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David Mumford | |
|---|---|
 David Mumford in 2010 | |
| Born | (1937-06-11)11 June 1937 (age 88) |
| Alma mater | Harvard University |
| Known for | Algebraic geometry Mumford surface Deligne-Mumford stacks Mumford–Shah functional[1] |
| Awards | Putnam Fellow (1955, 1956) Sloan Fellowship (1962) Fields Medal (1974) MacArthur Fellowship (1987) Shaw Prize (2006) Steele Prize (2007) Wolf Prize (2008) Longuet-Higgins Prize (2005, 2009) National Medal of Science (2010) BBVA Foundation Frontiers of Knowledge Award (2012) |
| Honours | |
| Scientific career | |
| Fields | Mathematics |
| Institutions | Brown University Harvard University |
| Thesis | Existence of the moduli scheme for curves of any genus (1961) |
| Doctoral advisor | Oscar Zariski |
| Doctoral students | Avner Ash Henri Gillet Tadao Oda Emma Previato Malka Schaps Michael Stillman Jonathan Wahl Song-Chun Zhu |
David Bryant Mumford (born 11 June 1937) is an Americanmathematician known for his work inalgebraic geometry and then for research intovision andpattern theory. He won theFields Medal and was aMacArthur Fellow. In 2010 he was awarded theNational Medal of Science. He is currently a University Professor Emeritus in the Division of Applied Mathematics atBrown University.
Mumford was born inWorth, West Sussex inEngland, of an English father and American mother. His father William started an experimental school inTanzania and worked for the then newly createdUnited Nations.[3]
He attendedPhillips Exeter Academy, where he received aWestinghouse Science Talent Search prize for his relay-based computer project.[4][5] Mumford then went toHarvard University, where he became a student ofOscar Zariski. At Harvard, he became aPutnam Fellow in 1955 and 1956.[6] He completed hisPhD in 1961, with a thesis entitledExistence of the moduli scheme for curves of any genus.[7]
Mumford's work in geometry combined traditional geometric insights with the latest algebraic techniques. He published onmoduli spaces, with a theory summed up in his bookGeometric Invariant Theory, on the equations defining anabelian variety, and onalgebraic surfaces. His booksAbelian Varieties (withC. P. Ramanujam) andCurves on an Algebraic Surface combined the old and new theories.
His lecture notes onscheme theory circulated for years in unpublished form. At the time, they were, beside the treatiseÉléments de géométrie algébrique, the only accessible introduction. Starting in 1967, the notes were mimeographed, bound in red cardboard, and distributed by Harvard's mathematics department under the titleIntroduction to Algebraic Geometry (Preliminary version of first 3 Chapters). Later (1988; 1999, 2nd ed.,ISBN 3-540-63293-X), they were published by Springer under theLecture Notes in Mathematics series asThe Red Book of Varieties and Schemes (though neither of the two published editions features a red cover).
Other work that was less thoroughly written up were lectures on varieties defined byquadrics, and a study ofGoro Shimura's papers from the 1960s.
Mumford's research did much to revive the classical theory oftheta functions, by showing that its algebraic content was large, and enough to support the main parts of the theory by reference to finite analogues of theHeisenberg group. This work on theequations defining abelian varieties appeared in 1966–7. He published some further books of lectures on the theory.
He also is one of the founders of thetoroidal embedding theory; and sought to apply the theory toGröbner basis techniques, through students who worked in algebraic computation.
In a sequence of four papers published in theAmerican Journal of Mathematics between 1961 and 1975, Mumford explored pathological behavior inalgebraic geometry, that is, phenomena that would not arise if the world of algebraic geometry were as well-behaved as one might expect from looking at the simplest examples. These pathologies fall into two types: (a) bad behavior in characteristic p and (b) bad behavior in moduli spaces.
Mumford's philosophy in characteristicp was as follows:
A nonsingular characteristicp variety is analogous to a general non-Kähler complex manifold; in particular, a projective embedding of such a variety is not as strong as aKähler metric on a complex manifold, and the Hodge–Lefschetz–Dolbeault theorems onsheaf cohomology break down in every possible way.
In the first Pathologies paper, Mumford finds an everywhere regular differential form on a smooth projective surface that is not closed, and shows that Hodge symmetry fails for classicalEnriques surfaces in characteristic two. This second example is developed further in Mumford's third paper on classification of surfaces in characteristicp (written in collaboration withE. Bombieri). This pathology can now be explained in terms of thePicard scheme of the surface, and in particular, its failure to be areduced scheme, which is a theme developed in Mumford's book "Lectures on Curves on an Algebraic Surface". Worse pathologies related to p-torsion incrystalline cohomology were explored byLuc Illusie (Ann. Sci. Ec. Norm. Sup. (4) 12 (1979), 501–661).
In the second Pathologies paper, Mumford gives a simple example of a surface in characteristicp where thegeometric genus is non-zero, but the second Betti number is equal to the rank of theNéron–Severi group. Further such examples arise inZariski surface theory. He also conjectures that theKodaira vanishing theorem is false for surfaces in characteristicp. In the third paper, he gives an example of anormal surface for which Kodaira vanishing fails. The first example of a smooth surface for which Kodaira vanishing fails was given byMichel Raynaud in 1978.
In the second Pathologies paper, Mumford finds that theHilbert scheme parametrizing space curves of degree 14 and genus 24 has a multiple component. In the fourth Pathologies paper, he finds reduced and irreducible complete curves which are not specializations of non-singular curves.
These sorts of pathologies were considered to be fairly scarce when they first appeared. ButRavi Vakil in his paper "Murphy's law in algebraic geometry" showed that Hilbert schemes of nice geometric objects can be arbitrarily "bad", with unlimited numbers of components and with arbitrarily large multiplicities (Invent. Math. 164 (2006), 569–590).
In three papers written between 1969 and 1976 (the last two in collaboration withEnrico Bombieri), Mumford extended theEnriques–Kodaira classification of smoothprojective surfaces from the case of the complexground field to the case of analgebraically closed ground field of characteristicp. The final answer turns out to be essentially the same as the answer in the complex case (though the methods employed are sometimes quite different), once two important adjustments are made. The first is that one may get "non-classical" surfaces, which come about whenp-torsion in thePicard scheme degenerates to a non-reduced group scheme. The second is the possibility of obtainingquasi-elliptic surfaces in characteristics two and three. These are surfaces fibred over a curve where the general fibre is a curve ofarithmetic genus one with a cusp.
Once these adjustments are made, the surfaces are divided into four classes by theirKodaira dimension, as in the complex case. The four classes are:a) Kodaira dimension minus infinity. These are theruled surfaces.b) Kodaira dimension 0. These are theK3 surfaces,abelian surfaces, hyperelliptic andquasi-hyperelliptic surfaces, andEnriques surfaces. There are classical and non-classical examples in the last two Kodaira dimension zero cases.c) Kodaira dimension 1. These are the elliptic andquasi-elliptic surfaces not contained in the last two groups.d) Kodaira dimension 2. These are thesurfaces of general type.
Mumford was awarded aFields Medal in 1974. He was aMacArthur Fellow from 1987 to 1992. He won theShaw Prize in 2006. In 2007 he was awarded theSteele Prize for Mathematical Exposition by theAmerican Mathematical Society. In 2008 he was awarded theWolf Prize;[8] on receiving the prize in Jerusalem fromShimon Peres, Mumford announced that he was donating half of the prize money toBirzeit University in thePalestinian territories and half toGisha, an Israeli organization that promotes the right to freedom of movement of Palestinians in the Gaza Strip.[9][10] In 2010 he was awarded theNational Medal of Science.[11] In 2012 he became a fellow of theAmerican Mathematical Society.[12]
There is a long list of awards and honors besides the above, including
He was elected President of theInternational Mathematical Union in 1995 and served from 1995 to 1999.[8]
