Early life
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. [3]
He attended Phillips Exeter Academy, where he received a Westinghouse Science Talent Search prize for his relay-based computer project. [4] [5] 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. [6] He completed his PhD in 1961, with a thesis entitled Existence of the moduli scheme for curves of any genus. He married Erika Mumford (1935-1988), an author and poet, in 1959 and they had four children, Stephen, Peter, Jeremy, and Suchitra. He currently has seven grandchildren.
Work in algebraic geometry
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 His lecture notes on scheme theory circulated for years in unpublished form, at a time when they were, beside the treatise Éléments de géométrie algébrique, the only accessible introduction.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 ( ISBN 3-540-63293-X).
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 is 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.
Work on pathologies in algebraic geometry
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.
Characteristic-p pathologies
Mumford's philosophy in characteristic p was as follows:
A nonsingular characteristic p variety is analogous to a general non-Kähler complex manifold; in particular, a projective embedding of such a variety is not as strong as a Kähler metric on a complex manifold, and the Hodge–Lefschetz–Dolbeault theorems on sheaf 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 classical Enriques surfaces in characteristic two. This second example is developed further in Mumford's third paper on classification of surfaces in characteristic p (written in collaboration with E. Bombieri). This pathology can now be explained in terms of the Picard scheme of the surface, and in particular, its failure to be a reduced scheme, which is a theme developed in Mumford's book "Lectures on Curves on an Algebraic Surface". Worse pathologies related to p-torsion in crystalline cohomology were explored by Luc 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 characteristic p where the geometric genus is non-zero, but the second Betti number is equal to the rank of the Néron–Severi group. Further such examples arise in Zariski surface theory. He also conjectures that the Kodaira vanishing theorem is false for surfaces in characteristic p. In the third paper, he gives an example of a normal surface for which Kodaira vanishing fails. The first example of a smooth surface for which Kodaira vanishing fails was given by Michel Raynaud in 1978.
Pathologies of 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 mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed versions of the complex plane: locally near every point they look like patches of the complex plane, but the global topology can be quite different. For example, they can look like a sphere or a torus or several sheets glued together.
In mathematics, complex geometry is the study of geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry is concerned with the study of spaces such as complex manifolds and complex algebraic varieties, functions of several complex variables, and holomorphic constructions such as holomorphic vector bundles and coherent sheaves. Application of transcendental methods to algebraic geometry falls in this category, together with more geometric aspects of complex analysis.
In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions. Abelian varieties are at the same time among the most studied objects in algebraic geometry and indispensable tools for research on other topics in algebraic geometry and number theory.
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Modern definitions generalize this concept in several different ways, while attempting to preserve the geometric intuition behind the original definition.
In algebraic geometry, a projective variety is an algebraic variety that is a closed subvariety of a projective space. That is, it is the zero-locus in of some finite family of homogeneous polynomials that generate a prime ideal, the defining ideal of the variety.
In mathematics, in particular algebraic geometry, a moduli space is a geometric space whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such spaces frequently arise as solutions to classification problems: If one can show that a collection of interesting objects can be given the structure of a geometric space, then one can parametrize such objects by introducing coordinates on the resulting space. In this context, the term "modulus" is used synonymously with "parameter"; moduli spaces were first understood as spaces of parameters rather than as spaces of objects. A variant of moduli spaces is formal moduli. Bernhard Riemann first used the term "moduli" in 1857.
In relation to the history of mathematics, the Italian school of algebraic geometry refers to mathematicians and their work in birational geometry, particularly on algebraic surfaces, centered around Rome roughly from 1885 to 1935. There were 30 to 40 leading mathematicians who made major contributions, about half of those being Italian. The leadership fell to the group in Rome of Guido Castelnuovo, Federigo Enriques and Francesco Severi, who were involved in some of the deepest discoveries, as well as setting the style.
Pierre René, Viscount Deligne is a Belgian mathematician. He is best known for work on the Weil conjectures, leading to a complete proof in 1973. He is the winner of the 2013 Abel Prize, 2008 Wolf Prize, 1988 Crafoord Prize, and 1978 Fields Medal.
In mathematics, a complex analytic K3 surface is a compact connected complex manifold of dimension 2 with а trivial canonical bundle and irregularity zero. An (algebraic) K3 surface over any field means a smooth proper geometrically connected algebraic surface that satisfies the same conditions. In the Enriques–Kodaira classification of surfaces, K3 surfaces form one of the four classes of minimal surfaces of Kodaira dimension zero. A simple example is the Fermat quartic surface
Kunihiko Kodaira was a Japanese mathematician known for distinguished work in algebraic geometry and the theory of complex manifolds, and as the founder of the Japanese school of algebraic geometers. He was awarded a Fields Medal in 1954, being the first Japanese national to receive this honour.
Michael Artin is an American mathematician and a professor emeritus in the Massachusetts Institute of Technology Mathematics Department, known for his contributions to algebraic geometry.
In mathematics, the Picard group of a ringed space X, denoted by Pic(X), is the group of isomorphism classes of invertible sheaves (or line bundles) on X, with the group operation being tensor product. This construction is a global version of the construction of the divisor class group, or ideal class group, and is much used in algebraic geometry and the theory of complex manifolds.
In algebraic geometry, the Kodaira dimensionκ(X) measures the size of the canonical model of a projective variety X.
In algebraic geometry, a moduli space of (algebraic) curves is a geometric space whose points represent isomorphism classes of algebraic curves. It is thus a special case of a moduli space. Depending on the restrictions applied to the classes of algebraic curves considered, the corresponding moduli problem and the moduli space is different. One also distinguishes between fine and coarse moduli spaces for the same moduli problem.
In mathematics, the Enriques–Kodaira classification groups compact complex surfaces into ten classes, each parametrized by a moduli space. For most of the classes the moduli spaces are well understood, but for the class of surfaces of general type the moduli spaces seem too complicated to describe explicitly, though some components are known.
In algebraic geometry, a moduli scheme is a moduli space that exists in the category of schemes developed by French mathematician Alexander Grothendieck. Some important moduli problems of algebraic geometry can be satisfactorily solved by means of scheme theory alone, while others require some extension of the 'geometric object' concept.
In mathematics, the concept of abelian variety is the higher-dimensional generalization of the elliptic curve. The equations defining abelian varieties are a topic of study because every abelian variety is a projective variety. In dimension d ≥ 2, however, it is no longer as straightforward to discuss such equations.
This is a glossary of algebraic geometry.
In mathematics, a Siegel modular variety or Siegel moduli space is an algebraic variety that parametrizes certain types of abelian varieties of a fixed dimension. More precisely, Siegel modular varieties are the moduli spaces of principally polarized abelian varieties of a fixed dimension. They are named after Carl Ludwig Siegel, the 20th-century German number theorist who introduced the varieties in 1943.
