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Donald C. Spencer

## Summary

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.

Donald C. Spencer
Born
Donald Clayton Spencer

April 25, 1912
DiedDecember 23, 2001 (aged 89)
NationalityAmerican
MIT
Trinity College, Cambridge[1]
AwardsBôcher Memorial Prize (1948)
National Medal of Science (1989)
Scientific career
InstitutionsPrinceton University
Doctoral advisorJ. E. Littlewood and G.H. Hardy
Doctoral studentsPierre Conner
Patrick X. Gallagher
Phillip Griffiths
Robert Hermann
Roger Horn
Louis Howard
Joseph J. Kohn
Suresh H. Moolgavkar

## Career

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.

## Legacy

After his death, a mountain peak outside Silverton, Colorado was named in his honor.[2]

## Publications

• Schaeffer, A. C.; Spencer, D. C. (1950), Coefficient Regions for Schlicht Functions, American Mathematical Society Colloquium Publications, Vol. 35, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-1035-4, MR 0037908
• Schiffer, M. M.; Spencer, D. C. (1955), Functionals of Finite Riemann Surfaces, Princeton University Press[3]
• Nickerson, H. K.; Spencer, D. C.; Steenrod, N. E. (1959), Advanced Calculus, Princeton, N.J.: Van Nostrand[4]Nickerson, H. K.; Spencer, D. C.; Steenrod, Norman Earl (2011). Dover reprint. ISBN 978-0-4864-8090-9; pbk{{cite book}}: CS1 maint: postscript (link)
• Kumpera, A.; Spencer, D. C. (1972), Lie Equations: Volume I, General Theory, AM-73, Annals of Mathematical Studies, Princeton University Press, ISBN 978-0-6910-8111-3; pbk{{citation}}: CS1 maint: postscript (link)
• Kumpera, A.; Spencer, D. C. (1974), Systems of Linear Partial Differential Equations and Deformation of Pseudogroup Structures, Les Presses de l'Université de Montréal

## References

1. ^ Sylvia Nasar, 'Donald C. Spencer, 89, Pioneering Mathematician, Dies', The New York Times, 1 January 2002. [1]
2. ^ Pankratz, Howard (2008-08-18). "Spencer peak added to Colorado mountain lexicon". Denver Post. Retrieved 2011-07-23.
3. ^ Ahlfors, Lars V. (1955). "Review of Functionals of finite Riemann surfaces. By M. M. Schiffer and D. C. Spencer" (PDF). Bull. Amer. Math. Soc. 61 (6): 581–584. doi:10.1090/s0002-9904-1955-09998-1.
4. ^ Allendoerfer, C. B. (1960). "Review of Advanced Calculus. By H. K. Nickerson, D. C. Spencer and N. E. Steenrod" (PDF). Bull. Amer. Math. Soc. 66 (3): 148–152. doi:10.1090/s0002-9904-1960-10411-9.