Math prizes: moonshine to quantum logic.

For research ranging from string theory to chaos, four mathematicians have won the Fields Medal, the most prestigious award in mathematics.

First presented in 1936 at the International Congress of Mathematicians (ICM), the Fields Medal is given every 4 years to mathematicians age 40 and younger "in recognition of work already done and as an encouragement for future achievements."

The recipients--announced this week at ICM 98 in Berlin--are Richard E. Borcherds and William T. Gowers of the University of Cambridge in England, Maxim Kontsevich of the Institut des Hautes Etudes Scientifiques in Buressur-Yvette, France, and Curtis T. McMullen of Harvard University.

Borcherds, 38, proved a conjecture establishing an unexpected connection between a class of geometric equations and an algebraic entity called a group. That entity consists of a set of objects for which chosen arithmetic rules apply. For example, all whole numbers and their sums constitute a group.

Simple groups are the building blocks from which all other groups are assembled. In the late 1970s, two mathematicians proposed that one particular simple but huge group--dubbed the monster--shares an important characteristic with mathematical expressions called elliptic curves.

That suggestion seemed so outrageous at the time that experts dubbed it "monstrous moonshine." To establish the link, Borcherds used mathematical ideas from string theory, which posits that particles of matter are made up of tiny, extended objects called strings (SN: 2/27/93, p. 136).

Kontsevich, 34, made important contributions to string theory, demonstrating the mathematical equivalence of two quantum-gravity models (SN: 6/13/98, p. 376). He also discovered a compact way to characterize mathematical knots (SN: 3/21/92, p. 186).

McMullen, 40, investigated the Mandelbrot set (SN: 11/23/91, p. 331), which compactly summarizes the chaotic behavior of certain equations, including those often used to describe phenomena such as flowing liquids and changing weather systems. His research provided a precise characterization of the Mandelbrot set's convoluted boundary.

Gowers, 35, proved important conjectures in the field of functional analysis. He extensively applied combinatorial techniques to explore sets of functions.

Also awarded this week was the Nevanlinna Prize, which goes to mathematicians who make significant contributions to theoretical computer science. The recipient is Peter W. Shor, 39, of AT&T Labs Research in Florham Park, N.J.

In 1994, Shor discovered a mathematical technique that takes advantage of the peculiarities of quantum mechanics to speed up the factoring of large composite numbers into their prime-number components (SN: 5/14/94, p. 308). That finding sparked a major, ongoing effort to develop a computer based on quantum logic (SN: 1/14/95, p. 30).