Math prizes: fields for further study.

Math prizes: Fields for further study

The surprising discovery of deep, hitherto hidden links among vastly different mathematical fields is one of the strongest threads that tie together the research of three young mathematicians who this week were each awarded a Fields Medal. To mathematicians, this award, named for Canadian mathematician John C. Fields, carries the prestige, if not the monetary value, of a Nobel Prize.

Michael H. Freedman, 35, of the University of California at San Diego, was honored for his work on classifying four-dimensional shapes or manifolds, part of the study of topology (SN:7/17/82, p.42). Freedman's methods for constructing the startling variety of forms possible in four-dimensional space was a key element in the solution to this problem. His research brought together powerful ideas in both geometry and algebra.

Simon K. Donaldson, 29, of Oxford University in England, although also studying four-dimensional manifolds, took a very different approach. To provide a new geometric tool, he borrowed methods from theoretical physics--a set of nonlinear differential equations widely used for describing electromagnetic effects and other phenomena. Together with Freedman's work, his results revealed that four-dimensional space has more than one possible structure.

"When Donaldson produced his first few results on four [-dimensional] manifolds,' says Oxford's Michael Atiyah, a previous Fields Medal winner, "the ideas were so new and foreign to geometers and topologists that they merely gazed in bewildered admiration. Slowly the message has got across and now Donaldson's ideas are beginning to be used by others in a variety of ways.'

Says Donaldson, "New ideas, once developed, have a life of their own.'

West German Gerd Faltings, 32, now at Princeton (N.J.) University, solved the Mordell conjecture, a long-standing problem concerning polynomial equations (SN:7/23/83, p.58). His success depended on finding connections between number theory and algebraic curves.

Also awarded this week at the International Congress of Mathematicians, held in Berkeley, Calif., was the Nevanlinna Prize. This prize goes to mathematicians who make significant contributions to the theory that underlies computer science.

This year, the recipient was Leslie G. Valiant, 37, of Harvard University. Valiant's research encompasses a wide variety of topics in computer science, ranging from the development of rapid methods for recognizing sentences in languages described by context-free grammars (SN:11/16/85, p.314) to general ideas about what can and cannot be computed within a "reasonable' time.

Unlike a Nobel Prize, the mathematics awards go only to individuals who are less than or equal to 40 years of age. This emphasis on youth is designed to encourage recipients to continue their research while recognizing novel ideas that open up new mathematical fields for others to explore.

All four prizewinners note that hard work, persistence and luck played important roles in their discoveries. But, says Donaldson, "the main point of doing this is to have fun.'