TY  - BOOK
AU  - Moore,John Douglas
ED  - SpringerLink (Online service)
TI  - Lectures on Seiberg-Witten Invariants
T2  - Lecture Notes in Mathematics,
SN  - 9783540409526
AV  - QA614-614.97 
U1  - 514.74 23
PY  - 2001///
CY  - Berlin, Heidelberg
PB  - Springer Berlin Heidelberg
KW  - Mathematics
KW  - Geometry, algebraic
KW  - Global analysis
KW  - Systems theory
KW  - Global differential geometry
KW  - Mathematical optimization
KW  - Mathematical physics
KW  - Global Analysis and Analysis on Manifolds
KW  - Systems Theory, Control
KW  - Calculus of Variations and Optimal Control; Optimization
KW  - Algebraic Geometry
KW  - Differential Geometry
KW  - Mathematical and Computational Physics
N2  - 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
UR  - http://dx.doi.org/10.1007/978-3-540-40952-6
ER  -