Donald Clayton Spencer (April 25, 1912 – December 23, 2001) was an American
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
, known for work on
deformation theory
In mathematics, deformation theory is the study of infinitesimal conditions associated with varying a solution ''P'' of a problem to slightly different solutions ''P''ε, where ε is a small number, or a vector of small quantities. The infinitesima ...
of structures arising in
differential geometry
Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...
, and on
several complex variables
The theory of functions of several complex variables is the branch of mathematics dealing with functions defined on the complex coordinate space \mathbb C^n, that is, -tuples of complex numbers. The name of the field dealing with the properties ...
from the point of view of
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives.
The function is often thought of as an "unknown" that solves the equation, similar to ho ...
s. He was born in
Boulder, Colorado
Boulder is a List of municipalities in Colorado#Home rule municipality, home rule city in Boulder County, Colorado, United States, and its county seat. With a population of 108,250 at the 2020 United States census, 2020 census, it is the most ...
, and educated at the
University of Colorado
The University of Colorado (CU) is a system of public universities in Colorado. It consists of four institutions: the University of Colorado Boulder, the University of Colorado Colorado Springs, the University of Colorado Denver, and the U ...
and
MIT
The Massachusetts Institute of Technology (MIT) is a private research university in Cambridge, Massachusetts, United States. Established in 1861, MIT has played a significant role in the development of many areas of modern technology and sc ...
.
Career
He wrote a Ph.D. in
diophantine approximation
In number theory, the study of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus of Alexandria.
The first problem was to know how well a real number can be approximated ...
under
J. E. Littlewood and
G.H. Hardy at the
University of Cambridge
The University of Cambridge is a Public university, public collegiate university, collegiate research university in Cambridge, England. Founded in 1209, the University of Cambridge is the List of oldest universities in continuous operation, wo ...
, completed in 1939. He had positions at MIT and
Stanford
Leland Stanford Junior University, commonly referred to as Stanford University, is a private research university in Stanford, California, United States. It was founded in 1885 by railroad magnate Leland Stanford (the eighth governor of and th ...
before his appointment in 1950 at
Princeton University
Princeton University is a private university, private Ivy League research university in Princeton, New Jersey, United States. Founded in 1746 in Elizabeth, New Jersey, Elizabeth as the College of New Jersey, Princeton is the List of Colonial ...
. There he was involved in a series of collaborative works with
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 ...
on the
deformation of complex structures, which had some influence on the theory of
complex manifold
In differential geometry and complex geometry, a complex manifold is a manifold with a ''complex structure'', that is an atlas (topology), atlas of chart (topology), charts to the open unit disc in the complex coordinate space \mathbb^n, such th ...
s and
algebraic geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
, and the conception of
moduli space
In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme (mathematics), scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of suc ...
s.
He also was led to formulate the
''d-bar Neumann problem'', for the operator
(see
complex differential form
In mathematics, a complex differential form is a differential form on a manifold (usually a complex manifold) which is permitted to have complex coefficients.
Complex forms have broad applications in differential geometry. On complex manifolds, t ...
) in PDE theory, to extend
Hodge theory
In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold ''M'' using partial differential equations. The key observation is that, given a Riemannian metric on ''M'', every coho ...
and the ''n''-dimensional
Cauchy–Riemann equations
In the field of complex analysis in mathematics, the Cauchy–Riemann equations, named after Augustin-Louis Cauchy, Augustin Cauchy and Bernhard Riemann, consist of a system of differential equations, system of two partial differential equatio ...
to the non-compact case. This is used to show existence theorems for
holomorphic function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex de ...
s.
He later worked on
pseudogroup
In mathematics, a pseudogroup is a set of homeomorphisms between open sets of a space, satisfying group-like and sheaf-like properties. It is a generalisation of the concept of a transformation group, originating however from the geometric approac ...
s and their deformation theory, based on a fresh approach to
overdetermined system
In mathematics, a system of equations is considered overdetermined if there are more equations than unknowns. An overdetermined system is almost always inconsistent equations, inconsistent (it has no solution) when constructed with random coeffi ...
s of PDEs (bypassing the Cartan–Kähler ideas based on
differential form
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications ...
s by making an intensive use of
jets). Formulated at the level of various
chain complex
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is contained in the kernel o ...
es, this gives rise to what is now called
Spencer cohomology, a subtle and difficult theory both of formal and of analytical structure. This is a kind of
Koszul complex
In mathematics, the Koszul complex was first introduced to define a cohomology theory for Lie algebras, by Jean-Louis Koszul (see Lie algebra cohomology). It turned out to be a useful general construction in homological algebra. As a tool, its ho ...
theory, taken up by numerous mathematicians during the 1960s. In particular a theory for
Lie equations formulated by
Malgrange emerged, giving a very broad formulation of the notion of ''integrability''.
Legacy
After his death, a mountain peak outside Silverton, Colorado was named in his honor.
See also
*
Kodaira–Spencer mapping
*
Salem–Spencer set
In mathematics, and in particular in arithmetic combinatorics, a Salem-Spencer set is a set of numbers no three of which form an arithmetic progression. Salem–Spencer sets are also called 3-AP-free sequences or progression-free sets. They have ...
Publications
*
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*
References
External links
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{{DEFAULTSORT:Spencer, Donald C.
1912 births
2001 deaths
20th-century American mathematicians
Alumni of Trinity College, Cambridge
Massachusetts Institute of Technology alumni
National Medal of Science laureates
People from Boulder, Colorado
Princeton University faculty
University of Colorado alumni