# David B. Mumford

Wolf Prize Laureate in Mathematics 2008/9

**David B. Mumford**

#### Affiliation at the time of the award:

**Brown University****, USA**

#### Award citation:

**“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”.**

#### Prize share:

**David B. Mumford**

**Phillip A. Griffiths**

**Pierre R. Deligne**

Central to modern algebraic geometry 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.

Professor David B. Mumford revolutionized 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, Mumford showed that moduli spaces, though often very complicated, exist except for what, after his work, are well-understood exceptions. This framework is critical for the work by Griffiths and Deligne. Classically, the moduli space of curves was parameterized by using periods of the abelian integrals on them. Mathematicians, e.g., the Wolf Prize winner Andre Weil, have unsuccessfully tried to generalize the periods to higher dimensions.

Professor Phillip A. Griffiths 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, e.g., his basic work with Deligne, John Morgan, and Dennis Sullivan on rational homotopy theory of compact Kaehler manifolds.

Building on Mumford´s and Griffiths’ work, Professor Pierre R. 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, which is called the Deligne-Mumford compactification. These ideas assisted Deligne in proving several other major results, e.g., the Riemann-Hilbert correspondence and the Weil conjectures.