Three Mathematicians Share 2008 Wolf Prize

January 25, 2008

The 2008 Wolf Prize will be awarded to Pierre R. Deligne (Institute for Advanced Study) for his work on mixed Hodge theory, the Weil conjectures, the Riemann-Hilbert correspondence, and for contributions to arithmetic; to Phillip A. Griffiths (Institute for Advanced Study) for his work on variations of Hodge structures, the theory of periods of abelian integrals, and for contributions to complex differential geometry; and to David B. Mumford (Brown University) for his work on algebraic surfaces, on geometric invariant theory, and for laying the foundations of the modern algebraic theory of moduli of curves and theta functions.

"Central to modern algebraic geometry," wrote the prize committee, "is the theory of moduli, i.e., variation of algebraic or analytic structure. This theory was traditionally mysterious and problematic. In critical special cases, i.e., curves, it made sense, i.e., the set of curves of genus greater than one had a natural algebraic structure. In dimensions greater than one, there was some sort of structure locally, but globally everything remained mysterious. The two main (and closely related) approaches to moduli were invariant theory on the one hand and periods of abelian integrals on the other. This key problem was tackled and greatly elucidated by Deligne, Griffiths, and Mumford."

Mumford, born in the UK in 1937, is recognized for revolutionizing the algebraic approach through invariant theory, which he renamed "geometric invariant theory." With this approach, he provided a complicated prescription for the construction of moduli in the algebraic case. As one application he proved that there were a set of equations defining the space of curves, with integer coefficients. Most important, he showed that moduli spaces, though often very complicated, do exist except for, after his work, in well-understood exceptions.

This framework, noted the prize committee, is critical to the work of Griffiths and Deligne.

Griffiths, born in the U.S. in 1938, discovered that the Hodge filtration measured against the integer homology generalizes the classical periods of integrals. He realized that the period mapping had a natural generalization as a map into a classifying space for variations of Hodge structure, with a new non-classical restriction imposed by the Kodaira-Spencer class action. This led to a great deal of work in complex differential geometry, e.g., his basic work with Deligne, John Morgan, and Dennis Sullivan on rational homotopy theory of compact Kaehler manifolds.

Deligne, born in Belgium in 1944, demonstrated how to extend the variation of Hodge theory to singular varieties. This advance, called mixed Hodge theory, allowed explicit calculation on the singular compactification of moduli spaces that came up in Mumford´s geometric invariant theory, which is called the Deligne-Mumford compactification. These ideas assisted Deligne in proving several major results, e.g., the Riemann-Hilbert correspondence and the Weil conjectures.

Source: Wolf Foundation