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Topic: Sheaf (mathematics)

	Sheaf (mathematics) - Wikipedia, the free encyclopedia
		In mathematics, a sheaf F on a topological space X is something that assigns a structure F(U) (such as a set, group, or ring) to each open set U of X.
		This corresponds to the notion of germ of a function used elsewhere in mathematics.
		In early developments of sheaf theory, it was shown that giving a sheaf F on X is as good as giving a certain topological space E together with a continuous map from E to X.
	en.wikipedia.org /wiki/Sheaf_space (2990 words)

	Mathematics
		Closure (mathematics) In mathematics, the closure C(X) of an object X is defined to be the smallest object that both inc...
		Degeneracy (mathematics) In mathematics, a degenerate case is a limiting case in which a class of object changes its nat...
		Mathematics Mathematics is commonly defined as the study of Philosophy of mathematics.
	www.brainyencyclopedia.com /topics/mathematics.html (2391 words)

	Sheaf (Site not responding. Last check: 2007-11-03)
		In mathematics a sheaf means, firstly, a topological construction F on a given space X giving one a set or richer structure F(U) for each open set U of X and subject to a patching condition.
		In early developments of sheaf theory, it was shown that giving a sheaf F as a functor is as good as giving a certain topological space Y together with a mapping from Y to X.
		The space associated to the sheaf of sections of Z, with its given continuous mapping to X, is therefore endowed with a local homeomorphism to X, and in a sense deals with all the 'ramification' in the mapping, in the 'best possible way'.
	www.termsdefined.net /sh/sheaf.html (773 words)

	PlanetMath: sheaf
		is actually a sheaf of rings, because continuous functions are uniquely specified by their values on an open cover.
		We conclude with some interesting examples of morphisms of sheaves, chosen to illustrate the unifying power of the language of schemes across various diverse branches of mathematics.
		This is version 9 of sheaf, born on 2002-04-28, modified 2003-12-08.
	planetmath.org /encyclopedia/Sheaf.html (708 words)

	Book Reviews
		Assuming the degree of conservatism of mathematicians and thus the percentage of all new preprints submitted in the electronic form is uniform across the spectrum of mathematical disciplines, those figures give a reasonably accurate picture of the level of current research activity in all the branches of mathematics.
		Among all branches of pure mathematics, algebraic geometry is nearly unique in that it maintains a fine and healthy balance between the abstract and the concrete.
		However, mathematics students should be particularly encouraged to study algebraic geometry, through stressing both its importance in pure mathematics where it forms the area of bubbling research activity, and its unusually high relevance for applications of mathematics, in particular those in computing.
	www.massey.ac.nz /~wwifs/mathnews/NZMS74/bookreviews.html (5417 words)

	Sheaf Theory (Graduate Texts in Mathematics) by Glen E. Bredon [ISBN: 0387949054] - Find Cheap Textbook Prices & Save ...
		The fact that sheaf theoretic cohomology satisfies the homotopy property is proved for general topological spaces.
		It would be very helpful to the newcomer to sheaf theory if the author could explain why the functor from open sets to abelian groups is a contravariant functor, instead of a covariant one.
		Sheaf cohomology is compared with other cohomology theories in the next chapter.
	www.gettextbooks.com /isbn_0387949054.html (588 words)

	PlanetMath: sheaf cohomology
		In this case, it does not matter if we take the derived functors in the category of sheaves of abelian groups or coherent sheaves.
		Sheaf cohomology can be explicitly calculated using Čech cohomology.
		This is version 10 of sheaf cohomology, born on 2003-08-14, modified 2005-05-15.
	planetmath.org /encyclopedia/SheafCohomology.html (219 words)

	Arizona Mathematics \| Events \| Weekly News \| Fall 2003 \| October 6 - October 12, 2003 (Site not responding. Last check: 2007-11-03)
		Abstract: I will explain my proof of the fact that the sheaf of locally exact differentials on a smooth, projective curve over fields of positive characteristics is a stable bundle.
		Given a set of such angular restraints, the mathematical problem is then to back-calculate the molecular structure, and to define the allowable solutions based on symmetry.
		Although progress has been made using a poor man's approach, the mathematical richness and importance to biochemistry and biotechnology suggests that a serious effort be undertaken to develop general methods.
	math.arizona.edu /~weeklynews/fall2003/2003_10_06.html (1267 words)

	ipedia.com: Sheaf Article (Site not responding. Last check: 2007-11-03)
		Alternate meanings: River Sheaf, King Sceaf In mathematics, a sheaf F on a given topological space X gives a set or richer structure F for each open set U of X. The structures F are compatible with th...
		The condition that F is a sheaf is exactly the F is the limit of the diagram.
		More precisely: to every sheaf F of sets on X there exists a local homeomorphism π: E → X such that F is isomorphic (in the sense of natural isomorphism, the isomorphism concept for functors) to the sheaf of sections of π that was described in the example section above.
	www.ipedia.com /sheaf.html (2683 words)

	Sceaf (Site not responding. Last check: 2007-11-03)
		In Norse mythology Sceaf (or Sheaf or Sheave) was a mythological child figure who to the coast on a ship with sheaf of grain.
		Sceaf was fostered by the people as of their own and later ruled over When grown up he used his sword extend his dominion forcing the rulers of neighbours to take him for a lord.
		This book is a rigorous overview of an approach to the study of schemes that uses a generalization of the complex topology called the etale topology.
	www.freeglossary.com /Sceaf (482 words)

	Practical Foundations of Mathematics
		Plainly we all write mathematics in a symbolic fashion, so there is a need for fluent translations that render symbols and diagrams as part of the same language.
		The category of contexts and substitutions (a phrase chosen to echo the mathematical examples) is generated by an elementary sketch (Sections 4.2-4.3) whose equations are defined by the substitution lemma in Section 1.1.
		The mathematical idea is to form the closure of the given system of equations to form a parsing congruence, but we observe that closure under the operations and transitivity is redundant.
	www.cs.man.ac.uk /~pt/Practical_Foundations/html/summary.html (7546 words)

	Mathematics Notes (Site not responding. Last check: 2007-11-03)
		Relative Affine Schemes: (RAS) Affine morphisms, the Spec construction, the sheaf associated to a sheaf of quasi-coherent modules over an algebra.
		The Relative Proj Construction: (TRPC) Associating a Proj with a sheaf of graded algebras.
		The sheaf associated to a sheaf of graded modules, the graded module associated to a quasi-coherent sheaf, functorial properties, ideal sheaves and closed subschemes, the duple embedding, twisting with invertible sheaves.
	www.users.on.net /~dmurfet/TheRisingSea/maths/maths.html (1406 words)

	Lycos - Offers for search topology mathematics - Compare Cheap Prices and Buy
		This introduction to modern algebraic topology treats the cohomology of spaces with sheaf (as opposed to constant)...
		This introduction to modern algebraic topology treats the cohomology of spaces with sheaf (as opposed to constant) coefficients.
		Despite the extensive use of the concepts of topology, differential geometry and other areas of mathematics in developments in theoretical physics, there are few books accessible to postgraduate students in physics.
	shopping.lycos.co.uk /search/topology_mathematics.html (1053 words)

	Mathematics (Site not responding. Last check: 2007-11-03)
		There is a general definition, due to Grothendieck, of the sheaf of differential operators on a general variety X. This is a coherent sheaf with many nice properties when X is a smooth variety, but in general it is only quasi-coherent.
		One line of investigation is to study this sheaf, or its ring of sections in the local case, and the corresponding representations or D-modules.
		Although nice results are known about this sheaf in a variety of singular situations, it is known to behave quite badly "mostly".
	www.maths.warwick.ac.uk /~eriksen/Math/math.html (404 words)

	Arizona Mathematics \| About \| Newsletters \| Mathematics Newsletter 1999-2000
		The Department of Mathematics and the Program in Applied Mathematics have jointly developed a new Professional Master's degree in Mathematical Science, which is intended specifically for individuals intending to make careers in industry and commerce, or in the public sector.
		Together, the mathematics departments at these universities share a common belief that the distance that has developed between teaching and research is artificial, and with the right leadership this great distance can be successfully closed.
		Having students work on mathematical research projects under the direction of a faculty member is an enormous help in encouraging them to continue their education in graduate school.
	math.arizona.edu /department/magazine/mathnews_1999-2000.html (6229 words)

	Amazon.com: Etale Cohomology. (PMS-33): Books (Site not responding. Last check: 2007-11-03)
		When reading this section, it is best to think of the fundamental group from ordinary topology in terms of the universal covering space instead of simple connectedness as it is the former concept that is employed to define the fundamental group of a scheme.
		The author turns to sheaf theory in the next chapter and shows how etale topology does give exactness for sequences of sheaves that are not exact in the Zariski topology.
		He convinces the reader right away that the sheaf category is more general than the abelian category by showing that the former does not have enough projectives.
	www.amazon.com /exec/obidos/tg/detail/-/0691082383?v=glance (1441 words)

	Tropes
		One extremely suggestive revelation is that sheaf theory is equally at home in category theory as in set theory.
		It may be wondered if the move to kernel tropism in trope bundle and sheaf theory is absolutely necessary.
		The angle of sheaf theory that Mormann particularly develops and exploits is intimately connected with topology.
	plato.stanford.edu /entries/tropes (5411 words)

	Seminars at Weizmann Institute of Science Faculty of Mathematical Sciences: Wed Nov 10 03:00:01 2004 (Site not responding. Last check: 2007-11-03)
		Grothendieck's {\it Sheaf to Functions Correspondence} suggests that there exists a geometric object (an $\ell$-adic sheaf) $\Fscr$ from which the function $F$ can be derived: $$ \Fscr\ \rsarrow\ F. The sheaf $\Fscr$ is defined over the algebraic closure, and can be approached by standard cohomological techniques.
		In this case the sheaf $\Fscr$ is the trace of the {\it Weil representation sheaf} $\Kscr$.
		The construction of that sheaf carried out in an unpublished letter from Deligne to Kazhdan in 1982.
	weizmann.ac.il /usersfiles/math/html/sem.2004:11:10:11:00:5:15997.shtml (167 words)

	Laura Taalman : Curriculum Vitae (Site not responding. Last check: 2007-11-03)
		Calculus with Functions I and II (a course which I developed and wrote the textbook for), Calculus I and II, Discrete Mathematics and Introduction to Proof, and Abstract Algebra.
		Over such a resolution, the Nash sheaf is generated by the differentials of monomial functions whose exponents are a Hsiang-Pati ordered set.
		Fellowship for new mathematicians and teachers of mathematics, to attend a series of conferences and professional development workshops.
	www.math.jmu.edu /~taal/CV2005.html (1230 words)

	Gerbe - Art History Online Reference and Guide (Site not responding. Last check: 2007-11-03)
		In mathematics, a gerbe is a construct in homological algebra.
		Gerbes were invented by Giraud, as a way of generalising to higher sheaf cohomology and group cohomology groups H
		The initial presentation of the idea made heavy use of category theory: a gerbe, roughly speaking, is a particular type of sheaf of categories.
	www.arthistoryclub.com /art_history/Gerbe (146 words)

	Amazon.com: Sheaf Theory (Graduate Texts in Mathematics): Books: Glen E. Bredon (Site not responding. Last check: 2007-11-03)
		An Introduction to Homological Algebra (Cambridge Studies in Advanced Mathematics) by Charles A. Weibel
		Algebraic Geometry 2: Sheaves and Cohomology (Translations of Mathematical Monograph, Vol 197) by Kenji Ueno
		Abelian l-adic Representations and Elliptic Curves (Research Notes in Mathematics (a K Peters), Vol 7) by Jean Pierre Serre
	www.amazon.com /exec/obidos/tg/detail/-/0387949054?v=glance (1123 words)

	TEXTBOOKS, LECTURE NOTES AND TUTORIALS IN MATHEMATICS (Site not responding. Last check: 2007-11-03)
		The theory of the foundations of mathematics - 1870 to 1940
		Linear Methods of Applied Mathematics, by Evans Harrell and James Herod.
		Mathematics of Rubik's cube by David Joyner is a good source for discrete group theory with applications to permutation puzzles.
	www.geocities.com /alex_stef/mylist.html (2404 words)

	Sheaf Theory (Graduate Texts in Mathematics) - DVD @ US MacLinks.net Apple Store (Site not responding. Last check: 2007-11-03)
		This is a very rigorous and detailed account of a very esoteric subject that historically began with the concept of analytic continuation in complex analysis.
		Abelian l-adic Representations and Elliptic Curves (Research Notes in Mathematics (a K Peters), Vol 7)
		Algebraic Geometry 2: Sheaves and Cohomology (Translations of Mathematical Monograph, Vol 197)
	maclinks.net /shop/us/dvd/details/0387949054.html (613 words)

	Categorical Foundations : Special Topics in Order, Topology, Algebra, and Sheaf Theory (Encyclopedia of Mathematics and ... (Site not responding. Last check: 2007-11-03)
		The Categorical Foundations : Special Topics in Order, Topology, Algebra, and Sheaf Theory (Encyclopedia of Mathematics and its Applications) is part of our discount Book catalog.
		Used Categorical Foundations : Special Topics in Order, Topology, Algebra, and Sheaf Theory (Encyclopedia of Mathematics and its Applications) are in stock for only $74.99.
		Discount pricing is subject to change, in order to get the Book Categorical Foundations Special Topics in Order Topology Algebra and Sheaf Theory Encyclopedia of Mathematics and its Applications at this reduced price, you must buy now!
	evieseden.com /amazon/asin.0521834147.Book_Categorical_Foundations_Special_Topics_in_Order_Topology_Algebra_and_Sheaf_Theory_Encyclopedia_of_Mathematics_and_its_Applications_.html (337 words)

	- Cambridge University Press
		The book offers categorical introductions to order, topology, algebra and sheaf theory, suitable for graduate students, teachers and researchers of pure mathematics.
		This is an outstanding scholarly attempt to bring together all significant mathematical constants in one place.
		This is a self-contained introduction to the basics of the theory of information and coding.
	www.cambridge.org /uk/browse?subjectID=1078541 (147 words)

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