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	Symplectic topology - Wikipedia, the free encyclopedia
		Symplectic topology (also called symplectic geometry although the terms are not completely synonymous) is a branch of differential topology/geometry which studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form.
		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).
	en.wikipedia.org /wiki/Symplectic_geometry (345 words)

	Symplectic topology
		Symplectic topology is that part of mathematics concerned with the study of symplectic manifolds.
		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.
	www.mik.fastload.org /sy/Symplectic_topology.html (542 words)

	PlanetMath: symplectic complement
		For the symplectic complement, we have the following dimension theorem.
		symplectic complement, isotropic subspace, coisotropic subspace, symplectic subspace, Lagrangian subspace
		This is version 5 of symplectic complement, born on 2003-04-02, modified 2004-02-28.
	planetmath.org /encyclopedia/SymplecticComplement.html (91 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.
		Symplectic manifolds arise naturally in abstract formulations of classical mechanics and analytical mechanics as the cotangent bundles of manifolds, e.g.
		A symplectic manifold endowed with a metric that is compatible with the symplectic form is a Kähler manifold.
	en.wikipedia.org /wiki/Symplectic_manifold (662 words)

	Developments in Symplectic Topology, by S. Donaldson (Site not responding. Last check: 2007-10-10)
		A symplectic structure on a manifold is a closed non-degenerate 2-form.
		We describe work of the speaker on the existence of codimension-2 symplectic submanifolds and work of C. Taubes relating the new Seiberg-Wihen invariants of symplectic 4-manifolds to Gromov’s pseudo-holomophic curves.
		Important features of both of these developments are the extension of ideas from Kahler geometry to the symplectic situation, and the role of curvature of complex line bundles.
	www.maths.tcd.ie /EMIS/mirror/IMU/bulletin/39/Donaldson.html (139 words)

	Differential geometry and topology - Gurupedia
		In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds.
		This is an analog of symplectic geometry which works for manifolds of odd dimension.
		A symplectic manifold is a differentiable manifold equipped with a symplectic form (that is, a
	www.gurupedia.com /d/di/differential_geometry.htm (938 words)

	Amazon.ca: Introduction to Symplectic Topology: Books: Dusa McDuff,Dietmar Salamon (Site not responding. Last check: 2007-10-10)
		Symplectic structures underlie the equations of classical mechanics and their properties are reflected in the behaviour of a wide range of physical systems.
		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. Read the first page
		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.
	amazon.ca /Introduction-Symplectic-Topology-Dusa-McDuff/dp/0198504519 (524 words)

	Topology Festival Abstracts (Site not responding. Last check: 2007-10-10)
		This talk will be about Lefschetz fibrations (i.e., fibrations over the 2-sphere with at most nodal fibers), their relation to symplectic 4-manifolds, and their characterization in terms of quasipositive factorizations in mapping class groups.
		This talk will be concerned with the study of families of algebraic varieties by means of symplectic topology.
		We shall explain how symplectic invariants give rise to new restrictions on algebraic families, which apparently cannot be detected on a purely algebraic level.
	www.math.cornell.edu /~festival/2005/abstracts.html (846 words)

	UCSC Mathematics Research
		Symplectic geometry is the geometry underlying classical mechanics and is important to quantum mechanics and low-dimensional topology, and is an active area of research.
		Her work is an interplay between group theory, symplectic geometry, and uses a good deal of symbolic manipulation.
		Professor Tamanoi works in algebraic topology, particularly elliptic cohomology theory and relations between algebraic topology and conformal field theory.
	www.math.ucsc.edu /research/index.html (911 words)

	Contact and Symplectic Geometry - Cambridge University Press
		Symplectic methods are one of the most active areas of research in mathematics currently, and this volume will attract much attention.
		Topology and analysis of contact circles H. Geiges and J. Gonzalo; 5.
		Relative Floer and quantum cohomology and the symplectic topology of Lagrangian submanifolds Y. Oh; 11.
	www.cambridge.org /uk/catalogue/catalogue.asp?isbn=0521570867 (375 words)

	Spring Semester -- 2005 (Site not responding. Last check: 2007-10-10)
		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.
		We will explore the topology of symplectic manifolds, with emphasis on explicit constructions, especially for 4-manifolds and their contact 3-manifold boundaries.
	www.math.utexas.edu /users/bgarcia/gradcse/Gompf-M392C1.html (254 words)

	People Topology Math Science
		With textbooks on Algebraic Topology, Vector Bundles and K-theory, and 3-manifolds.
		Topology and combinatorics: hyperplane arrangements, the topology and geometry of manifolds, the homology of discrete groups, the homotopy theory of high-dimensional knots.
		Quantum algebra and topology; bibliography of Vassiliev invariants.
	www.iaswww.com /ODP/Science/Math/Topology/People (409 words)

	Professional Webpage of Joe Johns (Site not responding. Last check: 2007-10-10)
		This conjecture is essential to understanding the topology of Lagrangian submanifolds of an arbitrary symplectic manifold.
		On the other hand it can be used as an abstract tool which translates problems in symplectic topology into the framework of homological algebra, where they may be easier to solve.
		Other topics in symplectic topology which interest me are more general questions about the topology of Lagrangian submanifolds and the phenomena of subsets which cannot be made disjoint from themselves by a Hamiltonian isotopy.
	www.math.uchicago.edu /~johns (428 words)

	Toric Topology 2006: abstracts
		There is a deep rigidity phenomenon in symplectic topology: certain subsets of a symplectic manifolds cannot be completely displaced from themselves by a Hamiltonian isotopy while it is possible to do so by just a smooth isotopy.
		Associated to a symplectic manifold (or even a stable almost complex manifold) with a finite group action is a stringy cohomology ring due to Fantechi-Goettsche whose coinvariants yield the so-called Chen-Ruan orbifold cohomology of the quotient symplectic orbifold.
		A near-symplectic manifold is a four-manifold equipped with a two-form that is symplectic on the complement of a union of circles and that vanishes ``nicely'' along the circles.
	www.math.toronto.edu /~megumi/ToricTopology/abstracts.html (5542 words)

	Differential geometry and topology - Wikipedia, the free encyclopedia
		Symplectic geometry is the study of symplectic manifolds.
		A symplectic manifold is a differentiable manifold equipped with a symplectic form ω(that is, a non-degenerate, bilinear, skew-symmetric and closed 2-form).
		In particular, a Kähler manifold is a complex and a symplectic manifold.
	en.wikipedia.org /wiki/Differential_geometry (1848 words)

	Symplectic capacities of toric manifolds and related results, Guangcun Lu
		Other related results are impacts of symplectic blow-up on symplectic capacities, symplectic packings in symplectic toric manifolds, the Seshadri constant of an ample line bundle on toric manifolds, and symplectic capacities of symplectic manifolds with $S^{1}$-action.
		V. Ginzburg, The Weinstein conjecture and the theorems of nearby and almost existence, The breadth of symplectic and Poisson geometry, 139--172, Progr.
		G. Lu, Gromov-Witten invariants and pseudo symplectic capacities, math.SG/0103195, v6, 6 September 2001, and v9, 3 December 2004, to appear in Israel Journal of Mathematics.
	projecteuclid.org /Dienst/UI/1.0/Display/euclid.nmj/1142344535 (673 words)

	Geometry / Topology at Michigan State University
		Ron Fintushel's areas of research are gauge theory, smooth 4-manifold topology and knot theory.
		Effie Kalfagianni does research on low dimensional topology, knot theory, and 3-manifolds.
		John McCarthy works on surface mapping class groups and symplectic topology.
	www.math.msu.edu /gt/Faculty.htm (110 words)

	Fields Institute - Symplectic and Contact Topology Workshop
		This two-week workshop will bring together researchers from symplectic topology, algebraic geometry, and mathematical physicists working in gauge theory and quantum field theory.
		Its main goal is to discuss the recent developments in the construction and computations of invariants of symplectic and contact manifolds and their automorphism groups, using methods of the theory of J-holomorphic curves, as well as those from gauge theory and dynamical Hamiltonian systems.
		Cet atelier de deux semaines réunira des chercheurs de topologie symplectique, de géométrie algébrique, et de physique mathématique travaillant en théorie de jauge et en théorie quantique des champs.
	www.fields.utoronto.ca /programs/scientific/00-01/symplectic/sym_topology (657 words)

	Floer Homology, Gauge Theory, and Low Dimensional Topology
		Gauge theory as a tool for studying topological properties of four-manifolds was pioneered by the fundamental work of Simon Donaldson in the early 1980's, and was revolutionized by the introduction of the Seiberg—Witten equations in the mid-1990's.
		The interaction between gauge theory, low—dimensional topology, and symplectic geometry has led to a number of striking new developments in these fields.
		The aim of this volume is to introduce graduate students and researchers in other fields to some of these exciting developments, with a special emphasis on the very fruitful interplay between disciplines.
	www.claymath.org /publications/Floer_Homology (356 words)

	Stanford Symplectic Geometry Seminar 2004-2005
		Symplectic Floer theories, Hilbert schemes, and the Jones polynomial
		Abstract: We shall discuss an existence theorem for foliated bundles with symplectic total holonomy, and its relationship to the homology of symplectomorphism groups.
		It turns out that the classical flux homomorphism on the identity component of the group of symplectomorphisms can sometimes be extended to the full symplectomorphism group not as a homomorphism, but as a crossed homomorphism that is related to the geometry of foliated bundles.
	math.stanford.edu /~lipshitz/seminar0405 (2052 words)

	Amazon.com: Introduction to Symplectic Topology (Oxford Mathematical Monographs): Books: Dusa McDuff,Dietmar Salamon (Site not responding. Last check: 2007-10-10)
		Symplectic structures underlie the equations of classical mechanics, and their properties are reflected in the behavior of a wide range of physical systems.
		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.
		nonsqueezing theorem, symplectic trivialization, symplectic action functional, flux homomorphism, exact symplectic manifold, symplectic circle action, symplectic isotopy, unitary trivialization, linear symplectomorphism, monotone twist condition, symplectic fibration, symplectic complement, symplectic rigidity, symplectic vector bundle, variational family, symplectic ball, symplectic cylinder, symplectic bundle, coisotropic submanifold, symplectomorphism groups, symplectic embeddings, symplectically embedded, symplectic capacity, symplectic connection, symplectic topology
	www.amazon.com /Introduction-Symplectic-Topology-Mathematical-Monographs/dp/0198511779 (1025 words)

	Geometry / Topology at Michigan State University
		Slava works in 4-dimensional topology and geometry.He is currently an Assistant Professor at the University of Virginia.
		His area of interest is the geometric topology of low-dimensional manifolds.
		His area of interest is stringy geometry and topology of orbifolds.
	www.mth.msu.edu /gt/Postdocs.htm (587 words)

	Symplectic Geometry Seminar
		Journees Peter Shalen, a conference on 3-dimensional topology and its role in mathematics, on the occasion of Peter Shalen's sixtieth birthday, June 12-15, 2006, at CRM in Montreal, Quebec.
		The purpose of the conference is to highlight the interactions of group theory with topology and geometry.
		So it is our aim to bring together leading researchers from these branches of mathematics where transformation groups appear, and so the notion of group and the concept of symmetry play important roles.
	www.math.toronto.edu /symplec/announce.html (785 words)

	Colloquim: Leonid Polterovich - Quasi-morphisms and quasi-states in symplectic topology.
		It proved to be a useful tool in geometry, topology and dynamics.
		In the present talk I focus on quasi-morphisms on groups of symplectic diffeomorphisms.
		They come from Floer theory -- the cornerstone of modern symplectic topology.
	www.math.psu.edu /colloquium/spring05/polterovich.html (114 words)

	University of Michigan Department of Mathematics: Symplectic Geometry and Hamiltonian Mechanics
		The permanent faculty members who specialize in Symplectic Geometry and Hamiltonian Mechanics are Anthony Bloch, Dan Burns, Alejandro Uribe, Michael Weinstein.
		Symplectic geometry and Hamiltonian dynamics spans from the core areas of symplectic topology to applied areas such as control theory and robotics.
		This area has seen tremendous growth in the last decade, due to new ground-breaking foundational work on the topology of symplectic manifolds, better understanding of their symmetries and new applications in physics and engineering.
	www.math.lsa.umich.edu /~fornaess/symplectic.html (360 words)

	Geometry and Topology Seminar
		In order to get a handle on the geometry and topology of moduli spaces of such "cut-and-paste" surfaces, we study the basic building blocks of these surfaces, the unduloids themselves.
		On the other hand, one only begins to understand the topology of 4-manifolds in a more systematic way after work of Freedman and Donaldson in the beginning of 1980's.
		I will review some of the basic questions, ideas and tools available in both subjects (group actions and topology of 4-manifolds), and survey some of the major results about finite group actions on 4-manifolds obtained in the past two decades.
	www.math.umass.edu /~sullivan/geotop.html (873 words)

	[No title]
		Symplectic manifolds play a central role in modern topology.
		Fundamental results on the topology of symplectic manifolds have also been obtained by M. Gromov, Y. Eliashberg, R. Gompf, D. McDuff, D. Salamon and G. Tian, among many others.
		More recently, the existence result obtained by Donaldson for symplectic Lefschetz pencil structures on symplectic manifolds has opened a completely new direction in low dimensional symplectic topology, followed by work of Auroux and Katzarkov on the topology of 4-dimensional symplectic manifolds viewed as finite ramified coverings of CP2.
	www.ipam.ucla.edu /programs/sgpws1 (302 words)

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