Donald C. Spencer
Encyclopedia
Donald Clayton Spencer was an American
mathematician
, known for major work on deformation theory
of structures arising in differential geometry, and on several complex variables
from the point of view of partial differential equation
s. He was born in Boulder, Colorado
, and educated at the University of Colorado
and MIT.
He wrote a Ph.D. in diophantine approximation
under J. E. Littlewood at the University of Cambridge
, completed in 1939. He had positions at MIT and Stanford before his appointment in 1950 at Princeton University
. There he was involved in a major series of collaborative works with Kunihiko Kodaira on the deformation of complex structures, which had a profound influence on the theory of complex manifold
s and algebraic geometry
, and the conception of moduli space
s.
He also was led to formulate the d-bar Neumann problem, for the operator
(see complex differential form
) in PDE theory, to extend Hodge theory
and the n-dimensional Cauchy-Riemann equations
to the non-compact case. This is used to show existence theorems for holomorphic function
s.
He later worked on pseudogroup
s and their deformation theory, based on a fresh approach to overdetermined system
s of PDEs (bypassing the Cartan-Kähler ideas based on differential form
s by making an intensive use of jets
). Formulated at the level of various chain complex
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
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.
After his death, a mountain peak outside of Silverton, Colorado was named in his honor.
United States
The United States of America is a federal constitutional republic comprising fifty states and a federal district...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
, known for major work on deformation theory
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 Pε, where ε is a small number, or vector of small quantities. The infinitesimal conditions are therefore the result of applying the approach...
of structures arising in differential geometry, and on several complex variables
Several complex variables
The theory of functions of several complex variables is the branch of mathematics dealing with functionson the space Cn of n-tuples of complex numbers...
from the point of view of partial differential equation
Partial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
s. He was born in Boulder, Colorado
Boulder, Colorado
Boulder is the county seat and most populous city of Boulder County and the 11th most populous city in the U.S. state of Colorado. Boulder is located at the base of the foothills of the Rocky Mountains at an elevation of...
, and educated at the University of Colorado
University of Colorado at Boulder
The University of Colorado Boulder is a public research university located in Boulder, Colorado...
and MIT.
He wrote a Ph.D. in diophantine approximation
Diophantine approximation
In number theory, the field of Diophantine approximation, named after Diophantus of Alexandria, deals with the approximation of real numbers by rational numbers....
under J. E. Littlewood at the University of Cambridge
University of Cambridge
The University of Cambridge is a public research university located in Cambridge, United Kingdom. It is the second-oldest university in both the United Kingdom and the English-speaking world , and the seventh-oldest globally...
, completed in 1939. He had positions at MIT and Stanford before his appointment in 1950 at Princeton University
Princeton University
Princeton University is a private research university located in Princeton, New Jersey, United States. The school is one of the eight universities of the Ivy League, and is one of the nine Colonial Colleges founded before the American Revolution....
. There he was involved in a major series of collaborative works with Kunihiko Kodaira on the deformation of complex structures, which had a profound influence on the theory of complex manifold
Complex manifold
In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic....
s and algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, and the conception of moduli space
Moduli space
In 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...
s.
He also was led to formulate the d-bar Neumann problem, for the operator
(see complex differential form
Complex differential form
In mathematics, a complex differential form is a differential form on a manifold which is permitted to have complex coefficients....
) in PDE theory, to extend Hodge theory
Hodge theory
In mathematics, Hodge theory, named after W. V. D. Hodge, is one aspect of the study of the algebraic topology of a smooth manifold M. More specifically, it works out the consequences for the cohomology groups of M, with real coefficients, of the partial differential equation theory of generalised...
and the n-dimensional Cauchy-Riemann equations
Cauchy-Riemann equations
In mathematics, the Cauchy–Riemann differential equations in complex analysis, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential equations which must be satisfied if we know that a complex function is complex differentiable...
to the non-compact case. This is used to show existence theorems for holomorphic function
Holomorphic function
In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...
s.
He later worked on pseudogroup
Pseudogroup
In mathematics, a pseudogroup is an extension of the group concept, but one that grew out of the geometric approach of Sophus Lie, rather than out of abstract algebra...
s and their deformation theory, based on a fresh approach to overdetermined system
Overdetermined system
In mathematics, a system of linear equations is considered overdetermined if there are more equations than unknowns. The terminology can be described in terms of the concept of counting constraints. Each unknown can be seen as an available degree of freedom...
s of PDEs (bypassing the Cartan-Kähler ideas based on differential form
Differential form
In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a better definition for integrands in calculus...
s by making an intensive use of jets
Jet (mathematics)
In mathematics, the jet is an operation which takes a differentiable function f and produces a polynomial, the truncated Taylor polynomial of f, at each point of its domain...
). Formulated at the level of various chain complex
Chain complex
In mathematics, chain complex and cochain complex are constructs originally used in the field of algebraic topology. They are algebraic means of representing the relationships between the cycles and boundaries in various dimensions of some "space". Here the "space" could be a topological space or...
es, this gives rise to what is now called Spencer cohomology
Spencer cohomology
In mathematics, Spencer cohomology is cohomology of a manifold with coefficients in the sheaf of solutions of a linear partial differential operator. It was introduced by ....
, a subtle and difficult theory both of formal and of analytical structure. This is a kind of Koszul complex
Koszul complex
In mathematics, the Koszul complex was first introduced to define a cohomology theory for Lie algebras, by Jean-Louis Koszul...
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.
After his death, a mountain peak outside of Silverton, Colorado was named in his honor.
External links
- Donald C. Spencer (1912-2001), Notices of the American Mathematical Society; vol. 51, no. 1 (January 2004).
- Donald C. Spencer, 89, Pioneering Mathematician, Dies, The New York TimesThe New York TimesThe New York Times is an American daily newspaper founded and continuously published in New York City since 1851. The New York Times has won 106 Pulitzer Prizes, the most of any news organization...
; January 1, 2002.