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, structure, space, models, and change.
History
On ...
, known for work on
deformation theory of structures arising in
differential geometry, and on
several complex variables from the point of view of
partial differential equations. He was born in
Boulder, Colorado, 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: University of Colorado Boulder, University of Colorado Colorado Springs, University of Colorado Denver, and the University o ...
and
MIT
The Massachusetts Institute of Technology (MIT) is a private land-grant research university in Cambridge, Massachusetts. Established in 1861, MIT has played a key role in the development of modern technology and science, and is one of the m ...
.
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 by r ...
under
J. E. Littlewood and
G.H. Hardy
Godfrey Harold Hardy (7 February 1877 – 1 December 1947) was an English mathematician, known for his achievements in number theory and mathematical analysis. In biology, he is known for the Hardy–Weinberg principle, a basic principle of pop ...
at the
University of Cambridge
The University of Cambridge is a public collegiate research university in Cambridge, England. Founded in 1209 and granted a royal charter by Henry III in 1231, Cambridge is the world's third oldest surviving university and one of its most pr ...
, completed in 1939. He had positions at MIT and
Stanford before his appointment in 1950 at
Princeton University
Princeton University is a private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth as the College of New Jersey, Princeton is the fourth-oldest institution of higher education in the United States and one of the ...
. 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 manifolds and
algebraic geometry, and the conception of
moduli spaces.
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 manifol ...
) in PDE theory, to extend
Hodge theory and the ''n''-dimensional
Cauchy–Riemann equations
In the field of complex analysis in mathematics, the Cauchy–Riemann equations, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential equations which, together with certain continuity and differenti ...
to the non-compact case. This is used to show existence theorems for
holomorphic functions.
He later worked on
pseudogroup In mathematics, a pseudogroup is a set of diffeomorphisms between open sets of a space, satisfying group-like and sheaf-like properties. It is a generalisation of the concept of a group, originating however from the geometric approach of Sophus Lie ...
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 (it has no solution) when constructed with random coefficients. However, an over ...
s of PDEs (bypassing the Cartan–Kähler ideas based on
differential forms 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 included in the kernel of t ...
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 ...
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 a ...
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