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	Morse theory - Wikipedia, the free encyclopedia
		Morse theory allows one to find CW structures and handle decompositions on manifolds and to obtain substantial information about their homology.
		Morse theory can be used to prove some strong results on the homology of manifolds.
		Morse homology is a particularly nice approach to the homology of smooth manifolds that is defined using a generic choice of Morse function and Riemannian metric, and has an infinite dimensional analog known as Floer homology.
	en.wikipedia.org /wiki/Morse_theory (1527 words)

	Morse (Site not responding. Last check: 2007-11-07)
		Morse taught briefly at Harvard before entering military service for the period of World War I. For the duration of the war he served as a private in the U.S. Army in France and for his outstanding work in the Ambulance Corps he was awarded the Croix de Guerre with Silver Star.
		This is now called Morse theory and it grew out of a major discovery which Morse made not long after returning to mathematics after the war and published in his important paper Relations between the critical points of a real function of n independent variables in 1925.
		Morse theory is important in the field of global analysis which is the study of ordinary and
	www-groups.dcs.st-and.ac.uk /~history/Mathematicians/Morse.html (1021 words)

	Schlumberger Fellowship Awarded to Student Studying Morse Theory
		Morse theory was developed in the 1920s by mathematician Marston Morse.
		Original Morse theory, which applied to a class of mathematical objects called smooth manifolds-such as a plane, a circle and the surface of a sphere-provides a general tools of attacking this problem.
		An important application of Morse theory is a catastrophe theory developed by several mathematicians such as R. Thom and others.
	hypatia.math.uri.edu /~kulenm/mth381pr/morseth/morseth.html (656 words)

	Morse Theory for Hamiltonian Systems
		It introduces a new Morse (index) theory that is easier to use, less technical, and more flexible than existing theories and features techniques and results that, until now, have appeared only in scattered journals.
		Morse Theory for Hamiltonian Systems provides a detailed description of the Maslov index, introduces the notion of relative Morse index, and describes the functional setup for the variational theory of Hamiltonian systems, including a new proof of the equivalence between the Hamiltonian and the Lagrangian index.
		Morse Theory for Hamiltonian Systems not only offers clear, well-written prose and a unified account of results and techniques, but it also stimulates curiosity by leading readers into the fascinating world of symplectic topology.
	www.ramex.com /ch/ch-4632.html (203 words)

	Morse Code and Phonetic Alphabets (Site not responding. Last check: 2007-11-07)
		The Java Morse code translator translates to and from Morse code and can play the Morse to you as sound.
		The CGI Morse code translator translates to and from Morse code and can play the Morse to you as sound.
		Morse 2000 Outreach - Morse in rehabilitation and education
	morsecode.scphillips.com (120 words)

	3.3 Fermat’s principle and Morse theory in globally hyperbolic spacetimes
		The standard application of Morse theory is the geodesic problem for Riemannian (i.e., positive definite) metrics: given two points in a Riemannian manifold, to find the geodesics that join them.
		A critical point (geodesic) is non-degenerate if the two end-points are not conjugate to each other, and its Morse index is the number of conjugate points in the interior, counted with multiplicity (“Morse index theorem”).
		In this way Morse theory provides us with precise mathematical versions of the statements “A fl hole produces infinitely many images” and “An isolated transparent gravitational lens produces an odd number of images”.
	univie.ac.at /EMIS/journals/LRG/Articles/lrr-2004-9/articlesu12.html (1205 words)

	Reidemeister torsion in circle-valued Morse theory (Site not responding. Last check: 2007-11-07)
		In Morse theory, one starts with a generic real-valued smooth function on a closed smooth manifold.
		The three papers below develop the Morse theoretic part of this story, which works in any number of dimensions and does not require the function to be harmonic.
		On a three-manifold with positive first Betti number, we conjecture that a finer version of the Morse theory invariant is equal to the Seiberg-Witten invariant, by analogy with Taubes' ``Seiberg-Witten = Gromov'' theorem in four dimensions.
	math.berkeley.edu /~hutching/pub/rt.html (529 words)

	Robin Forman, Scholarly Interests, Rice University (Site not responding. Last check: 2007-11-07)
		"Morse Theory and Evasiveness," Topological Combinatorics, a special session at the Joint Meeting of the American Mathematics Society and the Mathematical Association of America, San Antonio, Texas (January, 1999).
		"Discrete Morse Theory and Index Theory," Recent Developments in Index Theory, a special session at the regional meeting of the Americal Mathematical Society, Austin, Texas (October, 1999.).
		Katherine Crowley, Doctoral Thesis, "Discrete Morse Theory and the Geometry of Nonpositively Curved Simplicial Complexes" (2001).
	dacnet.rice.edu /faculty?FDSID=637 (1177 words)

	Morse Theory.nb
		Here we try to demonstrate this one example: Morse theory on a closed surface (in our case a torus).
		Basically Morse Theory can be thought of as a generalization of the classical theory of critical points (maxima, minima and saddle points) of smooth functions on Euclidean spaces.
		A function all of whose critical points are isolated and non-degenerate is called a Morse function.
	www.mimuw.edu.pl /~akoz/Mathematica/IMS2001/Morse (1664 words)

	Morse Theory (L24) (Site not responding. Last check: 2007-11-07)
		Morse Theory, developed by Marston Morse as a calculus of variations for paths in a manifold, became a powerful tool in the study of the topology of manifolds.
		The basic idea is that the gradient flow of a generic function determines a decomposition of the manifold into cells, and the ``attaching maps'' can be deduced from local information about the function.
		We shall study the basic constructions and results of the theory, discuss some classical applications (many of them due to Bott), and conclude by proving some well-known theorems in mathematics (possibly the Bott periodicity theorem and/or the Narasimhan-Seshadri theorem) by Morse theory methods.
	www.maths.cam.ac.uk /CASM/courses/02-03/descriptions/node37.html (210 words)

	A User's Guide To Discrete Morse Theory - Forman (ResearchIndex) (Site not responding. Last check: 2007-11-07)
		Abstract: this paper we present an adaptation of Morse Theory that may be applied to any simplicial complex (or more general cell complex).
		There have been other adaptations of Morse Theory that can be applied to combinatorial spaces.
		For example, a Morse theory of piecewise linear functions appears in [26] and the very powerful \Strati ed Morse Theory" was developed by Goresky and MacPherson [19],[20].
	citeseer.ist.psu.edu /forman01users.html (632 words)

	Morse functions (Site not responding. Last check: 2007-11-07)
		Morse, M.: Topological Methods in the Theory of Functions of a Complex Variable....
		Morse and Melnikov Functions for NLS Pde's 1...
		AMCA: Equivalence of Morse functions and Morse-Smale flows on 3-manifolds by Ale...
	www.scienceoxygen.com /math/677.html (151 words)

	Morse theory
		In differential topology, the techniques of Morse theory give a very direct way of visualising and possibly computing the homology of a manifold.
		The Morse lemma says that near such a non-degenerate critical value, co-ordinates may be taken locally on M making f into just the quadratic form one gets by diagonalising the Hessian.
		Given the apparatus, the basic assertion of Morse theory, that the Betti numbers can be read from f, isn't technically so hard.
	news-server.org /m/mo/morse_theory.html (616 words)

	Digital Morse Theory with Suggested Applications (Site not responding. Last check: 2007-11-07)
		It is based on an adaptation of Morse theory to volume data.
		The methods work without any assumption that the underlying interpolation function for the volume data is Morse, in the traditional sense.
		Rather, we identify and exploit the properties shared by interpolants that are consistent with typical isosurface extraction methods.
	cis.poly.edu /seminars/fall99/abstract8fall99.htm (331 words)

	IEEE Joint Seminar (Site not responding. Last check: 2007-11-07)
		Recently I came across with Morse theory (coined by Marston Morse, not S.F.B. Morse who invented the Morse code) and was greatly intrigued by its power and beauty.
		By analyzing functions defined on a space Morse theory gives information about the shape of the space and hence provides a link between analysis and topology.
		The classical Morse theory is "continuous" and I thought a "discrete" variation would be useful to computer-related problems.
	www.acae.cuhk.edu.hk /seminar/ieee/20021003A.html (218 words)

	Reply to Chuck Morse, "Theory of the Anti-Globalization Movement"
		Reply to Chuck Morse, "Theory of the Anti-Globalization Movement"
		Theory of the Anti-Globalization Movement,” New Formulation 1, no. 1 (November 2001): 22–31.
		We ask not only because we do not know the way (we do not), but also because asking the way is part of the revolutionary process itself” (Ibid.).
	www.newformulation.org /2brecherreply.htm (2429 words)

	Combinatorial Novikov-Morse Theory (ResearchIndex) (Site not responding. Last check: 2007-11-07)
		Abstract: this paper we extend the theory to include a combinatorial analog of Novikov's theory.
		We begin in x1 by defining, and beginning the investigation of, the appropriate notion of a combinatorial differential form.
		23 Supersymmetry and Morse theory (context) - Witten - 1982
	citeseer.ist.psu.edu /315969.html (524 words)

	Morse Theory, Minimax Theory and their Applications to Nonlinear Differential Equations
		Morse Theory, Minimax Theory and their Applications to Nonlinear Differential Equations
		This volume cotains both survey and creative papers dealing with Morse Theory, Minimax theory, Iteration theory of Maslov-type index and critical minimization problems.
		Some Advances in Morse Theory and Minimax Theory – Shujie Li On a Class of Elliptic Eigenvalue Problems with Constraint – Yongqing Li Iteration Theory of Maslov-type Index and its Applications – Chungen Liu
	www.intlpress.com /books/math/morse.html (336 words)

	Morse theory -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-07)
		Morse theory -- Facts, Info, and Encyclopedia article
		According to the basic insights of (Click link for more info and facts about Marston Morse) Marston Morse, a differentiable function on a manifold will, in a typical case, reflect the topology quite directly.
		Consider, for purposes of illustration, a mountainous landscape M.
	www.absoluteastronomy.com /encyclopedia/M/Mo/Morse_theory.htm (1423 words)

	Morse theory
		In differential topology, the techniques of Morse theory give a very direct way of visualising and possibly computing the homology (mathematics) of a manifold.
		Assuming a compact manifold M, such a function f, taking real values, will have a minimum value which starts M with a bowl; and a maximum value which caps off M.
		What Morse did was to identify this sequence, reading from the bottom up, namely bowl-saddle point-saddle point-cap with the Betti number sequence of the torus, namely 1, 2, 1, 0, 0,...
	www.ukpedia.com /m/morse-theory.html (605 words)

	Morse Theory Tutorial - Fall 2003 (Site not responding. Last check: 2007-11-07)
		Morse theory is an extremely simple tool which has revolutionized fields of mathematics several times over.
		Morse himself developed the theory and applied it to mathematical physics.
		Morse theory on loop spaces with the energy functional;
	www.math.princeton.edu /~cmanoles/morse.html (237 words)

	Amazon.ca: Books: Morse Theory for Hamiltonian Systems (Site not responding. Last check: 2007-11-07)
		Within six succinct chapters, Morse Theory for Hamiltonian Systems provides a detailed description of the Maslov index, introduces the notion of relative Morse index, and describes the functional setup for the variational theory of Hamiltonian systems, including a new proof of the equivalence between the Hamiltonian and the Lagrangian index.
		Provides mathematicians with a self-contained treatment of morse theory for Hamiltonian systems, with full proofs.
		Offers a clear account of the results found and the techniques used in the text, including the introduction of a Morse index theory that is more flexible than existing theories.
	www.amazon.ca /exec/obidos/ASIN/1584882026 (342 words)

	BioGeometry Research Project
		The structural properties of such functions are studied in Morse theory, which rests on the insight that every smooth function is near a generic smooth function whose critical points are isolated.
		We take another step and extend the differential notions from classical to combinatorial Morse theory, which studied piecewise linear real-valued functions on manifolds.
		This step is necessary because we are limited to finite measurements of smooth functions.
	biogeometry.duke.edu /research/year3/combinatorial.morse/index.html (301 words)

	Geometric Functional Analysis Seminar Abstract (Site not responding. Last check: 2007-11-07)
		ABSTRACT: In this talk we survey some recent work on a combinatorial version of Morse theory, which can be applied to very general cell complexes.
		This theory has recently been used to solve a variety of problems in combinatorial topology.
		If time permits, we will show how these arguments can be generalized to yield a combinatorial analogue of Novikov's extension of Morse theory to multi-valued functions.
	www.math.psu.edu /gfa/PastSeminars/SP99Abstracts/forman.html (126 words)

	Math Seminars (Site not responding. Last check: 2007-11-07)
		A Morse Theory Approach to Studying Configuration Spaces
		The (labeled) configuration space C_k(M) of k distinct points in a manifold M can be thought of as the k-fold product MxMx...xM with all of its diagonals removed.
		I will introduce the basic concepts of Morse theory and show how they extend to manifolds with corners.
	www.math.psu.edu /dynsys/abstracts-2003/handron.html (121 words)

	Infinite Dimensional Morse Theory and Multiple Solution Problems (Site not responding. Last check: 2007-11-07)
		The author not only employs Morse theory as a tool to study multiple solutions to differential equations arising in the calculus of variations, but covers a broad range of applications to semilinear elliptic PDE, to dynamical systems and symplectic geometry, and to geometry of harmonic maps and minimal surfaces.
		Topics covered include basic Morse theory and its various extensions, minimax principles in Morse theory, and applications of semilinear boundary value problems, periodic solutions of Hamiltonian systems, and harmonic maps.
		Containing several new results, this volume will be attractive and germaine to researchers and graduate students working in nonlinear analysis, nonlinear functional analysis, partial differential equations, ordinary differential equations, differential geometry, and topology.
	isbn.nu /0817634517 (446 words)

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