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Lecture Notes on Seiberg-Witten Invariants (Revised Second Edition)

Summary: Lecture Notes on Seiberg-Witten Invariants
(Revised Second Edition)
John Douglas Moore
July, 2010
Riemannian, symplectic and complex geometry are often studied by means
of solutions to systems of nonlinear 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 the present state of knowledge, understanding smooth structures on


Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara


Collections: Mathematics