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Topic: Sheaf theory

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	CTheory.net
		For de Saussure, the relationship between the signifier and the signified was merely historical and therefore arbitrary.
		The letters b, o, o and k could have signified a flying animal, but were instead doomed to represent a bound sheaf of printed papers too rarely capable of flight.
		These usual suspects condescend toward religion at the same time that they mandate tolerance for all lifestyles and teach postmodern theories suggesting that received beliefs tend to be arbitrary or self-serving or both.
	www.ctheory.net /articles.aspx?id=380 (3444 words)

	PhilSci Archive - Continuity and logical completeness: an application of sheaf theory and topoi
		Continuity and logical completeness: an application of sheaf theory and topoi
		Awodey, Steve (2000) Continuity and logical completeness: an application of sheaf theory and topoi.
		Topos theory permits one to apply this same idea to logic, and to consider continuously variable sets (sheaves).
	philsci-archive.pitt.edu /archive/00000175 (182 words)

	NationMaster - Encyclopedia: Sheaf
		More precisely: to every sheaf F of sets on X there exists a local homeomorphism In topology, a local homeomorphism is a map from one topological space to another that respects locally the topological structure of the two spaces.
		In mathematics, sheaf cohomology is the aspect of sheaf theory, concerned with sheaves of abelian groups, that applies homological algebra to make possible effective calculation of the global sections of a sheaf F. This is the main step, in numerous areas, from sheaf theory as a description of a geometric...
		In mathematics, especially in algebraic geometry and the theory of complex manifolds, a coherent sheaf F on a locally ringed space X is a sheaf isomorphic with the cokernel of a morphism of OX_modules OXm → OXn.
	www.nationmaster.com /encyclopedia/Sheaf (6573 words)

	PlanetMath: structure sheaf
		Those who are fans of topos theory will recognize this map as an isomorphism of topos.
		Cross-references: isomorphic, isomorphism, theory, topos, bijection, embedding, prime spectrum, affine variety, equivalence of categories, inclusion map, map, restriction, sections, sheaf, neighborhood, open, independent, fraction field, ring, coordinate, open subset, point, fix, Zariski topology, field, variety, algebraic, irreducible
		This is version 1 of structure sheaf, born on 2002-05-14.
	planetmath.org /encyclopedia/StructureSheaf.html (184 words)

	Lectures on vector bundles over Riemann surfaces, (Annals of mathematics studies) 1967 \| Browse \| All Categories \| page ...
		The use of sheaf theory will seem quite natural to the reader who has read the first book, and even those who have not but have a background in complex function theory will recognize sheaf theory as a generalization of the concept of analytic continuation.
		A 'free sheaf' of rank m is defined as being one isomorphic to the direct sum of m R-modules.
		Techniques from the theory of several complex variables are utilized without review to study the complex analytic equivalence of flat vector bundles, and he shows that every flat vector bundle is analytically equivalent to an 'irreducible' flat vector bundle.
	gigapedia.org /item.id:47289,title:lectures-on-vector-bundles-over-riemann-surfaces,-(annals-of-mathematics-studies)-1967,cat_id:0,cat_page:1.html (953 words)

	Springer Online Reference Works (Site not responding. Last check: )
		Abstract potential theory arose in the middle of the 20th century from the efforts to create a unified axiomatic method for treating a vast diversity of properties of the different potentials that are applied to solve problems of the theory of partial differential equations.
		The extension of the theory to a wide class of parabolic equations was obtained by H.
		as hyperharmonic sheaf is a harmonic subspace of
	eom.springer.de /p/p074150.htm (1418 words)

	DVD Recorders and Players: Sheaves in Geometry and Logic: A First Introduction to Topos Theory (Universitext) - $60.76
		This introduction to topos theory begins with a number of illustrative examples that explain the origin of these ideas and then describes thе sheafification process and the properties of an elementary topos.
		An understanding оf sheaf theory and category theory will definitely help when attempting to learn topos theory, but the book could be read without such a background.
		A reader interested in understanding how topos theory is used in this research should concentrate on the chapter on properties of elementary topoi, thе one on basic categories of topoi, and the chapter on localic topoi.
	www.dvdplayers-store.info /good30333837393737313034.html (1349 words)

	SheafWheat
		The syntax of the hologram theory of consciousness is clearly redundant: The universe is the result of my perception of a universe that already includes the perception that produces it.
		By holding up the cut sheaf, he showed the initiates not only who they were but what they had come to know through instruction by the Light, the key illuminist insight: as the wheat is given to us by Demeter, so is our cognition of the natural world, the place where it grows.
		The gesture of the sheaf of cut wheat is an example of what may be called a functional symbol: an object or image that symbolizes a process in nature and, at the same time, demonstrates the very process it symbolizes.
	www.metahistory.org /SheafWheat.php (4198 words)

	Category Theory
		Category theory is a general mathematical theory of structures and sytems of structures.
		Category theory reveals that many of these constructions are in fact special cases of objects in a category with what is called a "universal property".
		Given these simple facts, it remains to be seen whether category theory should be "on the same plane", so to speak, with set theory, whether it should be considered seriously as providing a foundational alternative to set theory or whether it is foundational in a different sense altogether.
	www.seop.leeds.ac.uk /archives/spr2001/entries/category-theory (3068 words)

	stell
		One of the appealing aspects of category theory is itsuse of diagrams (originating from the underlying graphs).
		The category-theoretic tools of sheaf theoryare likely to helpful in providing a formal framework for dealing suchquestions, but their application to this area has not yet been exploredin any detail.
		Sheaf theory allows us to model `local' properties of structures, andtheir relationship to `global' properties.
	www2.sis.pitt.edu /~cogmap/ncgia/stell.html (2106 words)

	Tropes
		Trope-cluster theory can be further developed to include a treatment of compound universals (also requiring further complications in the structure of individuals and universals) and propositions.
		The seemingly parsimonious trope-cluster theory, as we saw, is pushed to acknowledge at least a second category besides tropes, the second-level relations.
		The angle of sheaf theory that Mormann particularly develops and exploits is intimately connected with topology.
	www.seop.leeds.ac.uk /archives/sum2006/entries/tropes (5414 words)

	NationMaster - Encyclopedia: Continuously differentiable
		From what has just been said, partitions of unity don't apply to holomorphic functions; their different behaviour relative to existence and analytic continuation is one of the roots of sheaf theory.
		In mathematics, a smooth function is one that is infinitely differentiable, i.e., has derivatives of all finite orders.
		When one needs to talk about the set of all infinitely differentiable functions, and how elements of that space behave when differentiated and integrated, summed and taken limits of, then it turns out that the space of all smooth functions is an inappropriate choice, as it fails to be complete and closed under these operations.
	www.nationmaster.com /encyclopedia/Continuously-differentiable (708 words)

	Tropes (Stanford Encyclopedia of Philosophy)
		Trope-cluster theory can be further developed to include a treatment of compound universals (also requiring further complications in the structure of individuals and universals) and propositions.
		The seemingly parsimonious trope-cluster theory, as we saw, is pushed to acknowledge at least a second category besides tropes, the second-level relations.
		The angle of sheaf theory that Mormann particularly develops and exploits is intimately connected with topology.
	plato.stanford.edu /entries/tropes (5413 words)

	FOM: Thinking with universes
		For many of these theories this framework is used for proving the general theorems, even though it is formally dispensible, and even though one or another particular calculation of cohomology can be made with far less apparatus.
		That is for each g in G and each x in S we define g(x) in such a way that given also g' in G we have g'(g(s)) equal to (g'g)(s), and for e the unit element of G we have e(x)=x.
		A sheaf map from S to a sheaf S' is just an equivariant function f:S-->S', that is a function such that for each g in G and x in S we have f(g(x))=g(f(x)).
	www.cs.nyu.edu /pipermail/fom/1999-April/002989.html (963 words)

	[No title]
		A beautiful exposition of the theory of characteristic classes from an axiomatic point of view.
		Includes such topics as Frobenius theorem, the relation between Lie groups and Lie algebras, de Rham theory, foundations of sheaf theory, elliptic operators and the Hodge theorem.
		Covers sheaf theory, holomorphic vector bundles and their characteristic classes, elliptic operators and Hodge theory on Kahler manifolds, and Kodaira's vanishing and embedding theorems.
	www.theory.caltech.edu /~kapustin/Ph229/recbooks.html (897 words)

	POLIMETRICA Publisher
		The text is a graduate level presentation of sheaf theory over topological spaces and its generalizations to presheaves over semilattices, Heyting algebras and frames.
		There is a comprehensive discussion of the lattice theoretic basis of sheaf theory, including partial orders, distributive lattices, Heyting and Boolean algebras, as well as their complete counterparts.
		Presheaves and sheaves over topological spaces, an important geometrical construction, sets the stage for the theory of L-sets, L a semilattice, in which we discuss quantifiers, logical connectives and the laws governing change of base along a semilattice...
	www.polimetrica.com (729 words)

	Jean-Pierre Serre Summary
		From a very young age he was an outstanding figure in the school of Henri Cartan, working on algebraic topology, several complex variables and then commutative algebra and algebraic geometry, in the context of sheaf theory and homological algebra techniques.
		The problem was that the cohomology of a coherent sheaf over a finite field couldn't capture as much topology as singular cohomology with integer coefficients.
		Amongst his most original contributions were: the concept of algebraic K-theory; the Galois representation theory for l-adic cohomology and the conceptions that these representations were "large"; and the Serre conjecture on mod-p representations that made Fermat's last theorem a connected part of mainstream arithmetic geometry.
	www.bookrags.com /Jean-Pierre_Serre (2520 words)

	Mathematics Department and Graduate School Colloquia 2000-2001
		In the past 30 years, the wave equation has played an increasingly important role in harmonic analysis, both as a tool for studying problems in spectral theory, as well as by motivating the development of estimates for linear wave equations that are vital for the study of nonlinear wave equations.
		The theory of a priori estimates for fully nonlinear equations with a convexity condition is well understood.
		Recent developments in Gromov-Witten theory have established a connection between intersection theory on the moduli space of curves and branched covers of the sphere, bringing to bear ideas from the fields of mathematics mentioned in the title.
	www.math.washington.edu /Seminars/Archives/coll2000-2001.php?printer=friendly (2275 words)

	Alexander Grothendieck: A biography - Helium
		His theory of schemes has become established as the best universal foundation for this major field, because of its great expressive power as well as technical depth.
		Perhaps Grothendieck's deepest single accomplishment is the invention of the étale and l-adic cohomology theories, which explain an observation of André Weil's, that there is a deep connection between the topological characteristics of a variety and its diophantine (number theoretic) properties.
		He gave lectures on category theory in the forests surrounding Hanoi while the city was being bombed, to protest against the Vietnam war.
	www.helium.com /tm/55009/alexander-grothendieck-biography (1627 words)

	OUP: UK General Catalogue
		Rapid developments in multivariable spectral theory have led to important and fascinating results which also have applications in other mathematical disciplines.
		In this book, classical results from the cohomology theory of Banach algebras, multidimensional spectral theory, and complex analytic geometry have been freshly interpreted using the language of homological algebra.
		Various concepts from function theory and complex analytic geometry are drawn together and used to give a new approach to concrete spectral computations.
	www.oup.com /uk/catalogue/?ci=9780198536673&view=00&promo=jan0575 (255 words)

	Current Research
		Professor Ross H. Street: Category theory and its application to algebraic topology, homotopy theory, knots, combinatorics, group representations and computer science.
		Dominic Verity: Higher-dimensional category theory and its applications to simplicial/cubical geometries, homotopy theory, non-abelian cohomology, category theory and sheaf representations of rings.
		Associate Professor John V. Corbett: Mathematical aspects of quantum theory, non-statistical interpretations of quantum theory that use intuitionistic logic.
	www.maths.mq.edu.au /math_research.html (339 words)

	Amazon.com: Sheaf theory: Books: Glen E Bredon
		Primarily concerned with the study of cohomology theories of general topological spaces with "general coefficient systems", the parts of sheaf theory covered here are those areas important to algebraic topology.
		Among the many innovations in this book, the concept of the "tautness" of a subspace is introduced and exploited; the fact that sheaf theoretic cohomology satisfies the homotopy property is proved for general topological spaces; and relative cohomology is introduced into sheaf theory.
		This book is primarily concerned with the study of cohomology theories of general topological spaces with general coefficient systems.
	www.amazon.com /Sheaf-theory-Glen-E-Bredon/dp/B0006BOETQ (662 words)

	LISPWIRE :: LISP NEWS AND RESOURCES
		Its front end is a language for problems in algebraic topology and number theory.
		A sheaf is a choice of sparse matrices, one for each face relation, with various commuting properties.
		A linear map of sheaves is a choice of sparse matrix for each cell commuting with the matrices from the face relations.
	www.lispwire.com /entry-math-sheafhom-des (939 words)

	Applications of Sheaf Cohomology in Twistor Theory
		For example, in the representation of zero rest mass fields on spacetime, by means of twistor functions, the field equations are essentially replaced merely by a condition of holomorphicity (Penrose 1975, 1969) and in the nonlinear graviton (Penrose 1976) curved vacuum spacetimes are generated by deforming the complex structure of flat twistor space.
		One of the natural paths along which complex analysis and contour integral techniques can be developed leads to sheaf theory and sheaf cohomology which, as a result, takes up a fundamental role in the mathematical apparatus of twistor theory.
		Perhaps these results depending essentially on the structure of complex numbers give some clues towards an under- standing of the fundamental role of complex structures in physics but a full understanding, which perhaps incorporates the union of quantum mechanics and relativity, is yet to emerge.
	users.ox.ac.uk /~tweb/00003/index.shtml (584 words)

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