Sheaves in Topology
Title | Sheaves in Topology PDF eBook |
Author | Alexandru Dimca |
Publisher | Springer Science & Business Media |
Pages | 253 |
Release | 2012-12-06 |
Genre | Mathematics |
ISBN | 3642188680 |
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
Title | Topology of Singular Spaces and Constructible Sheaves PDF eBook |
Author | Jörg Schürmann |
Publisher | Birkhäuser |
Pages | 461 |
Release | 2012-12-06 |
Genre | Mathematics |
ISBN | 3034880618 |
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
Title | Sheaf Theory through Examples PDF eBook |
Author | Daniel Rosiak |
Publisher | MIT Press |
Pages | 454 |
Release | 2022-10-25 |
Genre | Mathematics |
ISBN | 0262362376 |
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
Title | Sheaf Theory PDF eBook |
Author | Glen E. Bredon |
Publisher | |
Pages | 296 |
Release | 1967 |
Genre | Sheaf theory |
ISBN |
Categories and Sheaves
Title | Categories and Sheaves PDF eBook |
Author | Masaki Kashiwara |
Publisher | Springer Science & Business Media |
Pages | 496 |
Release | 2005-12-19 |
Genre | Mathematics |
ISBN | 3540279504 |
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
Title | Intersection Homology & Perverse Sheaves PDF eBook |
Author | Laurenţiu G. Maxim |
Publisher | Springer Nature |
Pages | 270 |
Release | 2019-11-30 |
Genre | Mathematics |
ISBN | 3030276449 |
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
Title | Global Calculus PDF eBook |
Author | S. Ramanan |
Publisher | American Mathematical Soc. |
Pages | 330 |
Release | 2005 |
Genre | Mathematics |
ISBN | 0821837028 |
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.