# Simon Kirwan Donaldson

# Simon Kirwan Donaldson

**1957-**

**English Mathematician**

While still a graduate student, English mathematician Simon Donaldson published advances in topology (a division of geometry concerning the mathematical properties of deformed space) heralded by scholars as an important contribution to the understanding of four-dimensional "exotic" space. Besides sparking worldwide attention, Donaldson's work earned a 1986 Fields Medal for advancing understanding of four-dimensional space and, in particular, the topology of four-dimensional manifolds. Donaldson's work established that four-dimensional space has properties that are different from all other dimensions.

Donaldson used ideas taken from theoretical physics (for example, Yang-Mills equations and gauge theories) to develop classifications for four-dimensional space. Using elegant methodologies involving nonlinear partial differential equations and algebraic geometry, Donaldson's discoveries are regarded by many mathematicians as a seminal new synthesis between pure and applied mathematics—that is, between advancements of mathematical theory and mathematics used to solve practical problems in science, engineering, and economics. In essence, Donaldson utilized mathematical concepts used by physicists to solve purely mathematical topological problems.

Donaldson undertook his undergraduate education at Cambridge and completed his doctoral work at Oxford University. After earning his doctorate, Donaldson did further postdoctoral work at Oxford, completed an appointment at the Institute for Advanced Study at Princeton, then returned to Oxford as a professor of mathematics. In 1986 Donaldson was elected a Fellow of the Royal Society. In 1997 Donaldson took on an additional academic appointment at Stanford University.

Donaldson's work was based, in part, on the work of French mathematician Jules Henri Poincaré's (1854-1912) system of classifying manifolds that contain local coordinate systems that are related to each other by coordinate transformations belonging to a specified class. The concept of manifolds was first advanced by German mathematician Georg Friedrich Bernhard Riemann (1826-1866) to describe relationships in topological deformed space, where relative position and general shapes are defined rather than absolute distances or angles. Grounded in this mathematical heritage, Donaldson's work then proceeded to reverse the normal flow of ideas from pure math to applied math by utilizing physics applications—for example, the mathematics used to describe the interactions among subatomic particles—to fuel the engine of theoretical advancement.

Although a complete understanding of Donaldson's work requires a familiarity with mathematics beyond calculus (particularly a graduate level understanding of differential topology, geometry, and differential equations), Donaldson's work has been widely recognized for its significant advancement of low-dimensional topological geometry through the application of mathematical concepts used by particle physicists to describe short-lived subatomic particle-like wave packets called instantons.

Donaldson's important discoveries regarding the unique applications of instantons were substantially based on equations used in quantum mechanics—for example, equations published by physicists Chen Ning Yang (1922- ) and Robert L. Mills—that were originally derived from the fundamental electromagnetic equations put forth by Scottish physicist James Clerk Maxwell (1831-1879). Although the Yang-Mills equations utilized solutions termed instantons to calculate interactions among nuclear particles, Donaldson insightfully used the assumptions about the concepts of dimensional space underlying instantons to help analyze four-dimensional space. Subsequently, easier and more direct methods also derived from theoretical physics (such as monopoles and string theory) have been found that confirm Donaldson's conclusions and contributions.

Donaldson has also made significant contributions to the differential geometry of holomorphic vector bundles, applications of gauge theory. Donaldson's publications include studies on symplectic topology, the application of gauge theory to four-dimensional topology, complex algebraic surfaces and stable vector bundles, the intersection forms of four-manifolds, infinite determinants, and a co-authored work with P. B. Kronheimer titled *The Geometry Of Four-Manifolds* (1990).

**ADRIENNE WILMOTH LERNER**

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**Simon Kirwan Donaldson**