Background
He was born in Boulder, Colorado, and educated at the University of Colorado and Massachusetts Institute of Technology.
(This advanced monograph on finite Riemann surfaces, based...)
This advanced monograph on finite Riemann surfaces, based on the authors' 1949–50 lectures at Princeton University, remains a fundamental book for graduate students. The Bulletin of the American Mathematical Society hailed the self-contained treatment as the source of "a plethora of ideas, each interesting in its own right," noting that "the patient reader will be richly rewarded." Suitable for graduate-level courses, the text begins with three chapters that offer a development of the classical theory along historical lines, examining geometrical and physical considerations, existence theorems for finite Riemann surfaces, and relations between differentials. Subsequent chapters explore bilinear differentials, surfaces imbedded in a given surface, integral operators, and variations of surfaces and of their functionals. The book concludes with a look at applications of the variational method and remarks on generalization to higher dimensional Kahler manifolds.
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mathematician university professor
He was born in Boulder, Colorado, and educated at the University of Colorado and Massachusetts Institute of Technology.
He wrote a Doctor of Philosophy in diophantine approximation under J. East. Littlewood and G.H. Hardy at the University of Cambridge, completed in 1939.
He had positions at Massachusetts Institute of Technology 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
(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 of Silverton, Colorado was named in his honor.
(This advanced monograph on finite Riemann surfaces, based...)