Geometry of Vector Sheaves

Geometry of Vector Sheaves
Author: Anastasios Mallios
Publisher: Springer Science & Business Media
Total Pages: 468
Release: 1998
Genre: Mathematics
ISBN: 9780792350040

This text is part of a two-volume monograph which obtains fundamental notions and results of the standard differential geometry of smooth manifolds, without using differential calculus. Here, the sheaf-theoretic character is emphasized. This has theoretical advantages such as greater perspective, clarity and unification, but also practical benefits ranging from elementary particle physics, via gauge theories and theoretical cosmology (differential spaces), to non-linear PDEs (generalized functions). Thus, more general applications, which are no longer smooth in the classical sense, can be coped with. The treatise might also be construed as a new systematic endeavour to confront the ever-increasing notion that the world around us is far from being smooth enough.

Geometry of Vector Sheaves

Geometry of Vector Sheaves
Author: Anastasios Mallios
Publisher: Springer Science & Business Media
Total Pages: 457
Release: 2012-12-06
Genre: Mathematics
ISBN: 9401150060

This two-volume monograph obtains fundamental notions and results of the standard differential geometry of smooth (CINFINITY) manifolds, without using differential calculus. Here, the sheaf-theoretic character is emphasised. This has theoretical advantages such as greater perspective, clarity and unification, but also practical benefits ranging from elementary particle physics, via gauge theories and theoretical cosmology (`differential spaces'), to non-linear PDEs (generalised functions). Thus, more general applications, which are no longer `smooth' in the classical sense, can be coped with. The treatise might also be construed as a new systematic endeavour to confront the ever-increasing notion that the `world around us is far from being smooth enough'. Audience: This work is intended for postgraduate students and researchers whose work involves differential geometry, global analysis, analysis on manifolds, algebraic topology, sheaf theory, cohomology, functional analysis or abstract harmonic analysis.

Geometry of Principal Sheaves

Geometry of Principal Sheaves
Author: Efstathios Vassiliou
Publisher: Springer Science & Business Media
Total Pages: 454
Release: 2006-03-30
Genre: Mathematics
ISBN: 1402034164

The book provides a detailed introduction to the theory of connections on principal sheaves in the framework of Abstract Differential Geometry (ADG). This is a new approach to differential geometry based on sheaf theoretic methods, without use of ordinary calculus. This point of view complies with the demand of contemporary physics to cope with non-smooth models of physical phenomena and spaces with singularities. Starting with a brief survey of the required sheaf theory and cohomology, the exposition then moves on to differential triads (the abstraction of smooth manifolds) and Lie sheaves of groups (the abstraction of Lie groups). Having laid the groundwork, the main part of the book is devoted to the theory of connections on principal sheaves, incorporating connections on vector and associated sheaves. Topics such as the moduli sheaf of connections, classification of principal sheaves, curvature, flat connections and flat sheaves, Chern-Weil theory, are also treated. The study brings to light fundamental notions and tools of the standard differential geometry which are susceptible of the present abstraction, and whose role remains unexploited in the classical context, because of the abundance of means therein. However, most of the latter are nonsensical in ADG.

The Geometry of Moduli Spaces of Sheaves

The Geometry of Moduli Spaces of Sheaves
Author: Daniel Huybrechts
Publisher: Cambridge University Press
Total Pages: 345
Release: 2010-05-27
Genre: Mathematics
ISBN: 1139485822

This edition has been updated to reflect recent advances in the theory of semistable coherent sheaves and their moduli spaces. The authors review changes in the field and point the reader towards further literature. An ideal text for graduate students or mathematicians with a background in algebraic geometry.

Modern Differential Geometry in Gauge Theories

Modern Differential Geometry in Gauge Theories
Author: Anastasios Mallios
Publisher: Springer Science & Business Media
Total Pages: 303
Release: 2006-07-27
Genre: Mathematics
ISBN: 0817644741

This is original, well-written work of interest Presents for the first time (physical) field theories written in sheaf-theoretic language Contains a wealth of minutely detailed, rigorous computations, ususally absent from standard physical treatments Author's mastery of the subject and the rigorous treatment of this text make it invaluable

Manifolds, Sheaves, and Cohomology

Manifolds, Sheaves, and Cohomology
Author: Torsten Wedhorn
Publisher: Springer
Total Pages: 366
Release: 2016-07-25
Genre: Mathematics
ISBN: 3658106336

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.

Algebraic Geometry 2

Algebraic Geometry 2
Author: Kenji Ueno
Publisher: American Mathematical Soc.
Total Pages: 196
Release: 1999
Genre: Mathematics
ISBN: 9780821813577

Algebraic geometry is built upon two fundamental notions: schemes and sheaves. The theory of schemes was explained in Algebraic Geometry 1: From Algebraic Varieties to Schemes. In this volume, the author turns to the theory of sheaves and their cohomology. A sheaf is a way of keeping track of local information defined on a topological space, such as the local holomorphic functions on a complex manifold or the local sections of a vector bundle. To study schemes, it is useful to study the sheaves defined on them, especially the coherent and quasicoherent sheaves.

Vector Bundles in Algebraic Geometry

Vector Bundles in Algebraic Geometry
Author: N. J. Hitchin
Publisher: Cambridge University Press
Total Pages: 359
Release: 1995-03-16
Genre: Mathematics
ISBN: 0521498783

This book is a collection of survey articles by the main speakers at the 1993 Durham symposium on vector bundles in algebraic geometry.

Differential Geometry of Complex Vector Bundles

Differential Geometry of Complex Vector Bundles
Author: Shoshichi Kobayashi
Publisher: Princeton University Press
Total Pages: 317
Release: 2014-07-14
Genre: Mathematics
ISBN: 1400858682

Holomorphic vector bundles have become objects of interest not only to algebraic and differential geometers and complex analysts but also to low dimensional topologists and mathematical physicists working on gauge theory. This book, which grew out of the author's lectures and seminars in Berkeley and Japan, is written for researchers and graduate students in these various fields of mathematics. Originally published in 1987. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

The Geometry of Schemes

The Geometry of Schemes
Author: David Eisenbud
Publisher: Springer Science & Business Media
Total Pages: 265
Release: 2006-04-06
Genre: Mathematics
ISBN: 0387226397

Grothendieck’s beautiful theory of schemes permeates modern algebraic geometry and underlies its applications to number theory, physics, and applied mathematics. This simple account of that theory emphasizes and explains the universal geometric concepts behind the definitions. In the book, concepts are illustrated with fundamental examples, and explicit calculations show how the constructions of scheme theory are carried out in practice.