Simon K. Donaldson | Biography

Simon Donaldson shocked the mathematical world during the 1980s with a series of papers on the structure of four-dimensional spaces. Researchers had produced a collection of results during the previous decade that outlined a general understanding of the properties of spaces of five or more dimensions, and of course, the cases of one- and two-dimensional spaces were well known. Ironically, three- and four-dimensional spaces were the hardest to interpret, even though they are the most applicable to physical space (if time is considered to be the fourth dimension). Great progress was made in three-dimensions by William Thurston,who received a Fields Medal in 1982 for his efforts. That same year, Donaldson published his most remarkable result: four-dimensional space has highly unusual properties that are found in no other dimension. Speaking on the occasion of Donaldson's presentation with the 1986 Fields Medal, Michael Atiyah commented, "When Donaldson produced his first few results on four-manifolds [four-dimensional topological surfaces], the ideas were so new and foreign to geometers and topologists that they merely gazed in bewildered admiration . . . Donaldson has opened up an entirely new area; unexpected and mysterious phenomena about the geometry of four dimensions have been discovered."

Simon K. Donaldson was born on August 20, 1957, in Cambridge, England. He attended Pembroke College in Cambridge University and received his B.A. degree in 1979. During his second year of graduate studies at Worcester College in Oxford University, Donaldson made the spectacular discovery of "exotic" or nonstandard differential structures of four-dimensional Euclidean space. In other words, he found that there were different ways of orienting a mathematical structure in ordinary space with the addition of a fourth dimension. Because the standard differential structure is the only one possible in all other dimensions, the mathematical community was amazed at the exceptions created by the addition of the fourth dimension. After completing his doctorate in 1984, Donaldson spent a year at Princeton University's Institute for Advanced Study (IAS) and was a visiting scholar at Harvard University during the spring of 1985. He then returned to England where he holds an appointment at the Mathematics Institute in Oxford.

Reverses Yang-Mills Equations

In 1954, Chen Ning Yang and Robert Mills collaborated on derivations of mathematical formulas that combined the branch of mathematics known as topology (the study of the ways in which coordinate structures attach at a point) and the branch of physics called quantum electrodynamics (the study of electromagnetic phenomena under the rules of quantum mechanics). In doing so, they built upon the work of James Clerk Maxwell, who had introduced equations in the 19th century to describe the behavior of electromagnetic waves. The Yang-Mills equations generalize Maxwell's equations to more complex spaces. Since they are nonlinear partial differential equations, the Yang-Mills equations are very difficult to solve, even for specific cases. Work by Atiyah and others during the 1970s led to important connections between the equations and techniques from differential and algebraic geometry.

While others were working on methods for solving the Yang-Mills equations,Donaldson approached the topic from a fresh viewpoint. As described by John D. S. Jones in Nature, "Donaldson argues as follows: if we know something about the solutions of the Yang-Mills equations then we must be able to extract information about the underlying space . . . Donaldson starts by treating the solutions of the equations as, in some sense, the known quantity."

It is common for mathematicians to look to theoretical physics for problems to investigate. In this unusual reversal, Donaldson used the tools of physics to explore purely mathematical ideas. In four-dimensional Euclidean space, the absolute minimum solutions (under certain boundary conditions at infinity) of the Yang-Mills equations are called "instantons." Donaldson's inspiration was to look at the nonlinear space of parameters for these instantons as a lens through which he could examine the space on which the equations are defined.

One of the basic goals of topology is to classify multidimensional spaces into categories that have the same basic structure. Jones described this as being similar to the taxonomic classification system in biology. In mathematics, spaces can be classified in terms of their topology (connectedness) or their smoothness (lack of corners, as shown by continuity of derivatives). For example, the surface of a sphere belongs to a different topological class than that of a torus (doughnut-like shape) because of the existence of the hole. In three dimensions, there is no difference between the results of classifying spaces topologically or smoothly. In five or more dimensions, there are relatively minor differences that are well understood. Michael Freedman obtained clear results for topological classification of four-dimensional spaces about the same time that Donaldson was establishing very different results using smoothness criteria. In the words of Atiyah, this "shows that the differentiable and topological situations are totally different"--a situation that occurs only in spaces of four dimensions.

Another way of describing Donaldson's results was offered by John Baez in This Week's Finds in Mathematical Physics. He stated the basic question as being whether n-dimensional Euclidean space allows any smooth structure other than the usual one. He concluded, "The answer is no--EXCEPT if n=4, where there are uncountably many smooth structures!" This unexpected result generated great excitement within the mathematical community and earned Donaldson the Fields Medal, the most prestigious international mathematics award, in 1986.

Delves Deeper Into Four-Dimensional Space

Donaldson used intersection matrices as a tool for exploring four-dimensional spaces. Any four-dimensional space can be characterized by a matrix of integersin a way that describes how two-dimensional spaces intersect within it. This symmetric, invariant matrix will be the same for all topologically equivalent spaces. This means that if one finds two spaces with nonequivalent intersection matrices, those spaces will be topologically distinct. Conversely, Freedman showed that at most two topologically distinct spaces can be represented by equivalent intersection matrices.Examining the situation from a smoothness perspective, Donaldson found that there are unlimited possibilities for distinct spaces with equivalent intersection matrices.

It was in calculating these intersection matrices that Donaldson used the Yang-Mills equations. According to Jones, the instanton solutions are "concentrated in a very small ball and they behave like particles placed at the centre of the ball. This gives a way of recovering the points of the space from the solutions of the equations." With continued work, Donaldson was able to identify other invariants capable of distinguishing between two smoothly different, topologically equivalent manifolds.

One of the dramatic byproducts of Donaldson's discoveries was the description of exotic four-spaces. In Atiyah's description, these remarkable spaces "contain compact sets which cannot be contained inside any differentiably embedded 3-sphere!" In addition to producing startling results about the mathematics of four-dimensional spaces, Donaldson's work has also generated useful information for physicists. For instance, the earliest link between topology and quantum theory was Paul Dirac's idea that the electric charge of a particle (which is an integral multiple of the charge of a single electron) has a basis in magnetism. He described this in terms of a hypothetical "magnetic monopole'--a basic particle that radiates a magnetic field just as a charged particle radiates an electrical field. Donaldson established a direct link between the parameter space of monopoles having magnetic charge k and the space of rational functions of a complex variable of degree k.

In addition to continuing his research, Donaldson currently serves as one of nine voting members of the executive committee of the International Mathematical Union (IMU). He is a fellow of the Royal Society of London, from which he received the Royal Medal in 1992. In 1994 he shared withShing-Tung Yau the Swedish Academy of Science's Crafoord Prize, which is presented once every six years in the field of mathematics to provide financial support for research in an area of "particular interest and considerable activity."