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Topic: Symplectomorphisms

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	Symplectomorphism - Wikipedia, the free encyclopedia
		In mathematics, a symplectomorphism is an isomorphism in the category of symplectic manifolds.
		A Hamiltonian symplectomorphism is a symplectomorphism that arises as the flow of a Hamiltonian vector field, and hence from some Hamiltonian function.
		The flow of a symplectic vector field on a symplectic manifold is a symplectomorphism.
	en.wikipedia.org /wiki/Symplectomorphism (598 words)

	Hamiltonian mechanics - Wikipedia, the free encyclopedia
		By Liouville's theorem, each symplectomorphism preserves the volume form on the phase space.
		The collection of symplectomorphisms induced by the Hamiltonian flow is commonly called the Hamiltonian mechanics of the Hamiltonian system.
		Every smooth function G over the symplectic manifold generates a one-parameter family of symplectomorphisms and if { G, H } = 0, then G is conserved and the symplectomorphisms are symmetry transformations.
	en.wikipedia.org /wiki/Hamiltonian_mechanics (1321 words)

	AMCA: Loops in the group of Hamiltonian symplectomorphisms by Andres Vina (Site not responding. Last check: 2007-09-01)
		AMCA: Loops in the group of Hamiltonian symplectomorphisms by Andres Vina
		Let (M, \omega) be a symplectic manifold; the group of Hamiltonian symplectomorphisms of M is denoted by Ham(M).
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts.
	at.yorku.ca /c/a/f/w/09.htm (197 words)

	Vladimir Arnold - Wikipedia, the free encyclopedia
		Arnold can be said to have initiated the theory of symplectic topology as a distinct discipline.
		The Arnold conjecture on the number of fixed points of Hamiltonian symplectomorphisms and Lagrangian intersections were also a major motivation in the development of Floer homology.
		Arnold is well known for his lucid writing style, combining mathematical rigour with physical intuition, and an easy conversational style of teaching.
	en.wikipedia.org /wiki/Vladimir_Arnold (411 words)

	Topology Festival Abstracts (Site not responding. Last check: 2007-09-01)
		A symplectomorphism is a diffeomorphism of a manifold that preserves a symplectic form.
		Ever since Gromov showed that the group of symplectomorphisms of the product of two 2-spheres of equal size has the homotopy type of an extension of SO(3) x SO(3) by Z/2Z, people have been interested in understanding the special properties of groups of symplectomorphisms.
		The classification theory of hyperbolic 3-manifolds (with finitely generated fundamental group) hinges on Thurston's conjecture from the late 70's, that such a manifold is uniquely determined by its topological type and a finite number of invariants that describe the asymptotic structure of its ends.
	www.math.cornell.edu /~festival/2002/abstracts.html (392 words)

	Floer homology of families (Site not responding. Last check: 2007-09-01)
		My former student Tamas Kalman used this idea in his thesis to show that the inclusion of the space of Legendrian knots in R^3 into the space of smooth knots induces a noninjective map on the fundamental group.
		Hamiltonian isotopic symplectomorphisms, 3-manifolds, Legendrian knots, etc.) parametrized by a smooth manifold B. The invariant of a family consists of a spectral sequence whose E^2 term is the homology of B with twisted coefficients in the Floer homology of the fibers.
		The spectral sequence for families of symplectomorphisms will be constructed in a sequel.
	math.berkeley.edu /~hutching/pub/floerfamilies.html (416 words)

	Converse KAM Theory for monotone positive symplectomorphisms - Haro (ResearchIndex) (Site not responding. Last check: 2007-09-01)
		These are useful for symplectomorphisms in the annulus that satisfy weaker hypotheses than those usually required.
		They are introduced not only for the orbits of a symplectomorphism, but also for the so-called invariant Lagrangian graphs.
		Determination of an Exact Symplectomorphism From Its Primitive..
	citeseer.ist.psu.edu /197587.html (461 words)

	Schwarz' Abstract (Site not responding. Last check: 2007-09-01)
		An open conjecture by V.I.~Arnold is that the number of fixed points of exact symplectomorphisms should be at least the minimal number of critical points of smooth functions on $M$.
		In the case of nondegenerate fixed points the topological estimate that the number of fixed points is not less than the dimension of the rational homology of $M$ can now be proved by means of Floer homology for all closed symplectic manifolds.
		This equivalence between Floer homology and quantum cohomology is used in order to prove a quantum cup-length estimate for degenerate fixed points which unifies all previously known estimates.
	www.math.duke.edu /conferences/geomfest97/SchwarzAbstract.html (249 words)

	[No title]
		Denote by $Diff^{\infty}_{\omega}(M^2)$ the set of symplectomorphisms of $M^2$ equipped with the Whitney topology, i.e., the diffeomorphisms $f$ of $M^2$ which preserve the symplectic form $\omega$, i.e., $f^* \omega = \omega$.
		Since hyperbolic period points are $C^1$ structurally stable, Pixton's theorem yields a symplectomorphism $h \in U$ with a hyperbolic fixed point $z$ near $y$ such that the stable and unstable manifolds of $z$ intersect transversely.
		The KAM theorem implies that the set of ergodic symplectomorphisms is not dense.} \endroster \Refs \widestnumber\key{KW} \ref \key F \by B. Fuller \paper The Existence of Periodic Points \jour Annals of Math.
	www.ma.utexas.edu /mp_arc/html/papers/95-213 (1395 words)

	Topology Festival Abstracts
		In sharp contrast with Riemannian geometry, symplectic geometry has no local invariants (Darboux's theorem) and this gives rise to an infinite-dimensional group of transformations preserving the symplectic structure (symplectomorphisms).
		In particular, I will give a complete description of the rational cohomology ring of the symplectomorphism groups of rational ruled surfaces (due to Gromov, Abreu and Abreu-McDuff).
		This will show that, although in general these groups do not have the homotopy type of a finite-dimensional Lie group (flexibility), their topology reflects and is determined by the various different subgroups of (Kaehler) isometries they have (rigidity).
	www.math.cornell.edu /~festival/2001/abstracts.html (686 words)

	Citebase - D-branes, Symplectomorphisms and Noncommutative Gauge Theories (Site not responding. Last check: 2007-09-01)
		Authors: Martin, I. Ovalle, J. Restuccia, A. It is shown that the dual of the double compactified D=11 Supermembrane and a suitable compactified D=10 Super 4D-brane with nontrivial wrapping on the target space may be formulated as noncommutative gauge theories.
		It is shown that all these theories may be described in terms of symplectic connections on symplectic fibrations.
		The world volume being its base manifold and the (sub)group of volume preserving diffeomorphisms generate the symplectomorphisms which preserve the (infinite dimensional) Poisson bracket of the fibration.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:hep-th/0005095 (214 words)

	Lusternik-Schnirelmann Category in the New Millennium
		It is more surprising, however, that various off-shoots of category lead to a place for Hopf invariants in dynamics also.
		Finally, recent approaches to Arnold's conjecture that Hamiltonian symplectomorphisms on a symplectic manifold have at least as many fixed points as any function on the manifold has critical points reveal the strategic position Lusternik-Schnirelmann theory occupies in symplectic topology.
		The goal of this conference is to bring together mathematicians from areas on which category has had an impact to review the state of the art, set the course for future investigations and foster cross-fertilization among areas.
	www.csuohio.edu /math/oprea/LScatconf/lscatconf.html (485 words)

	[No title] (Site not responding. Last check: 2007-09-01)
		The investigator will develop a general procedure for translating problems in this area into combinatorial questions involving the monodromy of a family of codimension 2 submanifolds.
		It is expected that this will apply, in principle, both to the classification problem for symplectic manifolds and also to Lagrangian submanifolds and symplectomorphisms.
		Once the general foundations are in place applications will be considered: the question here will be to see if the combinatorial problems can be cast into a tractable form.
	www.cs.utexas.edu /users/yguan/NSFAbstracts/Abstracts/MPS/DMS.MPS.a9803192.txt (349 words)

	[No title] (Site not responding. Last check: 2007-09-01)
		Title: Homotopy decomposition of a group of symplectomorphisms of S^2\times S^2 Authors: Silvia Anjos and Gustavo Granja AMS Classification numbers: 57S05, 57R17, 55R35 Address of Authors: Departamento de Matematica Instituto Superior Tecnico Av.
		Rovisco Pais 1049-001 Lisboa Portugal Email address of Authors: sanjos@math.ist.utl.pt ggranja@math.ist.utl.pt Abstract: We continue the analysis started by Abreu, McDuff and Anjos of the topology of the group of symplectomorphisms of $S^2 \times S^2$ when the ratio of the areas of the two spheres lies in the interval (1,2].
		We use this to compute the homotopy type of the classifying space of the group of symplectomorphisms and the corresponding ring of characteristic classes for symplectic fibrations.
	www.math.purdue.edu /research/atopology/Anjos-Granja/homotopy.decomp.symplect.abstract (126 words)

	[No title]
		We use this to compute the homotopy type o* f the classifying space of the group of symplectomorphisms* and the correspondi* *ng ring of characteristic classes for symplectic fibrations.
		Let G~ denote the group of symplectomorphisms of M~.
		By now, much is known about the topology of the group G~ and, more generally, about symplectomorphism groups of ruled surfaces.
	www.math.purdue.edu /research/atopology/Anjos-Granja/homotopy.decomp.symplect.txt (1494 words)

	Seminars at Weizmann Institute of Science Faculty of Mathematical Sciences: Tue Nov 2 03:00:00 2004
		We find the local algebra behind this notion and show that the symplectic defect is in fact, an $\Ascr$-invariant of $f$ in the case of maximal isotropic map-germs.
		If the symplectic defect is positive on the plane then the classification of plane curves by symplectomorphisms differs from the classification by diffeomorphisms and the difference depends only on the $\Ascr$-equivalence class of the plane curves.
		Finally we provide the generic classification of symplectic bifurcations of curves together with some possible applications.
	www.weizmann.ac.il /usersfiles/math/html/sem.2004:11:02:14:00:0:22014.shtml (158 words)

	Matches for:
		One is related to the program of Homological Mirror Symmetry: the author defines a category of extended complex manifolds and studies its properties.
		The subject of the final paper is Non-commutative Symplectic Geometry, in particular the structure of the symplectomorphism group of a non-commutative complex plane.
		Pidstrygach -- On action of symplectomorphisms of the complex plane on pairs of matrices
	www.mathaware.org /bookstore?fn=20&arg1=ficseries&item=FIC-35 (314 words)

	18.966 - Geometry of Manifolds - Spring 2004
		Mon Feb 9: more symplectic linear algebra; symplectic manifolds; examples; symplectic structure on a cotangent bundle; Lagrangian submanifolds (pp.
		Wed Feb 11: conormal bundles; graphs of symplectomorphisms as Lagrangian submanifolds in products; isotopies and vector fields, Lie derivative; Hamiltonian vector fields (pp.
		Mon Feb 23: Weinstein's neighborhood theorem; tangent space to the group of symplectomorphisms (pp.
	www-math.mit.edu /~auroux/18.966 (731 words)

	Proceedings of the American Mathematical Society
		Using this fact, we prove that the analogous statement holds for groups of symplectomorphisms of certain blowups.
		M.Abreu, The topology of the group of symplectomorphisms of
		M.Abreu and D.McDuff, The topology of the groups of symplectomorphisms of ruled surfaces, J. Amer.
	www.mathaware.org /proc/2005-133-01/S0002-9939-04-07507-0/home.html (224 words)

	[No title] (Site not responding. Last check: 2007-09-01)
		Title : Mathematical Sciences: Topics in Topology Abstract : 9501701 Burghelea This project pursues research on the topology of the free loop spaces, on the homotopy type of the space of symplectomorphisms and on von Neumann topology.
		This research will provide new developments in areas like algebraic geometry, differential geometry, elliptic operators, and von Neumann algebras.
		It will also answer specific questions (solve ppoblems) of immediate concern in topology, geometry and mathematical physics.
	www.cs.utexas.edu /users/yguan/NSFAbstracts/Abstracts/MPS/DMS.MPS.a9501701.txt (102 words)

	[No title]
		Thus the symplectomorphism group of a manifold is always infinite dimensional.
		The special structures that arise in symplectic topology (Gromov-Witten invariants, quantum homology and so on) place as yet rather poorly understood restrictions on the topological properties of this group.
		This talk will explain some of these and describe some open questions.
	www.math.psu.edu /oldColloquium/030320.html (52 words)

	nlin.SI/0309017: Khesin, B., Levin, A., Olshanetsky, M. (Site not responding. Last check: 2007-09-01)
		We also generalize this bihamiltonian construction of integrable Euler-Arnold tops to several infinite-dimensional groups, appearing as certain large $N$ limits of GL(N).
		These are the group of a non-commutative torus (NCT) and the group of symplectomorphisms $SDiff(T^2)$ of the two-dimensional torus.
		The elliptic rotator on symplectomorphisms gives an elliptic version of an ideal 2D hydrodynamics, which turns out to be an integrable system.
	front.math.ucdavis.edu /nlin.SI/0309017 (305 words)

	Amazon.ca: Lectures on Symplectic Geometry: Books: Ana Cannas da Silva (Site not responding. Last check: 2007-09-01)
		The goal of these notes is to provide a fast introduction to symplectic geometry for graduate students with some knowledge of differential geometry, de Rham theory and classical Lie groups.
		This text covers symplectomorphisms, local forms, contact manifold, compatible almost complex structures, Kaehler manifolds, hamiltonian mechanics, moments maps, symplectic reduction and symplectic toric manifolds.
		It contains guided problems, called homework, designed to complement the exposition or extend the reader's understanding.
	www.amazon.ca /exec/obidos/ASIN/3540421955 (299 words)

	DC MetaData for: A 2-Cocycle on a Group of Symplectomorphisms
		DC MetaData for: A 2-Cocycle on a Group of Symplectomorphisms
		we construct a 2-cocycle on the group of symplectomorphisms
		First and second version had authors Mark Losik, Peter W. Michor and title "Extensions for a Group of Diffeomorphisms of a Manifold Preserving an Exact 2-form"
	www.esi.ac.at /Preprint-shadows/esi1525.html (97 words)

	Energy Citations Database (ECD) - Energy and Energy-Related Bibliographic Citations
		Energy Citations Database (ECD) Document #20462616 - Diffeomorphisms as symplectomorphisms in history phase space: Bosonic string model
		Availability information may be found in the Availability, Publisher, Research Organization, Resource Relation and/or Author (affiliation information) fields and/or via the "Full-text Availability" link.
		Diffeomorphisms as symplectomorphisms in history phase space: Bosonic string model
	www.osti.gov /energycitations/product.biblio.jsp?osti_id=20462616 (253 words)

	Introduction to Symplectic Geometry, fall 2004. (Site not responding. Last check: 2007-09-01)
		These may include, but are not restricted to, these topics:
		Review of differential forms and cohomology; Symplectomorphisms; Local normal forms; Hamiltonian mechanics;
		Group actions and moment maps; Geometric quantization; A glimpse of holomorphic techniques.
	www.math.toronto.edu /~karshon/grad (174 words)

	[No title] (Site not responding. Last check: 2007-09-01)
		Jerusalem Topology and Geometry Seminar This Sunday our seminar will be back!
		I am happy to report that Michael Entov from the Technion will speak on "Commutator Length of Symplectomorphisms" Sunday, 6 January 2002, 14:00 Mathematics, Room 209 Abstract: Any element of the commutant subgroup of a group can be represented as a product of commutators.
		The minimal number of factors in such a product is called the commutator length of the group element.
	www.math.technion.ac.il /~techm/20020106140020020106ent (97 words)

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