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.
