Math in the News
Deligne, Griffiths, and Mumford Share 2008 Wolf Prize in Mathematics
May 20, 2008
Deligne is honored for his work on mixed Hodge theory; the Weil conjectures; the Riemann-Hilbert correspondence; and for his contributions to arithmetic. Now at the Institute for Advanced Study in Princeton, N.J., he was born in Belgium in 1944.
Griffiths won recognition for his work on variations of Hodge structures; the theory of periods of Abelian integrals; and for his contributions to complex differential geometry. Born in the United States in 1938, he is also at the Institute for Advanced Study.
Mumford received the prize 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. Now at Brown University, he was born in the United Kingdom in 1937.
In its citation, the Wolf Foundation noted that the theory of moduli—variation of algebraic or analytic structure—is central to modern algebraic geometry. "This theory was traditionally mysterious and problematic," the citation said. In certain critical cases, the theory made sense. For example, the set of curves of genus greater than one had a natural algebraic structure. In dimensions greater than one, some sort of structure existed locally, but globally everything remained mysterious.
Mumford, Deligne, and Griffiths transformed the two main (and closely related) approaches to moduli—invariant theory on the one hand and periods of abelian integrals on the other—to elucidate the problem.
Mumford revolutionized the algebraic approach through invariant theory, which he renamed "geometric invariant theory." With this approach, Mumford provided a complicated prescription for the construction of moduli in the algebraic case. As one application, he proved that a set of equations defines the space of curves, with integer coefficients. Most important, Mumford showed that moduli spaces, though often very complicated, exist except for what, after his insights, are well-understood exceptions.
Griffiths, according to the Wolf Foundation, "had the fundamental insight that the Hodge filtration measured against the integer homology generalizes the classical periods of integrals. Moreover, 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. As one example of these new efforts, Griffiths worked with Deligne, John Morgan, and Dennis Sullivan on rational homotopy theory of compact Kaehler manifolds.
Building on Mumford´s and Griffiths´work, Deligne 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, now called the Deligne-Mumford compactification. These ideas assisted Deligne in proving several other major results, including the Riemann-Hilbert correspondence and the Weil conjectures.
Past Wolf Prize winners in mathematics have included Raoul Bott, Alberto P. Calderón, Andrei N. Kolmogorov, Mark G. Krein, Peter Lax, Hans Lewy, Laszlo Lovász, John Milnor, Jurgen Moser, Ilya Piatetski-Shapiro, Jean-Pierre Serre, Carl L. Siegel, Yakov Sinai, Elias M. Stein, Jacques Tits, Andre Weil, Hassler Whitney, Andrew Wiles, and Oscar Zariski. The award has tended to honor mathematicians late in their careers, recognizing a lifetime of achievement.
The prizes of $100,000 in each area are given annually in four out of five scientific fields on a rotating basis. The fields are agriculture, chemistry, mathematics, medicine, and physics. In the arts, the prize is given in one of four rotating categories: architecture, music, painting, and sculpture.
Source: Wolf Foundation, May 19, 2008.