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Sheaves in Topology

Author : Alexandru Dimca
Publisher : Springer Science & Business Media
Page : 253 pages
File Size : 13,42 MB
Release : 2012-12-06
Category : Mathematics
ISBN : 3642188680

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Constructible and perverse sheaves are the algebraic counterpart of the decomposition of a singular space into smooth manifolds. This introduction to the subject can be regarded as a textbook on modern algebraic topology, treating the cohomology of spaces with sheaf (as opposed to constant) coefficients. The author helps readers progress quickly from the basic theory to current research questions, thoroughly supported along the way by examples and exercises.

Topology of Singular Spaces and Constructible Sheaves

Author : Jörg Schürmann
Publisher : Birkhäuser
Page : 461 pages
File Size : 27,94 MB
Release : 2012-12-06
Category : Mathematics
ISBN : 3034880618

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This volume is based on the lecture notes of six courses delivered at a Cimpa Summer School in Temuco, Chile, in January 2001. Leading experts contribute with introductory articles covering a broad area in probability and its applications, such as mathematical physics and mathematics of finance. Written at graduate level, the lectures touch the latest advances on each subject, ranging from classical probability theory to modern developments. Thus the book will appeal to students, teachers and researchers working in probability theory or related fields.

Sheaf Theory through Examples

Author : Daniel Rosiak
Publisher : MIT Press
Page : 454 pages
File Size : 20,39 MB
Release : 2022-10-25
Category : Mathematics
ISBN : 0262362376

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An approachable introduction to elementary sheaf theory and its applications beyond pure math. Sheaves are mathematical constructions concerned with passages from local properties to global ones. They have played a fundamental role in the development of many areas of modern mathematics, yet the broad conceptual power of sheaf theory and its wide applicability to areas beyond pure math have only recently begun to be appreciated. Taking an applied category theory perspective, Sheaf Theory through Examples provides an approachable introduction to elementary sheaf theory and examines applications including n-colorings of graphs, satellite data, chess problems, Bayesian networks, self-similar groups, musical performance, complexes, and much more. With an emphasis on developing the theory via a wealth of well-motivated and vividly illustrated examples, Sheaf Theory through Examples supplements the formal development of concepts with philosophical reflections on topology, category theory, and sheaf theory, alongside a selection of advanced topics and examples that illustrate ideas like cellular sheaf cohomology, toposes, and geometric morphisms. Sheaf Theory through Examples seeks to bridge the powerful results of sheaf theory as used by mathematicians and real-world applications, while also supplementing the technical matters with a unique philosophical perspective attuned to the broader development of ideas.

Sheaf Theory

Author : Glen E. Bredon
Publisher :
Page : 296 pages
File Size : 41,27 MB
Release : 1967
Category : Sheaf theory
ISBN :

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Categories and Sheaves

Author : Masaki Kashiwara
Publisher : Springer Science & Business Media
Page : 496 pages
File Size : 24,87 MB
Release : 2005-12-19
Category : Mathematics
ISBN : 3540279504

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Categories and sheaves appear almost frequently in contemporary advanced mathematics. This book covers categories, homological algebra and sheaves in a systematic manner starting from scratch and continuing with full proofs to the most recent results in the literature, and sometimes beyond. The authors present the general theory of categories and functors, emphasizing inductive and projective limits, tensor categories, representable functors, ind-objects and localization.

Intersection Homology & Perverse Sheaves

Author : Laurenţiu G. Maxim
Publisher : Springer Nature
Page : 270 pages
File Size : 14,91 MB
Release : 2019-11-30
Category : Mathematics
ISBN : 3030276449

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This textbook provides a gentle introduction to intersection homology and perverse sheaves, where concrete examples and geometric applications motivate concepts throughout. By giving a taste of the main ideas in the field, the author welcomes new readers to this exciting area at the crossroads of topology, algebraic geometry, analysis, and differential equations. Those looking to delve further into the abstract theory will find ample references to facilitate navigation of both classic and recent literature. Beginning with an introduction to intersection homology from a geometric and topological viewpoint, the text goes on to develop the sheaf-theoretical perspective. Then algebraic geometry comes to the fore: a brief discussion of constructibility opens onto an in-depth exploration of perverse sheaves. Highlights from the following chapters include a detailed account of the proof of the Beilinson–Bernstein–Deligne–Gabber (BBDG) decomposition theorem, applications of perverse sheaves to hypersurface singularities, and a discussion of Hodge-theoretic aspects of intersection homology via Saito’s deep theory of mixed Hodge modules. An epilogue offers a succinct summary of the literature surrounding some recent applications. Intersection Homology & Perverse Sheaves is suitable for graduate students with a basic background in topology and algebraic geometry. By building context and familiarity with examples, the text offers an ideal starting point for those entering the field. This classroom-tested approach opens the door to further study and to current research.

Global Calculus

Author : S. Ramanan
Publisher : American Mathematical Soc.
Page : 330 pages
File Size : 19,16 MB
Release : 2005
Category : Mathematics
ISBN : 0821837028

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The power that analysis, topology and algebra bring to geometry has revolutionised the way geometers and physicists look at conceptual problems. Some of the key ingredients in this interplay are sheaves, cohomology, Lie groups, connections and differential operators. In Global Calculus, the appropriate formalism for these topics is laid out with numerous examples and applications by one of the experts in differential and algebraic geometry. Ramanan has chosen an uncommon but natural path through the subject. In this almost completely self-contained account, these topics are developed from scratch. The basics of Fourier transforms, Sobolev theory and interior regularity are proved at the same time as symbol calculus, culminating in beautiful results in global analysis, real and complex. Many new perspectives on traditional and modern questions of differential analysis and geometry are the hallmarks of the book. The book is suitable for a first year graduate course on Global Analysis.

Manifolds, Sheaves, and Cohomology

Author : Torsten Wedhorn
Publisher : Springer
Page : 366 pages
File Size : 48,3 MB
Release : 2016-07-25
Category : Mathematics
ISBN : 3658106336

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This book explains techniques that are essential in almost all branches of modern geometry such as algebraic geometry, complex geometry, or non-archimedian geometry. It uses the most accessible case, real and complex manifolds, as a model. The author especially emphasizes the difference between local and global questions. Cohomology theory of sheaves is introduced and its usage is illustrated by many examples.

Sheaves on Manifolds

Author : Masaki Kashiwara
Publisher : Springer Science & Business Media
Page : 522 pages
File Size : 33,17 MB
Release : 2013-03-14
Category : Mathematics
ISBN : 3662026619

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Sheaf Theory is modern, active field of mathematics at the intersection of algebraic topology, algebraic geometry and partial differential equations. This volume offers a comprehensive and self-contained treatment of Sheaf Theory from the basis up, with emphasis on the microlocal point of view. From the reviews: "Clearly and precisely written, and contains many interesting ideas: it describes a whole, largely new branch of mathematics." –Bulletin of the L.M.S.

Sheaves in Geometry and Logic

Author : Saunders Mac Lane
Publisher :
Page : 627 pages
File Size : 36,53 MB
Release : 1992
Category : Algebraische Geometrie - Garbentheorie
ISBN : 9783540977100

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An introduction to the theory of toposes which begins with illustrative examples and goes on to explain the underlying ideas of topology and sheaf theory as well as the general theory of elementary toposes and geometric morphisms and their relation to logic.