The Prize is awarded to Simon Donaldson, University of Oxford, England, for his fundamental investigations in four-dimensional geometry through application of instantons, in particular his discovery of new differential invariants: And Shing-Tung Yau, Harvard University, Cambridge, MA, USA, for his development of non-linear techniques in differential geometry leading to the solution of several outstanding problems.
The Crafoord Prize amounts to c. USD 300 000, and is divided equally between the two prizewinners. It will be awarded in September 1994.
Modern geometry is divided in two main directions, differential geometry which takes distances into account, and topology, which only cares about shape. Gradually it has become clear that there is in topology a clear division between geometry in five dimensions or more on the one hand, and geometry in lower dimensions on the other. In the fifties and sixties topology in five and higher dimensions was investigated and today could be said to be well understood. As for low dimensions, from one to four, they have all turned out to have their own peculiarities. One and two dimensions are, because of their perspicuity, relatively simple. Three-dimensional geometry, through considerably more complicated, behaves much like that of lower, particularly two, dimensions. Geometry of four dimensions seems to be a boundary case. It differs fundamentally from lower dimensions while the methods used for higher dimensions do not work.
The study of five or more dimensions shows that in topology there is reason to take into account how strongly the form of geometrical objects are permitted to be changed. Continuity is the notion that is used when one allows the sharpest possible changes, differentiability the one used when only smooth changes are accepted. The understanding of this distinction in these dimensions is part of the achievements obtained in the fifties and sixties while this distinction may not be seen in one, two and three dimensions.
Thanks to Shing-Tung Yau´s work over the past twenty years, the role and understanding of the basic partial differential equations in geometry has changed and expanded enormously within the field of mathematics. His work has had an impact on areas of mathematics and physics as diverse as topology, algebraic geometry, representation theory, and general relativity as well as differential geometry and partial differential equations. Yau is a student of legendary Chinese mathematician Shiing-Shen Chern, for whom he studied at Berkeley. As a teacher he is very generous with his ideas and he has had many students and also collaborated with many mathematicians.
Shing-Tung Yau was born in Guandong in southern China in 1949. He received his Ph. D. from the University of California at Berkeley in 1971, and his D. Sc. From the Chinese University of Hong Kong in 1980. During the years 1971-72 and 1979-80 he was a member of the Institute for Advanced Study in Princeton, NJ, USA, and was a professor there 1980-84. After that Yau was a professor at the University of California at San Diego, La Jolla, for some years, and is now (since the end of the 80´s) at Harvard University.
Simon Donaldson was born in Cambridge, UK in 1957. He was Junior Research Fellow at All Soul´s College in Oxford 1983-85 and Visiting Member at The Institute for Advanced Study, Princeton, NJ, USA 1983-84. Donaldson is since 1985 Wallis Professor of Mathematics at Oxford University.