David Mumford

fl. 1937, Worth, Sussex, England

AlgebraGeometryApplied and computational mathematicsFields Medal 1974Shaw Prize in Mathematical Sciences 2006

American-British mathematician who built the moduli theory of curves and abelian varieties via geometric invariant theory, then turned to pattern theory and computer vision; Fields Medal 1974, Shaw Prize 2006.

David Mumford was born in Worth, Sussex, to an English father and an American mother, and grew up largely in the United States. He studied at Harvard University, earning his PhD in 1961 under Oscar Zariski with a thesis on moduli of algebraic curves. He joined the Harvard faculty and spent the first half of his career reshaping algebraic geometry: his book "Geometric Invariant Theory" systematized quotients of varieties by reductive group actions, and with it he constructed the moduli spaces of curves and abelian varieties as algebraic objects.

In a series of papers Mumford studied the geometry of the moduli space $\mathcal{M}_g$ of genus- $g$ curves: with Deligne he introduced the stable-curve compactification (1969), with Harris he proved that $\mathcal{M}_g$ is of general type for odd genus $g \geq 25$ and even genus $g \geq 24$ , and his work on theta functions and abelian varieties bridged algebraic geometry and the theory of special functions. He also wrote influential surveys and textbooks, including "The Red Book of Varieties and Schemes".

From the mid-1980s Mumford moved to Brown University and changed fields entirely, founding with a school of collaborators the mathematical theory of pattern recognition — Bayesian approaches to image segmentation, the Mumford–Shah functional, and statistics of natural images. He received the Fields Medal in 1974, a MacArthur Fellowship in 1987, and the Shaw Prize in 2006 jointly with Wu Wen-tsun, and served as president of the International Mathematical Union from 1995 to 1998.

Workplaces: Harvard University, Brown University

United Kingdom, United States

Award-winning work

Fields Medal 1974Awarded for his work on algebraic surfaces and on moduli problems. Mumford developed geometric invariant theory into a practical tool: he determined which group actions admit good quotients and used this to construct coarse moduli spaces for algebraic curves and for polarized abelian varieties, giving concrete form to Grothendieck's program. His work on the moduli space of curves — including the first proof of the existence of the Deligne–Mumford compactification's key properties and the study of its cohomology — made

\mathcal{M}_g

a central object of algebraic geometry. His book "Geometric Invariant Theory" (1965) remains the standard reference on quotients in algebraic geometry.

Contributions, linked to the library

Library topicAlgebraic geometryCreated geometric invariant theory and the moduli spaces of curves and abelian varieties.