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A Revolution in Mathematics
During 1994 and 1995, mathematics witnessed a revolution in the study of four-dimensional topology. Sparking this revolution was a new set of equations, known as the Seiberg-Witten equations, which grew out of work in theoretical physics. These equation shave simplified much of the existing theory of four-dimensional shapes, known as manifolds, and have led to some spectacular new results.
To try to understand what a four-dimensional manifold is, it helps to consider lower-dimensional manifolds first. Two simple examples of two-dimensional manifolds are the surface of a ball or of a doughnut. Imagine you are standing on one of these surfaces and you are very small compared to the surface: you would see only a small patch of the surface, and that patch would look very much like a flat, two-dimensional plane. The definition of a two-dimensional manifold amounts to making mathematically precise the notion that small portions of the surface look like a flat plane. Similarly, a three-dimensional manifold---such as the interior of a sphere or of a doughnut---have the property that small portions of them look like three-dimensional space. Four-dimensional manifolds, though much more difficult to visualize, are nevertheless easily defined mathematically in an analogous way, as are higher-dimensional manifolds.
In the nineteenth century mathematicians already understood that two-dimensional manifolds can be classified according to the number of holes they have (e.g., the surface of a ball has zero holes, the surface of a doughnut has one hole, etc.). This means that if two, two-dimensional manifolds have the same number of holes, no matter how different they might otherwise look, they are, in a fundamental, mathematical sense, the same. A long-standing aim in topology---the branch of mathematics concerned with manifolds---has been to provide this kind of classification for manifolds in dimensions larger than two (in other dimensions the distinguishing characteristic would not necessarily be the number of holes, but some other mathematical property). These classifications are the least understood in exactly those dimensions that are important in physics---the usual three dimensions of our physical world, and the four dimensions of space-time. So it is natural that mathematicians have turned to ideas from physics, like the Seiberg-Witten equations, to help them understand manifolds of these dimensions.
For the past decade or so, one of the main tools for understanding such questions has been a theory developed by the mathematician Simon Donaldson (Oxford University) and based on ideas from gauge theory in physics. While Donaldson theory produced some spectacular results, it was from technically extremely difficult. When the Seiberg-Witten equations burst onto the scene in the fall of 1994, mathematicians were well prepared to put the new tools to immediate use: they had already cut their teeth on the much more difficult theory of Donaldson.
It was in a lecture in September 1994 that mathematical physicist Edward Witten (Institute for Advanced Study, Princeton) first conjectured that certain equations that arose out of his joint work with physicist Nathan Seiberg (Rutgers University) might contain all of the information found in Donaldson theory. This idea was quickly taken up by a number of mathematicians, most notably Peter B. Kronheimer (now at Harvard University), Tomasz S. Mrowka (California Institute of Technology), and Clifford H.Taubes (Harvard University). Within weeks startling results were found.
The Seiberg-Witten equations have been used to simplify and generalize most of the results obtained through Donaldson theory. One important new result shows that there are strong restrictions on the geometry and topology of an important class of manifolds called symplectic manifolds. The Seiberg-Witten equations have also been used to prove a long-standing question, known as the Thom Conjecture, about what kinds of two-dimensional surfaces can occur inside a four-dimensional manifold.
As important as these results have been in mathematics, the saga of the Seiberg-Witten equations is far from over. In fact, physics suggests a whole class of equations of which the Seiberg-Witten equations are only the simplest. This new class of equations is sure 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, which appeared in the March 1995 issue of the Notices of the AMS.