To try to understand what a four-dimensional manifold is, it helpsto consider lower-dimensional manifolds first. Two simple examplesof two-dimensional manifolds are the surface of a ball or of adoughnut. Imagine you are standing on one of these surfaces andyou are very small compared to the surface: you would see onlya small patch of the surface, and that patch would look very muchlike a flat, two-dimensional plane. The definition of a two-dimensionalmanifold amounts to making mathematically precise the notion thatsmall portions of the surface look like a flat plane. Similarly,a three-dimensional manifold---such as the interior of a sphereor of a doughnut---have the property that small portions of themlook like three-dimensional space. Four-dimensional manifolds,though much more difficult to visualize, are nevertheless easilydefined mathematically in an analogous way, as are higher-dimensionalmanifolds.
In the nineteenth century mathematicians already understood thattwo-dimensional manifolds can be classified according to the numberof holes they have (e.g., the surface of a ball has zero holes,the surface of a doughnut has one hole, etc.). This means thatif two, two-dimensional manifolds have the same number of holes,no matter how different they might otherwise look, they are, ina fundamental, mathematical sense, the same. A long-standing aimin topology---the branch of mathematics concerned with manifolds---hasbeen to provide this kind of classification for manifolds in dimensionslarger than two (in other dimensions the distinguishing characteristicwould not necessarily be the number of holes, but some other mathematicalproperty). These classifications are the least understood in exactlythose dimensions that are important in physics---the usual threedimensions of our physical world, and the four dimensions of space-time.So it is natural that mathematicians have turned to ideas fromphysics, like the Seiberg-Witten equations, to help them understandmanifolds of these dimensions.
For the past decade or so, one of the main tools for understandingsuch questions has been a theory developed by the mathematicianSimon Donaldson (Oxford University) and based on ideas from gaugetheory in physics. While Donaldson theory produced some spectacularresults, it was from technically extremely difficult. When theSeiberg-Witten equations burst onto the scene in the fall of 1994,mathematicians were well prepared to put the new tools to immediateuse: they had already cut their teeth on the much more difficulttheory of Donaldson.
It was in a lecture in September 1994 that mathematical physicistEdward Witten (Institute for Advanced Study, Princeton) firstconjectured that certain equations that arose out of his jointwork with physicist Nathan Seiberg (Rutgers University) mightcontain all of the information found in Donaldson theory. Thisidea was quickly taken up by a number of mathematicians, mostnotably Peter B. Kronheimer (now at Harvard University), TomaszS. Mrowka (California Institute of Technology), and Clifford H.Taubes (Harvard University). Within weeks startling results werefound.
The Seiberg-Witten equations have been used to simplify and generalizemost of the results obtained through Donaldson theory. One importantnew result shows that there are strong restrictions on the geometryand topology of an important class of manifolds called symplecticmanifolds. The Seiberg-Witten equations have also been used toprove a long-standing question, known as the Thom Conjecture,about what kinds of two-dimensional surfaces can occur insidea four-dimensional manifold.
As important as these results have been in mathematics, the sagaof the Seiberg-Witten equations is far from over. In fact, physicssuggests a whole class of equations of which the Seiberg-Wittenequations are only the simplest. This new class of equations issure to bring to light new discoveries and insights.
This research was described in the article, ''Gauge Theory is Dead!---Long Live Gauge Theory!''by D. Kotschick, whichappeared in the March 1995 issue of the Notices of the AMS.