Honors for connecting number theory, geometry, and algebra. (Math Prizes).

The Fields Medal, the world's highest honor mathematical research, has gone to two mathematicians who forged new links between different branches of mathematics. The recipients--announced this week the International Congress of Mathematicians in Beijing--are Laurent Lafforgue of the Institut des Hautes Etudes Scientifiques in Bures-sur-Yvette, France, and Vladimir Voevodsky of the Institute for Advanced Study in Princeton, N.J.

First awarded in 1936, the Fields Medal is now presented every 4 years by the International Mathematical Union to mathematicians of age 40 and younger "in recognition of work already done and as an encouragement for future achievements."

Lafforgue worked on a major component of a far-reaching mathematical effort known as the Langlands program. Formulated in the 1960s by Robert P. Langlands of the Institute for Advanced Study, the program a set of mathematical conjectures about how certain aspects of number theory might be related to one another and to other areas of mathematics. The proof in 1994 of Fermat's last theorem (SN: 11/5/94, p. 295) and subsequent work on other pieces of the Langlands puzzle (SN: 1/15/00, p. 47) have confirmed the value of Langlands' insights.

Lafforgue proved the so-called global Langlands correspondence not for ordinary numbers but for function fields, which are formulas that can be treated like numbers. Along the way, he invented a new geometric construction that may turn out to be useful in other mathematical areas.

Voevodsky's research largely concerned the development of novel ways to describe the geometric shapes of solutions to algebraic equations. "His research ha influenced the development of algebraic geometry and topology," says Philip A. Griffiths of the Institute for Advanced Study.

Also awarded this week at the progress was the Rolf Nevanlinna Prize, which goes to researchers who make significant contributions to mathematical aspects of computer science. The recipient is Madhu Sudan of the Massachusetts Institute of Technology.

Sudan helped advanced ingenious computer-based methods for checking the validity of mathematical proofs (SN: 6/6/92, p. 382), and he developed techniques that computers use to detect and correct errors automatically (SN: 4/6/02, p. 216).

"The achievements of the Fields medalists and Nevanlinna Prize winner show great depth and originality," says Jacob Palis of the Instituto de Matematica Pura e Aplicada in Rio de Janeiro. "Their choice of problems, their methods, and their results are quite different from one another, and this diversity exemplifies the vitality of the whole of the mathematical sciences."