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The Seiberg-Witten equations and 4-manifold topology

1996; American Mathematical Society; Volume: 33; Issue: 1 Linguagem: Inglês

10.1090/s0273-0979-96-00625-8

1088-9485

Simon Donaldson,

Algebraic Geometry and Number Theory

Since 1982 the use of gauge theory, in the shape of the Yang-Mills instanton equations, has permeated research in 4-manifold 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 4-manifold 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 differential-geometric equation by Seiberg and Witten: in the space of a few weeks long-standing 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 from this initial period of intense activity.The time period being comparatively short, it is hard to give complete references for some of the new material, and perhaps also to attribute some of the advances precisely.The author is grateful to a number of mathematicians, but most particularly to Peter Kronheimer, for explaining these new developments as they unfolded.