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Donald C. Spencer (1912–2001)

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Donald C. Spencer (1912–2001) Donald C. Spencer (1912–2001) Joseph J. Kohn, Phillip A. Griffiths, Hubert Goldschmidt, Enrico Bombieri, Bohous Cenkl, Paul Garabedian, Louis Nirenberg Joseph J. Kohn on sequences of integers; in applied mathematics on Donald C. Spencer died on December 23, 2001. He was fluid mechanics; and in the born on April 25, 1912, in Boulder, Colorado. He was theory of one complex vari- an undergraduate at the University of Colorado (B.A. able on univalent functions, in 1934) and at the Massachusetts Institute of Tech- on conformal mappings, nology (B.S. in Aeronautical Engineering in 1936) and and on Riemann surfaces. a graduate student in mathematics at Cambridge At Princeton, Spencer’s University (Ph.D. in 1939, Sc.D. in 1963). His doctoral research turned to several thesis, “On a Hardy-Littlewood problem of dio- complex variables and com- phantine approximation and its generalizations”, plex manifolds. He collabo- was written under the direction of J. E. rated with Kunihiko Littlewood and initiated Spencer’s remarkable math- Kodaira, who is also widely Donald C. Spencer ematical career. Spencer taught at M.I.T. (1939–42), recognized as one of the at Stanford (1942–50 and 1963–68), and at Princeton great mathematical pioneers of the twentieth cen- (1950–63 and 1968–78). He was the Eugene Higgins tury. Together, Spencer and Kodaira developed the Professor at Princeton (1971–72) and the Henry modern theory of deformations of complex mani- Burchard Fine Professor at Princeton (1972–78). He folds into a major tool, the basis for a large body received the degree Sc.D. honoris causa from of subsequent research. This work has had and Purdue University (1971) and was the joint recipi- continues to have tremendous influence in large seg- ent with A. C. Schaeffer of the Bôcher Memorial Prize ments of mathematics, including the theory of sev- (1948). He was a member of the U.S. National Acad- eral complex variables, differential geometry, al- emy of Sciences (from 1961), a fellow of the Ameri- gebraic geometry, and mathematical physics. In can Academy of Arts and Sciences (from 1967), and fact, Spencer’s work with Kodaira was one of the a recipient of the National Medal of Science (1989). most remarkable mathematical collaborations of the He was the author of numerous important research twentieth century; its only parallel is the famous papers and four influential books (see [1], [2], [3], Hardy-Littlewood work. To give a feeling for the ex- [4], [5]). citement generated by these efforts, I quote from Spencer’s mathematical work is truly impressive the introduction to Kodaira’s book (Complex and spans many fields in which he made funda- Manifolds and Deformations of Complex Structures, mental contributions. Before coming to Princeton, Springer-Verlag, 1981): he worked in number theory on lattice points and In order to clarify this mystery, Spencer and I developed the theory of defor- Joseph J. Kohn is professor of mathematics at Princeton University. His email address is [email protected]. mations of compact complex manifolds. JANUARY 2004 NOTICES OF THE AMS 17 Just as Spencer had an unfailing instinct for how to approach mathematical research, so he also had an unfailing instinct for how to inspire both stu- dents and fellow mathematicians. Recently Victor Guillemin, professor of mathematics at M.I.T., wrote the following tribute: When I first met Don Spencer, I was a twenty-two-year-old graduate student, and Spencer was in his mid-forties and at the height of his fame. Kodaira-Spencer was the most exciting development of that era in differential geometry, the early sixties equivalent of Seiberg-Witten, and I and a lot of my fellow graduate students were caught up in this excitement and After retirement in Durango, CO. working on Spencer sequences, Spencer The process of the development was cohomology, the Spencer theory of de- the most interesting experience in my formations, etc. This was the state of whole mathematical life. It was similar affairs when I first met Don in the fall of to an experimental science developed by 1960; and therefore, not surprisingly, the the interaction between experiments thing I most remember about him was (examination of examples) and theory. his incredible kindness and empathy for In this book I have tried to reproduce young people. I have had several older this interesting experience; however I colleagues who have been able, as one could not fully convey it. Such an ex- says, to “bridge the generation gap”, but perience may be a passing phenomenon there is no one I’ve ever known who had which cannot be reproduced. this quality to the extent that Spencer did. Some of us took advantage of it and Furthermore, in 1987 Kodaira wrote in his let- must at times have bored him silly with ter to the President’s Committee on the National the catalogue of our accomplishments Medal of Science: and aspirations. But he was unfailing non- Spencer’s contributions to mathematics judgmental and sympathetic. More than go far beyond his published papers. He that, one could feel that his interest in exerted tremendous influence on his col- these aspirations was the genuine com- laborators and students. His enthusiasm modity, and not just feigned to make us knew no limit and was contagious. In feel good. One’s ego at 22 is a fragile Princeton he was always surrounded by vessel, and Don instinctively knew this. a group of mathematicians who shared For that (as for much else) I remember his enthusiasm and collaborated in the Don as the kindest, nicest person I’ve research of complex analysis (I was one ever known. of them). In the 1950s the theory of com- In 1978 Spencer retired from Princeton and plex manifolds was developed extensively moved to Durango, Colorado. There he became in Princeton. The driving force behind very active in the conservation and ecology move- this development was in fact Spencer’s ments and also an avid hiker. He soon made many enthusiasm. friends and became well known in the area. The city At the same time, Spencer introduced the use of Durango designated April 25 as “Don Spencer of potential theory in the study of complex mani- Day”. After his retirement, Don Spencer kept in folds with boundaries, and in particular formu- touch with his many friends and colleagues, al- lated the “d-bar-Neumann problem”, which has led ways inspiring, supportive, and full of enthusiasm. to very important developments in both several Don was my thesis adviser, mentor, colleague, complex variables and partial differential equa- and friend. I consider myself fortunate to have tions. In the last decade of his career, Spencer been so close to such a remarkable human being. worked on overdetermined systems of partial dif- Here I will give a brief account of his mathemati- ferential equations and on pseudogroups. cal contributions to the theory of partial differen- In all his work Spencer shows remarkable origi- tial equations (P.D.E.). nality and insight. His influence on his students, his Don’s contributions to P.D.E. can be divided collaborators, and his many friends also has had a into three groups: (1) equations arising from fluid lasting impact on twentieth-century mathematics. mechanics, (2) equations arising from complex 18 NOTICES OF THE AMS VOLUME 51, NUMBER 1 analysis, and (3) the theory of linear overdeter- boundary-value problem mined systems. His work in (1) is discussed below for an elliptic operator by Paul Garabedian. The work in (2) is divided into (the boundary values are two parts: equations arising from one complex imposed by the require- variable and those from several complex variables. ment of belonging∗ to the Here I will concentrate on the work connected with domain of ∂ ). Spencer several complex variables. The contributions in (3) and I (see [10]) studied are discussed by Hubert Goldschmidt. this phenomenon by in- The development of the theory of P.D.E. is closely troducing the operator linked with advances in complex analysis; in fact, ∂τ = ∂ + τ∂, setting up Riemann’s approach to the study of conformal the analogous ∂τ -Neu- mapping via the Dirichlet principle led to the sys- mann problem, and tematic development of the theory of elliptic P.D.E. studying its behavior for and associated variational problems. The applica- small τ. In [10] we man- tion of these methods to the theory of several com- aged to solve the problem plex variables was initiated by Hodge in his theory on balls in Cn by use of of harmonic integrals on compact manifolds. It is spherical harmonics, but this work that led H. Weyl to prove the funda- this case is too special for mental hypoellipticity theorem, known as Weyl’s the important applica- lemma, which in turn led to the development of the tions. general theory of elliptic P.D.E. One of the main appli- The theory of harmonic integrals on compact cations of the ∂-Neumann manifolds is a crucial ingredient in the Kodaira- problem on (0, 1)-forms, Receiving National Medal of Science Spencer theory of deformations of complex envisioned by Spencer, is from President George Bush, 1989. structures. An example of this is the fundamental the construction of holo- existence theorem proved by Kodaira, Nirenberg, morphic functions with specified properties. It is and Spencer (see [6]), which is described below by this approach that C. B. Morrey used in [11] to P. A. Griffiths. show that a compact real-analytic manifold M can Spencer’s most spectacular contribution to the be embedded real analytically into euclidean space. theory of P.D.E., one which has been a major in- Thus Morrey wanted to use the ∂τ -Neumann prob- fluence in mathematical research, is setting down lem with Ω a thin tubular neighborhood of M in the program to generalize harmonic integrals to the complexification of M; if the ∂τ -Neumann prob- noncompact manifolds.
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