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for auction "Deformation Theory" Donald C Spencer Hand Written Letter Dated 1972. 

ES-9551

Donald

Clayton Spencer (April

25, 1912 – December 23, 2001) was an American mathematician,

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 and MIT. He wrote a Ph.D.

in diophantine approximation under J. E.

Littlewood and G.H. Hardy 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 series of collaborative works

with Kunihiko Kodaira 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 {\displaystyle {\bar

{\partial }}} (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 functions. He later worked

on pseudogroups and

their deformation theory, based on a fresh approach to overdetermined systems 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 complexes, 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 Silverton, Colorado was named in his honor.