Дэвид Мамфорд: Американ математигі, геометрия және көру теориясының шебері

American mathematician
the Dean of Brechin

David Bryant Mumford (born 11 June 1937) is an American mathematician known for his work in algebraic geometry and then for research into vision and pattern theory. He won the Fields Medal and was a MacArthur Fellow. In 2010 he was awarded the National Medal of Science. He is currently a University Professor Emeritus in the Division of Applied Mathematics at Brown University.

Mumford was born in Worth, West Sussex in England, of an English father and American mother. His father William started an experimental school in Tanzania and worked for the then newly created United Nations. He attended Phillips Exeter Academy, where he received a Westinghouse Science Talent Search prize for his relay based computer project. Mumford then went to Harvard University, where he became a student of Oscar Zariski. At Harvard, he became a Putnam Fellow in 1955 and 1956. He completed his PhD in 1961, with a thesis entitled Existence of the moduli scheme for curves of any genus. He married Erika, an author and poet, in 1959 and they had four children, Stephen, Peter, Jeremy, and Suchitra. He currently has seven grandchildren.

Mumford's work in geometry combined traditional geometric insights with the latest algebraic techniques. He published on moduli spaces, with a theory summed up in his book Geometric Invariant Theory, on the equations defining an abelian variety, and on algebraic surfaces. His books Abelian Varieties (with C. P. Ramanujam) and Curves on an Algebraic Surface combined the old and new theories. His lecture notes on scheme theory circulated for years in unpublished form, at a time when they were, beside the treatise Éléments de géométrie algébrique, the only accessible introduction. They are now available as The Red Book of Varieties and Schemes
Other work that was less thoroughly written up were lectures on varieties defined by quadrics, and a study of Goro Shimura's papers from the 1960s. Mumford's research did much to revive the classical theory of theta 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 the Heisenberg group. This work on the equations defining abelian varieties appeared in 1966–7. He published some further books of lectures on the theory. He also was one of the founders of the toroidal embedding theory; and sought to apply the theory to Gröbner basis techniques, through students who worked in algebraic computation.

In a sequence of four papers published in the American Journal of Mathematics between 1961 and 1975, Mumford explored pathological behavior in algebraic 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.

In the second Pathologies paper, Mumford finds that the Hilbert 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. But Ravi Vakil showed in his paper "Murphy's law in algebraic geometry" has shown 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 with Enrico Bombieri), Mumford extended the Enriques–Kodaira classification of smooth projective surfaces from the case of the complex ground field to the case of an algebraically closed ground field of characteristic p. 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 when p torsion in the Picard scheme degenerates to a non reduced group scheme. The second is the possibility of obtaining quasi elliptic surfaces in characteristics two and three. These are surfaces fibred over a curve where the general fibre is a curve of arithmetic genus one with a cusp. Once these adjustments are made, the surfaces are divided into four classes by their Kodaira dimension, as in the complex case. The four classes are:
a) Kodaira dimension minus infinity. These are the ruled surfaces. b) Kodaira dimension 0. These are the K3 surfaces, abelian surfaces, hyperelliptic and quasi hyperelliptic surfaces, and Enriques surfaces. There are classical and non classical examples in the last two Kodaira dimension zero cases. c) Kodaira dimension 1. These are the elliptic and quasi elliptic surfaces not contained in the last two groups. d) Kodaira dimension 2. These are the surfaces of general type.