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基本説明
Provides an excellent example of how global analysis techniques, can be applied to the solution of interesting geometrical problems.
Full Description
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.
Contents
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.
