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The SeibergWitten equations and 4–manifold topology
 Bull. Amer. Math. Soc
, 1996
"... Since 1982 the use of gauge theory, in the shape of the YangMills instanton equations, has permeated research in 4manifold topology. At first this use of differential geometry and differential equations had an unexpected and unorthodox flavour, but over the years the ideas have become more familia ..."
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Since 1982 the use of gauge theory, in the shape of the YangMills instanton equations, has permeated research in 4manifold topology. At first this use of differential geometry and differential equations had an unexpected and unorthodox flavour, but over the years the ideas have become more familiar; a body of techniques has built up through the efforts of many mathematicians, producing results which have uncovered some of the mysteries of 4manifold theory, and leading to substantial internal conundrums within the field itself. In the last three months of 1994 a remarkable thing happened: this research area was turned on its head by the introduction of a new kind of differentialgeometric equation by Seiberg and Witten: in the space of a few weeks longstanding problems were solved, new and unexpected results were found, along with simpler new proofs of existing ones, and new vistas for research opened up. This article is a report on some of these developments, which are due to various mathematicians, notably Kronheimer, Mrowka, Morgan, Stern and Taubes, building on the seminal work of Seiberg [S] and Seiberg and Witten [SW]. It is written as an attempt to take stock of the progress stemming
Gauge theory for embedded surfaces
 I, Topology
, 1993
"... (i) Topology of embedded surfaces. Let X be a smooth, simplyconnected 4manifold, and ξ a 2dimensional homology class in X. One of the features of topology in dimension 4 is the fact that, although one may always represent ξ as the fundamental class of some smoothly ..."
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(i) Topology of embedded surfaces. Let X be a smooth, simplyconnected 4manifold, and ξ a 2dimensional homology class in X. One of the features of topology in dimension 4 is the fact that, although one may always represent ξ as the fundamental class of some smoothly
Donaldson and SeibergWitten invariants of algebraic surfaces
, 1996
"... Donaldson theory and more recently SeibergWitten theory have led to dramatic breakthroughs in the study of smooth 4manifolds, and in particular of algebraic surfaces and their generalizations, symplectic 4manifolds. In this paper, we shall survey some of the main results. Many expositions of Seib ..."
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Donaldson theory and more recently SeibergWitten theory have led to dramatic breakthroughs in the study of smooth 4manifolds, and in particular of algebraic surfaces and their generalizations, symplectic 4manifolds. In this paper, we shall survey some of the main results. Many expositions of SeibergWitten theory have
A Symmetric Family Of YangMills Fields
"... . We examine a family of finite energy SO(3) YangMills connections over S 4 , indexed by two real parameters. This family includes both smooth connections (when both parameters are odd integers), and connections with a holonomy singularity around 1 or 2 copies of RP 2 . These singular YM connec ..."
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. We examine a family of finite energy SO(3) YangMills connections over S 4 , indexed by two real parameters. This family includes both smooth connections (when both parameters are odd integers), and connections with a holonomy singularity around 1 or 2 copies of RP 2 . These singular YM connections interpolate between the smooth solutions. Depending on the parameters, the curvature may be selfdual, antiselfdual, or neither. For the (anti)selfdual connections, we compute the formal dimension of the moduli space. For the nonselfdual connections we examine the second variation of the YangMills functional, and count the negative and zero eigenvalues. Each component of the nonselfdual moduli space appears to consist only of conformal copies of a single solution. 1. Introduction and Statement of Results 1.1 Main Results Until recently, the phrase "YangMills theory in four dimensions" essentially meant the study of smooth solutions to the (anti)selfduality equations F = \S...
Gauge theory for embedded surfaces, II
"... (i) The theorem and an outline of the proof. This paper is the second in a series of two, aimed at developing results about the topology of embedded surfaces Σ in a 4manifold X using some new YangMills moduli spaces associated to such pairs (X,Σ). The moduli spaces were ..."
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(i) The theorem and an outline of the proof. This paper is the second in a series of two, aimed at developing results about the topology of embedded surfaces Σ in a 4manifold X using some new YangMills moduli spaces associated to such pairs (X,Σ). The moduli spaces were
DERIVATIVES IN MATHEMATICS AND PHYSICS
"... Since the invention of the derivative of a function by Newton and Leibnitz in late seventeenth century, it has found innumerable applications to both mathematics and physics. Since mid nineteenth century, the idea of taking derivatives has gone through a lot of developments which are fundamental in ..."
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Since the invention of the derivative of a function by Newton and Leibnitz in late seventeenth century, it has found innumerable applications to both mathematics and physics. Since mid nineteenth century, the idea of taking derivatives has gone through a lot of developments which are fundamental in both mathematics and