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Lectures on Seiberg-Witten Invariants: Lecture Notes in Mathematics, cartea 1629

Autor John D. Moore
en Limba Engleză Paperback – 24 apr 2001
Riemannian, symplectic and complex geometry are often studied by means ofsolutions to systems ofnonlinear differential equations, such as the equa­ tions of geodesics, minimal surfaces, pseudoholomorphic curves and Yang­ Mills connections. For studying such equations, a new unified technology has been developed, involving analysis on infinite-dimensional manifolds. A striking applications of the new technology is Donaldson's theory of "anti-self-dual" connections on SU(2)-bundles over four-manifolds, which applies the Yang-Mills equations from mathematical physics to shed light on the relationship between the classification of topological and smooth four-manifolds. This reverses the expected direction of application from topology to differential equations to mathematical physics. Even though the Yang-Mills equations are only mildly nonlinear, a prodigious amount of nonlinear analysis is necessary to fully understand the properties of the space of solutions. . At our present state of knowledge, understanding smooth structures on topological four-manifolds seems to require nonlinear as opposed to linear PDE's. It is therefore quite surprising that there is a set of PDE's which are even less nonlinear than the Yang-Mills equation, but can yield many of the most important results from Donaldson's theory. These are the Seiberg-Witte~ equations. These lecture notes stem from a graduate course given at the University of California in Santa Barbara during the spring quarter of 1995. The objective was to make the Seiberg-Witten approach to Donaldson theory accessible to second-year graduate students who had already taken basic courses in differential geometry and algebraic topology.
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Specificații

ISBN-13: 9783540412212
ISBN-10: 3540412212
Pagini: 136
Ilustrații: VIII, 121 p.
Dimensiuni: 155 x 235 x 7 mm
Greutate: 0.2 kg
Ediția:2nd ed. 2001
Editura: Springer Berlin, Heidelberg
Colecția Springer
Seria Lecture Notes in Mathematics

Locul publicării:Berlin, Heidelberg, Germany

Public țintă

Research

Cuprins

Preliminaries: Introduction; What is a vector bundle? What is a connection? The curvature of a connection; Characteristic classes; The Thom form; The universal bundle; Classification of connections; Hodge theory. Spin geometry on four-manifolds: Euclidean geometry and the spin groups; What is a spin structure? Almost complex and spin-c structures; Clifford algebras; The spin connection; The Dirac operator; The Atiyah-Singer index theorem. Global analysis: The Seiberg-Witten equations; The moduli space; Compactness of the moduli space; Transversality; The intersection form; Donaldson's theorem; Seiberg-Witten invariants; Dirac operators on Kaehler surfaces; Invariants of Kaehler surfaces. Bibliography. Index.