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Topic: Combinatorial topology

	Topology Encyclopedia
		Topology (Greek topos, "place," and logos, "study") is a branch of mathematics that is an extension of geometry.
		In pointless topology one considers instead the lattice of open sets as the basic notion of the theory, while Grothendieck topologies are certain structures defined on arbitrary categories which allow the definition of sheaves on those categories, and with that the definition of quite general cohomology theories.
		The trade-off is that the accuracy of the topology map depends on the granularity of the polling...
	www.hallencyclopedia.com /topic/Topology.html (1819 words)

	Amazon.com: Combinatorial Topology: Books: P. S. Alexandrov (Site not responding. Last check: 2007-11-05)
		Fundamental topological facts, together with detailed explanations of the necessary technical apparatus, constitute this clearly written, well-organized 3-part text.
		Part 1 deals with certain classic problems without using the formal techniques of homology theory; parts 2 and 3 focus on the central concept of combinatorial topology, the Betti groups.
		Topology I: General Survey (Encyclopaedia of Mathematical Sciences) by S.P. Novikov in Back Matter
	www.amazon.com /Combinatorial-Topology-P-S-Alexandrov/dp/0486401790 (732 words)

	topology. The Columbia Encyclopedia, Sixth Edition. 2001-05 (Site not responding. Last check: 2007-11-05)
		Topology is sometimes referred to popularly as “rubber-sheet geometry” because a figure can be changed to that of an equivalent figure by bending, stretching, twisting, and the like, but not by tearing or cutting.
		Topology is concerned with those properties of geometric figures that are invariant under continuous transformations.
		A surface is a simple example of a topological space, the basic entity studied in topology.
	www.bartleby.com /65/to/topology.html (892 words)

	What Is Topology?
		Topology is a relatively new branch of mathematics; most of the work has been done since 1900.
		Combinatorial topology considers the global property of spaces, built up from a network of vertices, edges and faces.
		Topology is used in many branches of mathematics, such as differentiable equations, dynamical systems, knot theory, and Riemann surfaces in complex variables.
	www.math.uwaterloo.ca /PM_Dept/What_Is/Topology/topology.shtml (470 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-groups.dcs.st-and.ac.uk /history/Search/historysearch.cgi?SUGGESTION=Topology&CONTEXT=1 (12732 words)

	54: General topology
		Topology is the study of sets on which one has a notion of "closeness" -- enough to decide which functions defined on it are continuous.
		Thus a general theme in topology is to test the extent to which the axioms force the kind of structure one expects to use and then, as appropriate, introduce other axioms so as to better match the intended application.
		Since the axioms of topology are stated in terms of subsets of X, it should be no surprise that one branch of topology is closely related to set theory, particularly "descriptive set theory".
	www.math.niu.edu /~rusin/known-math/index/54-XX.html (2431 words)

	BioGeometry Recommended Literature (Site not responding. Last check: 2007-11-05)
		It is often said that topology started as a focused discipline with the work of Henri Poincare right before and after the end of the nineteenth century.
		Combinatorial studies in topology lead naturally to algebraic structures as discrete representations of continuous spaces.
		The algebraisation of topology initiated by Emmy Noether has led to a field dominated by the algebraists.
	biogeometry.duke.edu /education/literature/topology.html (416 words)

	topology -> Branches of Topology on Encyclopedia.com 2002
		Arbitrated Loop This ring topology is widely used and can connect up to 127 nodes without using a switch.
		LAN Bus 10Base5 "Thick" Ethernet and 10Base2 "Thin" Ethernet use a bus topology, which is a common cable between.
		Switch Fabric A switch fabric is the most flexible topology, enabling all servers and storage devices to communi.
	www.encyclopedia.com /html/section/topology_branchesoftopology.asp (567 words)

	Interactive Mathematics Miscellany and Puzzles
		Combinatorial topology has a wealth of applications, many of which result from connections with the theory of differential equations.
		Point set topology and group theory are developed as they are needed.
		In addition, a supplement surveying point set topology is included for the interested student and for the instructor who wishes to teach a mixture of point set and algebraic topology.
	www.cut-the-knot.com /books/henle/back.shtml (242 words)

	Topology - Wikibooks
		General Topology is based solely on set theory and concerns itself with structures of sets.
		Topology generalises many distance related concepts, such as continuity, compactness and convergence.
		Topology deals mostly with concepts such as open sets and continuous functions.
	en.wikibooks.org /wiki/Topology (252 words)

	Park City Mathematics Institute (Site not responding. Last check: 2007-11-05)
		Candidates should have completed basic graduate courses in topology and/or algebra (preferably both.) In general, these students will have completed their first year, and in some cases, may already be working on a thesis.
		Most existing proofs of polynomiality and positivity for q, t-quantities rely on a geometric interpretation involving the Hilbert scheme of points in the plane, which leads to a quasi-combinatorial interpretation of the coefficients as character multiplicities in a doubly-graded representation of the symmetric group.
		The M\"obius number of a poset, an important combinatorial invariant, is equal to the reducted Euler characteristic of the order complex, and important topological invariant.
	www.admin.ias.edu /ma/2004/gss2004.htm (1651 words)

	Topology
		Elegant, intuitive approach to topology from self-theoretic topology to Betti groups; how concepts of topology are useful in math and physics.
		Among the best available reference introductions to general topology, this volume is appropriate for advanced undergraduate and beginning graduate students.
		Fresh approach explains nontrivial applications of metric space topology to analysis; topics from elementary algebraic topology focus on concrete results with minimal formalism.
	store.doverpublications.com /by-subject-science-and-mathematics-mathematics-topology.html (607 words)

	[No title]
		In topology we discuss the Moebius strip without specifying just how rapidly it twists around itself, we specify "coffee cup" without discussing capacity or diameter, and so on: these are real geometric objects, and yet we focus on the underlying shape, free to make smooth distortions.
		I suppose this is called "Geometric Topology", and it includes all the parts of topology which are used to entice unsuspecting schoolchildren to investigate the subject: those topological puzzles with chains and loops, Klein bottles, Knot Theory, and so on.
		Finally, I might mention "Differential Topology", which is closely related to the geometric side, since most of the geometric shapes we play with are "nice" in the sense that they are curves or surfaces: every point has a neighborhood which looks just like the line or the plane.
	www.math.niu.edu /~rusin/known-math/99/what_top (2341 words)

	Topology Festival Abstracts
		Despite this local flexibility, there is a certain rigidity that controls global symplectic phenomena, and symplectic topology studies this global rigidity.
		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).
		We will show how combinatorial analogues of some ingredients of differential topology, such as Vector Fields and their corresponding Flows, can play an important role in the investigation of combinatorial spaces (i.e.
	www.math.cornell.edu /~festival/2001/abstracts.html (686 words)

	Jerusalem Mathematics Colloquium (Site not responding. Last check: 2007-11-05)
		Abstract: "Infinite Combinatorial Topology", despite having its roots back in the Cantor era, is a new field in mathematics.
		This field studies, from a combinatorial point of view, the diagonalization arguments which appear in topological properties and constructions.
		Many classical properties are put into a general framework, and combinatorial methods are used to obtain new insights into these.
	www.ma.huji.ac.il /~colloq/2002-03/col.030220.html (159 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).
		Combinatorial group theory, decision problems, automata theory and formal language theory, computational complexity.
	www.math.uiuc.edu /GraduateProgram/researchmath/geomtop.html (335 words)

	#38: New Perspectives in Algebraic Combinatorics (Site not responding. Last check: 2007-11-05)
		Algebraic combinatorics involves the use of techniques from algebra, topology and geometry in the solution of combinatorial problems, or the use of combinatorial methods to attack problems in these areas.
		The rich combinatorial problems arising from the study of these various areas are the subject of this book, which represents work done or presented at seminars during the program.
		It contains contributions on matroid bundles, combinatorial representation theory, lattice points in polyhedra, bilinear forms, combinatorial differential topology and geometry, Macdonald polynomials and geometry, enumeration of matchings, the generalized Baues problem, and Littlewood-Richardson semigroups.
	www.msri.org /publications/books/Book38/desc.html (242 words)

	Topology of Combinatorial Differential Manifolds - Anderson (ResearchIndex) (Site not responding. Last check: 2007-11-05)
		Topology of Combinatorial Differential Manifolds - Anderson (ResearchIndex)
		Abstract: We prove that all combinatorial differential manifolds involving only Euclidean oriented matroids are PL manifolds.
		20 Topology of oriented matroids (context) - Edmonds, Mandel - 1982
	citeseer.ist.psu.edu /201357.html (442 words)

	The Princeton Mathematics Community in the 1930s (PMC39) (Site not responding. Last check: 2007-11-05)
		In the late '30s I was working in combinatorial topology with not a great deal of results to show.
		I had an opportunity to teach an undergraduate course in topology, combinatorial topology that is, classification of 2-dimensional surfaces and that sort of thing.
		I was trying to write a book on combinatorial topology to go with the undergraduate course that I had been teaching at Princeton.
	libweb.princeton.edu /libraries/firestone/rbsc/finding_aids/mathoral/pmc39.htm (7510 words)

	קרן וולף · Wolf Foundation
		Infinite-combinatorial topology: This is a new branch with old roots in the field of set-theoretic topology.
		The approach is often to develop “dictionaries” translating topological properties into combinatorial ones, and then applying the known methods of infinite combinatorics.
		Algebraic geometry and combinatorial group theory: We found a method which yields a probabilistic solution to a given system of equations in “random” subgroups of the braid group.
	www.cs.biu.ac.il /~tsaban/cv.html (1205 words)

	Robin Forman, Scholarly Interests, Rice University (Site not responding. Last check: 2007-11-05)
		Robin Forman, "Combinatorial Differential Topology and Geometry," New Perspectives in Algebraic Combinatorics, L.J. Billera, et al.
		Robin Forman, "Applications of Combinatorial Differential Topology," in the Proceedings of the SullivanFest, the conference in honor of Dennis Sullivan's 60th birthday, (2004).
		Robin Forman, "Applications of combinatorial differential topology," Proceedings of the SullivanFest, a conference in honor of Dennis Sullivan's 60th birthday, (to appear).
	dacnet.rice.edu /faculty?FDSID=637 (1177 words)

	Jeff Erickson's Publications by Subject
		Combinatorial geometry focuses primarily on counting important features (such as vertices, facets, or incidences) of complicated geometric objects (such as triangulations, polyhedra, Voronoi diagrams, or approximation hierarchies).
		Along with their combinatorial duals, Delaunay triangulations, they have been an object of formal scientific study for well over a hundred years.
		The goal here is to identify useful combinatorial and geometric features of real geometric data, and then either analyze existing algorithms in light of those features, or develop new algorithms to exploit them.
	compgeom.cs.uiuc.edu /~jeffe/pubs/category.html (997 words)

	CR: MA/0141 (sec 001) Combinatorial Topology (Site not responding. Last check: 2007-11-05)
		“Combinatorial Topology” is the study of the properties of objects that do not change when they are deformed or stretched.
		For example, a sphere would be topologically equivalent to a cube, but neither one would be the same as a doughnut.
		The subject is quite abstract, but topology is one of the most fascinating fields of mathematics.
	www.brown.edu /Students/Critical_Review/2003.2004.1/MA0141_1GOO.html (329 words)

	Max Dehn (Site not responding. Last check: 2007-11-05)
		Dehn's interests later turned to topology and combinatorial group theory.
		In 1907 he wrote with Poul Heegaard the first book on the foundations of combinatorial topology, then known as analysis situs.
		In 1910 Dehn published a paper on three dimensional topology in which he introduced Dehn surgery and used it to construct homology spheres.
	www.worldhistory.com /wiki/M/Max-Dehn.htm (451 words)

	Chronological list of videos
		Tabachnikov Combinatorial Problems Arising in Knots and 3-manifolds #2/4 970121 MSRI Geometry Workshop, 1992 Combinatorics and low-dimensional topology X-S Lin & Homology and combinatorics of singular knots V.
		Gordon--Introductory Workshop in Combinatorics and Low-dimensional topology #18/20 960812-960823 Representation theory and symmetric functions, V B.
		Sagan Intoductory Workshop in Combinatorics and Low-dimensional Topology #15/20 960812-960823 Geometric Combinatorics, III L.
	www.msri.org /local/library/video_list.html (1717 words)

	Basic Library List-Topology
		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)

	Geometry and Topology for Mesh Generation - Cambridge University Press
		The book combines topics in mathematics (geometry and topology), computer science (algorithms), and engineering (mesh generation).
		The original motivation for these topics was the difficulty faced (both conceptually and in the technical execution) in any attempt to combine elements of combinatorial and of numerical algorithms.
		Mesh generation is a topic where a meaningful combination of these different approaches to problem solving is inevitable.
	www.cambridge.org /uk/catalogue/catalogue.asp?isbn=0521793092 (220 words)

	Cornell Math - Marshall M. Cohen
		I am a geometric topologist and a combinatorial group theorist.
		Much of my work has dealt with the introduction of combinatorial and algebraic themes into geometric problems or geometric themes into combinatorial and algebraic problems.
		Over the years this work has involved the intermingling of topological manifolds, combinatorial topology, the foundations of piecewise linear topology, simple-homotopy theory, automorphisms of free groups, spaces of length functions on groups and equations over groups.
	www.math.cornell.edu /People/Faculty/cohen.html (272 words)

	The Geomblog: Betti Numbers
		Topology is an area that has many intriguing connections with geometry and algorithms in general (Kneser's conjecture, the Kahn-Saks-Sturtevant partial resolution of the evasiveness conjecture), and we need books that can bridge the gap.
		You should go straight for a good book on Algebraic (not Combinatorial) Topology, is what I say.
		topology texts and have just ordered Intro to Topology by Bert Mendelson...perhaps there is a good defn in there but I doubt it.
	geomblog.blogspot.com /2005/01/betti-numbers_20.html (623 words)

	Scientific Project p04m03f (Site not responding. Last check: 2007-11-05)
		The creation of a separate subproject with this title reflects a growing interest in applications of topological ideas and methods in combinatorics, combinatorial geometry, theoretical computer science (discrete and computational geometry, complexity theory) etc.The project is closely linked with the Seminar for Geometry, Topology and Algebra (GTA seminar).
		Research on the highest level in the areas of combinatorial (algebraic) topology and geometry which have proven to be useful in other areas of mathematics, computer science and natural sciences.
		Applications in combinatorial (discrete) geometry (ham sandwich and equipartition theorems, common transversals, Tverberg type theorems, convex polytopes, graph and knot theory).
	www.mi.sanu.ac.yu /projects/scientific_p04m03/sproject_p04m03f.htm (220 words)

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