectic. This is version 4 of examples of symplectic manifolds, born on 2002-12-06, modified 2004-10-24. planetmath.org /encyclopedia/ExamplesOfSymplecticManifolds.html   (189 words)

 Symplectic topology   (Site not responding. Last check: 2007-11-07) A symplectic manifold is a pair (M,ω) of a smooth manifold M together with a closed non degenerate differential 2-form ω, the symplectic form. Fundamental examples of symplectic manifolds are given by the cotangent bundles of manifolds; these arise in classical mechanics, where the set of all possible configurations of a system is modelled as a manifold, and this manifold's cotangent bundle describes the phase space of the system. Unlike Riemannian manifolds, symplectic manifolds are extremely non-rigid: they have many symplectomorphisms coming from Hamiltonian vectorfields. www.termsdefined.net /sy/symplectic-topology.html   (690 words)

 [No title]   (Site not responding. Last check: 2007-11-07) Chapter 1, Section 2: Stein manifolds Starting references: Ellisberg/Gromov, Convex symplectic manifolds Ellisberg (notes by A) Symplectic geometry of plurisubharmonic functions Seidel/Smith, Symplectic geometry of Ramanujan's surface (unpublished note) More symplectic terminology (to work fruitfully with symplectic manifolds which are non-compact or have boundary): Def'n: Let $N^{2n-1}$ be an odd-dimensional manifold. A {\em contact 1-form} on $N$ is a nowhere-zero 1-form $\alpha$ such that $d\alpha$ is a symplectic structure on the subbundle $\xi=\ker(\alpha) \subset TN$. Idea: use this to introduce a suitable class of noncompact symplectic manifolds Def'n: Let $(M,\omega)$ be symplectic, $\theta$ a 1-form primitive of $\omega$, $\phi:M \to \R$ a proper exhausting function (i.e., a function which is proper and bounded below; thus $\{x\phi(x) \leq c\}$ is compact, and these sets exhaust the manifold). www.math.uchicago.edu /~msmukler/oct28   (732 words)

 An Interview with Izu Vaisman, CIM Bulletin #8 Since you want an explanation for a non expert, let me first tell that symplectic geometry and topology is a field of mathematics which studies symplectic manifolds and generalizations, these being objects which the non expert should think of as phase spaces of mechanical systems. Symplectic topology is the study of the category of symplectic manifolds and their natural equivalences (symplectomorphisms). In symplectic geometry, I include the general study of symplectic manifolds and their generalizations, under all the aspects which are of interest either in mathematics itself or in applications to mathematical physics. at.yorku.ca /i/a/a/h/05.htm   (1194 words)

 Spring Semester -- 2005   (Site not responding. Last check: 2007-11-07) Symplectic structures (closed, nondegenerate 2-forms) arose from classical physics and algebraic geometry, but in recent years they have developed deep connections with other areas such as low-dimensional topology. On 4-manifolds, the existence of symplectic structures is a delicate question perhaps analogous to hyperbolization of 3-manifolds. While symplectic manifolds are necessarily even dimensional, their natural boundaries are odd-dimensional contact manifolds. www.math.utexas.edu /users/bgarcia/gradcse/Gompf-M392C1.html   (254 words)

 Cornell Math - MATH 758, Spring 2000 A symplectic manifold is a smooth, even-dimensional manifold equipped with a smooth, non-degenerate, closed two-form. Such manifolds have played an important role in classical mechanics for well over a century, but only recently have they been studied extensively from a topological viewpoint. Many interesting examples of 4-manifolds are symplectic, as are Kaehler manifolds and (more generally) almost-complex manifolds. www.math.cornell.edu /~www/Courses/GradCourses/SP00/758.html   (131 words)

 Amazon.ca: Books: Torus Actions on Symplectic Manifolds   (Site not responding. Last check: 2007-11-07) Symplectic manifolds and torus actions are investigated, with numerous examples of torus actions, for instance on some moduli spaces. In Chapters II and III, symplectic geometry and Hamiltonian group actions are introduced, especially torus actions and action-angle variables. As a family of examples of symplectic manifolds, moduli spaces of flat connections are discussed in Chapter V. Then, Chapter VI centers on the Duistermaat-Heckman theorem. www.amazon.ca /exec/obidos/ASIN/3764321768   (307 words)

 E-Cats多言語対応OPAC Introduction to symplectic topology / Dusa McDuff and Dietmar Salamon. Generalized symplectic geometries and the index of families of elliptic problems / Liviu I. Nicolaescu. Symplectic geometry and quantization : two symposia on symplectic geometry and quantization problems, July 1993, Japan / Yoshiaki Maeda, Hideki Omori, Alan Weinstein, editors. runners.ritsumei.ac.jp /opac/disp-query?mode=2&con1=5&kywd1=%53%79%6D%70%6C%65%63%74%69%63%20%6D%61%6E%69%66%6F%6C%64%73&con2=3&con3=4&disp=1   (492 words)

 Abelian fibred holomorphic symplectic manifolds   (Site not responding. Last check: 2007-11-07) Irreducible holomorphic symplectic manifolds are higher dimensional generalizations of K3 surfaces. The purpose of this article is to describe a framework for understanding irreducible holomorphic symplectic manifolds, which hopefully will lead towards some kind of classification. There is evidence to suggest that if the 2n-dimensional irreducible holomorphic symplectic manifold X admits a non-trivial fibration, then the fibres must be abelian varieties and the base must be P^n (a large part of this has been proved by Matsushita). www.math.sunysb.edu /~sawon/abelianHK.shtml   (575 words)

 Convex polytopes and quantization of symplectic manifolds -- Vergne 93 (25): 14238 -- Proceedings of the National ... Quantum mechanics associate to some symplectic manifolds M a quantum model Q(M), which is a Hilbert space. A symplectic manifold of dimension 2n is a manifold M with a closed nondegenerate two-form In particular, the symplectic volume of M is calculated by integrating over P the locally polynomial function h(t). www.pnas.org /cgi/content/full/93/25/14238   (2513 words)

 [No title]   (Site not responding. Last check: 2007-11-07) A new tool to study reducibility of a weak symplectic form to a constant one is introduced and used to prove a version of Darboux theorem more general than previous ones. More precisely, at each point of the considered manifold a Banach space is associated to the symplectic form (dual of the phase space with respect to the symplectic form), and it is shown that Darboux theorem holds if such a space is locally constant. Consider a weak symplectic manifold $M$ on which Darboux theorem is assumed to hold (e.g. www.ma.utexas.edu /mp_arc/a/98-18   (129 words)

 Contact Program   (Site not responding. Last check: 2007-11-07) The birth of symplectic topology in eighties (see [3, 29, 7, 21], et al.), greatly influenced by Arnold's conjectures (see [2]) brought an exciting development in contact topology as well. Contact homology is a systematic way to bring Gromov's very successful theory of pseudoholomorphic curves [29] in symplectic manifolds into the arena of contact topology. With the new invariants of contact homology and symplectic field theory we hope to lay a foundation for the theory of contact structures on 5-manifolds. www.aimath.org /projects/contactgeometry/program.html   (1555 words)

 Quantum Geometry A very interesting class of examples of noncommutative *-algebras can be obtained by deforming the algebras of smooth functions over symplectic manifolds M, so that a quantum correspondence principle holds. This requirement is actually a central problem in the deformation quantization of symplectic manifolds. be the (commutative) algebra of smooth functions on a symplectic manifold M. www.matem.unam.mx /~micho/qgeom9.html   (285 words)

 Geometry and Topology of Manifolds Conference 2004 A Conference on Geometry and Topology of Manifolds will be held at the campus of McMaster University in Hamilton, Ontario, Canada, during May 14–18, 2004. The objective of this conference is to describe recent progress in low-dimensional topology arising from geometric techniques, such as gauge theory, symplectic and contact geometry, etc., as well as to discuss, and to compare questions, methods and applications. Geography of symplectic 4-manifolds with Kodaira dimension one www.math.mcmaster.ca /geomtop/conference   (510 words)

 SYMPOSIUM ON SYMPLECTIC GEOMETRY 1997--2000, REPORT The mathematical topics covered a wide range of subjects centered around symplectic topology and geometry, as well as its many applications and related areas such as Kähler and algebraic geometry or the use of symmetry in mechanics, and several joint activities with these groups were organised (programmes are appended). His approach is to fill as much as possible of the symplectic manifold by a disc bundle over the symplectic submanifold representing a multiple of the symplectic form. Their proof involves the study of symplectic fibrations over the 2-sphere, and is based on the work of Seidel [128]. www.maths.warwick.ac.uk /mrc/1997-98/report.html   (8037 words)

 RESEARCH in DIFFERENTIAL GEOMETRY Kähler manifolds (a class of complex manifolds which includes algebraic manifolds) possess a differential 2-form, w, which is non-degenerate (i.e., wÙw is never 0), and closed (i.e., dw = 0). It used to be believed that closed, symplectic manifolds were always complex. ), is a differentiable invariant of the manifold. facpub.stjohns.edu /~watsonw/diffgeom.htm   (1548 words)

 A Symplectic Analogue Of The Mostow-Palais Theorem - Gotay, Tuynman (ResearchIndex)   (Site not responding. Last check: 2007-11-07) A Symplectic Analogue Of The Mostow-Palais Theorem - Gotay, Tuynman (ResearchIndex) Gotay, M.J. and Tuynman, G.M., A symplectic analogue of the MostowPalais theorem, (to appear in Proceedings of the Seminaire SudRhodanien de Geometrie a Berkeley). 1 is a universal symplectic manifold for reduction (context) - Gotay, Tuynman - 1989 citeseer.csail.mit.edu /612483.html   (344 words)

 Symplectic Manifolds with No Kahler Structure (Lecture Notes in Mathematics): 紀伊國屋書店BookWeb This is a research monograph covering the majority of known results on the problem of constructing compact symplectic manifolds with no Kaehler structure with an emphasis on the use of rational homotopy theory. The aim of this text is to clarify the interrelations between certain aspects of symplectic geometry and homotopy theory in the framework of the problems mentioned above. Symplectic Structures in Total 137(36) Spaces of Bundles 5.1 Preliminaries on Homogeneous Spaces 137(4) 5.2 Compact Homogeneous Symplectic Manifolds 141(5) 5.3 The Weinstein Problem for Fiber Bundles 146(5) 5.4 Koszul Complexes and Minimal Models of 151(10) Homogeneous Spaces 5.5 Symplectic Fat Bundles and The 161(12) Formalizing Tendency of Symplectic Structures Chapter 6. bookweb.kinokuniya.co.jp /guest/cgi-bin/booksea.cgi?ISBN=3540631054   (403 words)

 Amazon.ca: Books: Riemannian Geometry of Contact and Symplectic Manifolds   (Site not responding. Last check: 2007-11-07) Chapter 4 focuses on the general setting of Riemannian metrics associated with both symplectic and contact structures, and Chapter 5 is devoted to integral submanifolds of the contact subbundle. Topics treated in the subsequent chapters include Sasakian manifolds, the important study of the curvature of contact metric manifolds, submanifold theory in both the K¿hler and Sasakian settings, tangent sphere bundles, curvature functionals, complex contact manifolds and 3 Sasakian manifolds. The book serves both as a general reference for mathematicians to the basic properties of symplectic and contact manifolds and as an excellent resource for graduate students and researchers in the Riemannian geometric arena. www.amazon.ca /exec/obidos/ASIN/0817642617   (383 words)

 Amazon.com: Books: Introduction to Symplectic Topology (Oxford Mathematical Monographs)   (Site not responding. Last check: 2007-11-07) Symplectic structures first arose in the study of classical mechanical systems such as the planetary system, and almost all the classical work on symplectic geometry was focused on the attempt to understand how these systems behave. An authoritative and comprehensive reference...McDuff and Salamon have done an enormous service to the symplectic community: their book greatly enhances the accessibility of the subject to students and researchers alike. The discussion begins with classic topology and cover a variety of final year undergraduate topics such as complex manifolds and inverse differential techniques before moving into the vastly complex world of Symplectic Topology. www.amazon.com /exec/obidos/tg/detail/-/0198504519?v=glance   (665 words)

 Amazon.com: Books: Structure of Dynamical Systems (Progress in Mathematics)   (Site not responding. Last check: 2007-11-07) The aim of the book is to treat all three basic theories of physics, namely, classical mechanics, statistical mechanics, and quantum mechanics from the same perspective, that of symplectic geometry, thus showing the unifying power of the symplectic geometric approach. In Chapter 3 a coherent symplectic description of Galilean and relativistic mechanics is given, culminating in the classification of elementary particles (relativistic and non-relativistic, with or without spin, with or without mass). In Chapter 4 statistical mechanics is put into symplectic form, finishing with a symplectic description of the kinetic theory of gases and the computation of specific heats. www.spinics.net /am/0817636951   (641 words)

 Constructing symplectic forms on 4-manifolds which vanish on circles Constructing symplectic forms on 4-manifolds which vanish on circles alpha > 0, a closed 2-form omega is constructed, Poincare dual to alpha, which is symplectic on the complement of a finite set of unknotted circles Z. The number of circles, counted with sign, is given by d = (c(1)(s)(2)-3sigma(X)-2chi(X))/4, where s is a certain spin(C) structure naturally associated to omega. D T. Gay and Robion Kirby, "Constructing symplectic forms on 4-manifolds which vanish on circles" (2004). repositories.cdlib.org /postprints/251   (127 words)

 Publications [#9565] of Robert L Bryant This is a series of nine elementary lectures on Lie groups and symplectic geometry that were the basis for a short course in the subject that I gave in Park City, Utah in 1990 as part of the Regional Geometry Institute Summer School. As a result of my using these lectures again as a resource for a graduate course in Spring 2003 and also as a result of Eugene Lerman pointing out some problems with Lecture 8 (wherein I discuss hyperKähler reduction), I have found some serious flaws in Lecture 8. A series of lectures on Lie groups and symplectic geometry, aimed at the beginning graduate student level. fds.duke.edu /db/aas/math/faculty/bryant/publications/9565   (246 words)

 The invariants and classification of complex, symplectic and smooth 4-manifolds   (Site not responding. Last check: 2007-11-07) The invariants and classification of complex, symplectic and smooth 4-manifolds More recent work of Donaldson allows many of the characteristic constructions of algebraic geometry to be carried out for symplectic 4-manifolds; he and Ivan Smith are currently working out Lefschetz fibrations. Another area of current progress based on Donaldson's foundational work is the symplectic analog of branched coverings between algebraic surfaces; this is closely related to topology and braid monodromy, and is being studied by Catanese and Teicher. euclid.mathematik.uni-kl.de /NEW/node24.html   (219 words)

 UK Nonlinear News 14 (November 1998): Book Review In this framework `elementary particles' are defined to be symplectic manifolds on which the appropriate symmetry group acts transitively, so that all motions can be transformed into each other by a change of reference frame. Souriau and other authors have shown that, under appropriate conditions, every such symplectic manifold is a covering space of an orbit of a natural action of the group on the dual of its Lie algebra. The aim is to produce a theory which assigns to any symplectic space of motions U a Hilbert space H of quantum states, and a mapping from an appropriate space of smooth function on U (the classical observables) to a corresponding space of operators on H (the quantum observables). www.amsta.leeds.ac.uk /Applied/news.dir/issue14.dir/art/review3.html   (1303 words)

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