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# Topic: Differential topology

 Differential geometry and topology - Wikipedia, the free encyclopedia Differential geometry is the study of geometry using calculus. Initially and up to the middle of the nineteenth century, differential geometry was studied from the extrinsic point of view: curves, surfaces were considered as lying in a Euclidean space of higher dimension (for example a surface in an ambient space of three dimensions). A symplectic manifold is a differentiable manifold equipped with a symplectic form (that is, a closed non-degenerate 2-form). en.wikipedia.org /wiki/Differential_geometry   (1106 words)

 Kids.net.au - Encyclopedia Differential topology -   (Site not responding. Last check: 2007-11-07) Differential topology is the field of mathematics dealing with differentiable functions on differentiable manifolds. A vector field is a function from a manifold to the disjoint union of its tangent spaces, such that at each point, the value is a member of the tangent space at that point. A symplectic manifold is a differentiable manifold equipped with a closed non-degenerate bi-linear alternating form (a closed 2-form for short). www.kids.net.au /encyclopedia-wiki/di/Differential_topology   (532 words)

 Differential geometry and topology - LearnThis.Info Enclyclopedia   (Site not responding. Last check: 2007-11-07) In mathematics, differential topology is the field dealing with differentiable functionss on differentiable manifolds. A vector field is a function from a manifold to the disjoint union of its tangent spaces (this union is itself a manifold known as the tangent bundle), such that at each point, the value is an element of the tangent space at that point. Finsler geometry has the Finsler manifold as the main object of study — this is a differential manifold with a Finsler metric, i.e. encyclopedia.learnthis.info /d/di/differential_geometry_and_topology.html   (998 words)

 Differential topology   (Site not responding. Last check: 2007-11-07) In mathematics, differential topology is the field dealingwith differentiable functions on differentiable manifolds. The apparatus of differential geometry is that of calculus on manifolds: this includes the study of manifolds, tangent bundles, cotangent bundles, differential forms, exterior derivatives, integrals of p-forms over p-dimensional submanifolds and Stokes' theorem, wedgeproducts, and Lie derivatives. A symplectic manifold is a differentiable manifold equipped with asymplectic form (that is, a closed non-degenerate 2- form). www.therfcc.org /differential-topology-31952.html   (874 words)

 Graduate Study in Geometry and Topology Modern differential geometry is concerned with the spaces on which calculus of several variables applies (differentiable manifolds) and the various geometrical structures which can be defined on them. Differential topology is the study of those properties of smooth manifolds that are invariant under smooth homeomorphisms with smooth inverses (diffeomorphisms). General topology has been an active research area for many years, and is broadly the study of topological spaces and their associated continuous functions. www.math.uiuc.edu /GraduateProgram/researchmath/gradgeomtop.html   (1213 words)

 Differential geometry and topology - InfoSearchPoint.com   (Site not responding. Last check: 2007-11-07) Initially and up to the middle of the nineteenth century, differential geometry was studied from the extrinsic point of view: curves, surfaces and other objects were considered as lying in a space of higher dimension (for example a surface in an ambient space of three dimensions). A special case of differential geometry is Riemannian manifolds (see also Riemannian geometry): geometrical objects such as surfaces which locally look like Euclidean space and therefore allow the definition of analytical concepts such as tangent vectors and tangent space, differentiability, and vector and tensor fields. A symplectic manifold is a differentiable manifold equipped with a closed 2-form (that is, a closed non-degenerate bi-linear alternating form). www.infosearchpoint.com /display/Differential_topology   (826 words)

 Search Results for Topology By topology we mean the doctrine of the modal features of objects, or of the laws of connection, of relative position and of succession of points, lines, surfaces, bodies and their parts, or aggregates in space, always without regard to matters of measure or quantity. His first-class achievements in topology and functional analysis, in the theory of ordinary and partial differential equations, in the mathematical problems of geophysics and electrodynamics, in computational mathematics and in mathematical physics are all widely known. Obituary: Vijay Kumar Patodi (1945-1976), Topology 16 (1) (1977), i. www-gap.dcs.st-and.ac.uk /~history/Search/historysearch.cgi?SUGGESTION=Topology&CONTEXT=1   (12732 words)

 MAT 566 Differential Topology Differential topology is a subject in which geometry and analysis is used to obtain topological invariants of spaces, often numerical. They are also the basic objects of study of differential geometry, but whereas differential geometry is concerned mainly with local invariants (for example, curvature), differential topology is concerned with global issues. In differential topology we count the number of zeros of a smooth vector field, weighted by their indices, and once again get two. www.math.sunysb.edu /~sawon/MAT566/index.shtml   (832 words)

 Klaus Wirthmueller - Topology The basics of topology (often referred to as point set topology) have the merit that they can be usefully applied in a wide variety of contexts. True topology is often called algebraic because it is based on the surprising fact that topology, an inherently geometric concept, is governed by rich algebraic structures. R.M. Switzer: Algebraic Topology - Homotopy and Homology. www.mathematik.uni-kl.de /~wirthm/E/Wirthmueller/Top   (810 words)

 docs/outreach/oi/waves/pinch.3dhtml Topology -- an important area of mathematics that has applications from subatomic physics to large-scale astronomy -- was born when mathematicians realized that there is something behind the notion of nearness that does not depend on measurements. Topology deals with all sorts of surfaces (and also curves). A subarea called differential topology concentrates on surfaces that are smooth. www.geom.uiuc.edu /docs/outreach/oi/waves/pinch.html   (369 words)

 Amazon.com: Books: Differential Topology (Graduate Texts in Mathematics, Vol 33)   (Site not responding. Last check: 2007-11-07) Topology from the Differentiable Viewpoint by John Milnor To achieve this, the author carefully studies the topology of the spaces of maps between differential manifolds. Those interested in applications of differential topology will be amply prepared to apply these results to the relevant areas, which are many. www.amazon.com /exec/obidos/tg/detail/-/0387901485?v=glance   (1545 words)

 Differential Topology   (Site not responding. Last check: 2007-11-07) A bit of analysis knowledge is nice, particularly in chapter four, and linear algebra (which seems to be a lost art, at least over here in the states) is absolutely critical. It has a lack of rigour that is not made up by being more intuitive or giving the reader insight into why differential topology is such a great subject. Differential topology has influenced many areas of mathematics, and also has many applications in physics, engineering, comptuer graphics, network engineering, and economics. www.onlinemerchantaccountnow.com /BookStore/isbn0132126052.html   (1023 words)

 Topology history Perhaps the first work which deserves to be considered as the beginnings of topology is due to Euler. A second way in which topology developed was through the generalisation of the ideas of convergence. Poincaré developed many of his topological methods while studying ordinary differential equations which arose from a study of certain astronomy problems. www-groups.dcs.st-and.ac.uk /~history/HistTopics/Topology_in_mathematics.html   (1455 words)

 Differential Topology (Graduate Texts in Mathematics, Vol 33) There are numerous excercises on many different levels ranging from practical applications of the theorems to significant further development of the theory and including some open research problems. Hirsch has assembled a very fine text which is suitable for a second year graduate mathematics course in differentiable manifolds. This book introduces the basic concepts in differential topology, a field that has taken on particular importance in medical imaging, game theory, and network optimization. www.literacyconnections.com /0_0387901485.html   (312 words)

 57: Manifolds and cell complexes Perhaps it is easiest to use classic literature to understand differential topology: Flatland; here are two Backup sites and the home page for Project Gutenberg. General topology is for spaces without the local Euclidean nature of the spaces in this section. Geometric topology is a natural language in which to study families of motions; applications include some topics in mechanics of moving particles and systems. www.math.niu.edu /~rusin/known-math/index/57-XX.html   (768 words)

 Combinatorial Differential Topology And Geometry - Forman (ResearchIndex)   (Site not responding. Last check: 2007-11-07) Combinatorial Differential Topology And Geometry - Forman (ResearchIndex) One derived from Morse theory, a standard tool in differential topology, and the other derived from Bochner's method, a standard tool in differential geometry and global analysis. Forman, Combinatorial differential topology and geometry, in New Perspectives in Algebraic Combinatorics (L. Billera, A. Bjorner, C. Greene, R. Simion, and R. Stanley, eds.), Cambridge University Press, to appear. citeseer.ist.psu.edu /257844.html   (988 words)

 Differential Topology and Quantum Field Theory:0125140762:Nash, Charles:eCampus.com The remarkable developments in differential topology and how these recent advances have been applied as a primary research tool in quantum field theory are presented here in a style reflecting the genuinely two-sided interaction between mathematical physics and applied mathematics. The author, following his previous work (Nash/Sen: Differential Topology for Physicists, Academic Press, 1983), covers elliptic differential and pseudo-differential operators, Atiyah-Singer index theory, topological quantum field theory, string theory, and knot theory. The explanatory approach serves to illuminate and clarify these theories for graduate students and research workers entering the field for the first time. www.ecampus.com /bk_detail.asp?isbn=0125140762   (137 words)

 Geometry & Topology   (Site not responding. Last check: 2007-11-07) An accessible introduction to topology, differential geometry, Lie groups and algebras, with illustrations from mechanics, relativity, electromagnetism, and Yang-Mills theory. If you have had no prior exposure to topology, this is a good place to start.  The topics include graphs, classical surfaces, map-coloring problems, knots, the fundamental group, some notions of homology, and fiber bundles. This is an introduction to differential geometry and Lie groups aimed at physicists. members.aol.com /JKempys/links-ge.htm   (1722 words)

 Geometry and Topology, Department of Mathematics, UIUC The document Graduate Study in Geometry and Topology outlines the general areas of geometry and topology studied here and describes the advanced undergraduate and graduate courses that are offered regularly. Topology (contact geometry/topology, Morse theory, braid theory), dynamics (flows, bifurcation theory, Conley index), and applications (fluids, robotics, computational topology). Differential geometry, foliation theory, gauge theory, moduli spaces, low dimensional geometry and topology, topological quantum field theory. www.math.uiuc.edu /GraduateProgram/researchmath/geomtop.html   (335 words)

 Basic Library List-Topology   (Site not responding. Last check: 2007-11-07) From Geometry to Topology Philadelphia, PA: Crane, Russak, 1974. Moise, Edwin E. Geometric Topology in Dimensions 2 and 3 New York, NY: Springer-Verlag, 1977. Differential Topology Englewood Cliffs, NJ: Prentice Hall, 1974. www.maa.org /BLL/topology.htm   (866 words)

 Amazon.com: Books: Differential Topology   (Site not responding. Last check: 2007-11-07) Differential Topology (Graduate Texts in Mathematics, Vol 33) by Morris W. Hirsch This is a perfect book for a first course in manifold theory, provided the student has studied basic multivariable calculus and the differential geometry of curves and surfaces in 3-space. It was only a year later, when taking a course on differential geometry, that an instructor, Doug Moore from UC Santa Barbara, bothered to make these ideas more clear in his excellent lecture notes. www.amazon.com /exec/obidos/tg/detail/-/0132126052?v=glance   (1805 words)

 Differential Topology S'04   (Site not responding. Last check: 2007-11-07) Another useful reference, both for the underlying calculus and for differential forms, is Spivak's Calculus on Manifolds. There will be roughly 2 weeks of preliminary material at the beginning of the term that is not in the text. Pick a theorem of differential topology and discuss it. rene.ma.utexas.edu /users/sadun/S04/prelim/blurb.html   (315 words)

 Allen Hatcher's Homepage This is the first in a series of three textbooks in algebraic topology having the goal of covering all the basics while remaining readable by newcomers seeing the subject for the first time. This is intended to be a readable introduction to spectral sequences, with emphasis on their applications to algebraic topology. The idea would be to present an introduction to differential topology that goes as far as developing the theory of exotic spheres, that is, nonstandard differential structures on the standard topological sphere. www.math.cornell.edu /~hatcher   (1123 words)

 57R: Differential topology Specific manifolds may be treated as Lie groups, etc; for example the classical groups (viewed as geometric objects) are considered in 51N30. Pointers to understanding the construction of exotic differentiable structures on R^4. Topology of the spaces of automorphisms (invertible self-maps, in various categories) of S^3 www.math.niu.edu /~rusin/known-math/index/57RXX.html   (486 words)

 Course Descriptions For example, all of the surfaces studied in the Topology of Surfaces course unit may be given a smooth structure. This structure enables us to do differential calculus and in particular we can apply calculus techniques to study the critical points of real valued functions defined on a smooth manifold, which may be maxima, minima or saddle points. The aim of this module is to introduce the basic ideas of differential topology by studying Morse functions, surgery and immersions. www.ma.man.ac.uk /DeptWeb/UGCourses/Syllabus/Level4/2003/MT4522.html   (397 words)

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